WO2003071367A1 - A system and method for machining an object - Google Patents

Abstract

A system for machining an object, the system including: an obtaining means for obtaining a desired profile of the object, and information about an edge of a shaping tool; a processor for processing the desired profile of the object and the information about the edge in order to determine a path for the object and/or edge to be moved along.

Description

A SYSTEM AND METHOD FOR MACHINING AN OBJECT

FIELD OF THE INVENTION

The present invention relates generally to a system and method for machining an object. The invention has been designed especially, but not exclusively, for micro-machining .

BACKGROUND OF THE INVENTION

Micro-machining involves machining an extremely- minute object (for example, an optical fibre) such that it has a required profile. Because the minute objects are so small, it is not possible to effectively machine minute objects using traditional machining techniques.

A number of techniques have been developed to shape minute objects into a required profile. However, these techniques generally do not provide sufficient control, and thus often provide unpredictable dimensional accuracy.

Being able to predict the dimensional accuracy is very important in micro-machining because it provides accurate machining, which in turn ensures that the minute object is machined as required. For example, dimensional accuracy is very important when machining a lens on the end of an optical fibre. Being able to provide dimensional accuracy ensures that the machining produces the desired lens. Significant dimensional accuracy is likely to result in an incorrect lens being machined on the end of the optical fibre.

Therefore it is desirable to have a system and method which provides sufficient control in order to minimize dimensional inaccuracies. SUMMARY OF THE INVENTION

According to a first aspect of the present invention, there is provided a system for machining an object, the system including: an obtaining means for obtaining a desired profile of the object, and obtaining information about an edge of a shaping tool; and a processor for processing the desired profile of the object and the information about the edge in order to determine a path for the object and/or edge to be moved along.

Preferably, the system further includes an assembly for providing a relative movement between the object and the edge such that a portion of the object is removed when the object and edge come into contact with each other.

Preferably, the system further includes an apparatus for moving the object and/or edge along the path whilst the relative movement is being provided, to thereby effect machining of the object into the desired profile.

Preferably, the obtaining means includes a camera for obtaining an image of the edge, the image being the information about the edge.

Preferably, the camera has a resolution of about

176nm.

Preferably, the obtaining means includes a terminal for allowing a user of the system to enter a function which defines the desired profile.

To determine the path, the processor is preferably arranged to identify parts of the edge of the shaping tool which are suited to cutting parts of the desired profile of the object.

Preferably, in order to determine the path the processor: processes the information about the edge in order to determine a profile thereof; selects a number of parts along the desired profile of the object; for each one of the number of parts, selects a location on the profile of the edge which corresponds to the one of the number of parts; and superimposes the number of parts along the desired profile onto the object, wherein the path is such that when the apparatus moves the object and/or edge along the path, each one of the number of parts superimposed onto the object comes into contact with the corresponding location on the profile of the edge.

Preferably, in order to determine the location on the profile of the edge, the processor determines a tangent for each one of the number of parts along the desired profile of the object, the processor selecting the location on the profile of the edge on the basis that it has a tangent that is the same as the tangent for the one of the number of parts along the desired profile.

Thus, this has the advantage of providing dimensional accuracy when machining the object.

Preferably, the processor includes an image- processing device for determining the outer limits of the edge, the processor using the outer limits to create the profile of the edge. Preferably, the processor determines the outer limits using an optimal linear operator for step edge detection.

Preferably, the assembly includes a drive motor which is coupled to a member for retaining the shaping tool, the drive motor and member being such that they are capable of rotating the shaping tool in order to provide the relative movement between the object and the edge.

Preferably, the drive motor is coupled to the member via a flexible coupling and the member is mounted to an air bearing such that eccentricities are kept to an acceptable level whilst the shaping tool is being rotated.

Preferably, the drive motor is such that the shaping tool can be rotated at about 10,000 rpm.

Preferably, the acceptable level of eccentricities is about lOnm.

Preferably, the drive motor is driven by compressed air.

Preferably, the apparatus includes a chuck for holding the object, and an actuation means coupled to the chuck, the actuation means being capable of moving the chuck in order to effect movement of the object along the path whilst being held by the chuck.

Preferably, the actuation means includes a feedback circuit capable of producing a signal that can be used by the actuation means to determine a location of the chuck.

Preferably, the actuation means is capable of providing incremental and/or translational movement of the chuck in at least x and z axes.

Preferably, the actuation means is capable of moving the chuck a distance of about 200 microns

Preferably, the actuation means includes a piezoelectric actuator.

Preferably, the feedback circuit includes a capacitive transducer.

Preferably, the apparatus includes a drive motor coupled to an adjustable device capable of holding the object, the drive motor effecting rotation of the adjustable device such that the object can be rotated when held by the device .

Preferably, the device is capable of being adjusted so that the object can be moved into a position such that a rotational axis of the object and a rotational axis of the device coincide.

Preferably, the device includes a chuck for holding the object, a shaft coupled to the chuck, a housing in which the shaft resides, and a plurality of adjustable supports for supporting the shaft in the housing, whereby adjusting the supports allows the fibre to be moved into the position.

Preferably, the system further includes a control circuit capable of detecting whether the rotational axis of the object and the rotational axis of the device coincide, and upon detecting that the rotational axis of the object and the rotational axis of the device do not coincide, adjusting the supports such that the rotational axis of the object and the rotational axis of the device coincide. Preferably, the control circuit includes a camera for capturing an image of the object whilst being rotated by the apparatus, and a processor for processing the image and in order to determine whether the rotational axis of the object coincides with the rotational axis of the device, the processing also being capable of adjusting the supports .

Preferably, the system is suitable for micro- machining.

According to a second aspect of the present invention, there is provided a method for machining an object, the method including the steps of: obtaining a desired profile of the object; obtaining information about an edge of a shaping tool; and processing the desired profile of the object and the information about the edge in order to determine a path for the object and/or edge to be moved along.

Preferably, the method further includes providing relative movement between the object and the edge such that a portion of the object is removed when the object and edge come into contact with each other.

Preferably, the method further includes the step of moving the object and/or edge along the path whilst the relative movement is being provided, to thereby effect machining of the object into the desired profile.

Preferably, the obtaining step includes using a camera to obtaining an image of the edge, the image being the information about the edge.

Preferably, the obtaining step includes using a terminal to enter a function which defines the desired profile .

Preferably, the processing step includes the steps of: processing the information about the edge in order to determine a profile thereof; selecting a number of parts along the desired profile of the object; for each one of the number of parts, selecting a location on the profile of the edge which corresponds to the one of the number of parts; and superimposing the number of parts along the desired profile onto the object, wherein the path is such that when the object and/or edge move along the path, each one of the number of parts superimposed onto the object comes into contact with the location on the profile of the edge.

Preferably, selecting the location on the profile of the edge includes determining a tangent for each one of the number of parts along the desired profile of the object, the location on the profile of the edge being selected on the basis that it has a tangent that is the same as the tangent for the one of the number of parts along the desired profile.

Preferably, processing the image of the edge includes determining the outer limits of the edge, the step of using the desired profile using outer limits to create the profile of the edge.

Preferably, the outer limits are determined using an optimal linear operator for step edge detection.

Preferably, the step of rotating the shaping tool in order to provide the relative movement between the object and the edge. Preferably, the shaping tool is rotated at about 10,000 rpm.

Preferably, the method further included the step of moving the object along the path.

Preferably, the object is moved incremental and/or translational in at least x and z axes.

Preferably, the movement is about 200 microns

Preferably, the method further included the step of rotating the object.

Preferably, the method is used for micro- machining.

According to a third aspect of the present invention, there is provided a micro-machining apparatus, including: a mounting device being adapted to provide mounting for a workpiece to be machined; and a machining head including a grinding element having a peripheral edge portion, the machine head being operatively coupled to the mounting device to permit relative movement between the machining head and the workpiece to provide contact therebetween, the grinding element being rotated and contacted with the workpiece whereby the peripheral edge portion effects machining of the workpiece in a predetermined profile.

According to a fourth aspect of the present invention, there is provided a micro-machining head including a grinding element having a peripheral edge portion, the grinding element being rotatable about a rotational axis and its peripheral edge portion being configured to contact a workpiece which on rotation of the grinding element is machined in a predetermined profile.

According to a fifth aspect of the present invention, there is provided a method of micro-machining a work-piece, the method including the steps of:

Providing a mounting device to which the workpiece is mounted, and a machining head having a grinding element and being operatively coupled to the mounting device;

Rotating the grinding element about a rotational axis; and

Providing relative movement between the machining head and the workpiece to effect contact between a peripheral edge portion of the grinding element and the workpiece to provide machining of said workpiece in a predetermined profile.

BRIEF DESCRIPTION OF THE DRAWINGS

Notwithstanding any other embodiments, which may fall within the scope of the present invention, a preferred embodiment of the present invention will now be described, by way of example only, with reference to the accompanying figures, in which:

figure 1 illustrates a system for machining an object in accordance to the preferred embodiment of the present invention;

figure 2 illustrates an optical fibre which the system shown in figure 1 can machine;

figure 3 illustrates profile information that is used to calculate a path which the system shown in figure 1 uses to machine the optical fibre; figure 4 illustrates an assembly which forms part of the system shown in figure 1;

figure 5 illustrates an apparatus which forms part of the system shown in figure 1;

figure 6 illustrates a cross-sectional view of an adjustable device which forms part of the apparatus shown in figure 5; and

figure 7 illustrates the rotational axis of an optical fibre and a chuck which forms part of the device shown in figure 6.

A PREFERRED EMBODIMENT OF THE INVENTION

Whilst the following description is in the context of machining an end of the optical fibre to form a lens, it will be appreciated that the invention can be used to machine a wide range of objects and is in no way limited to machine the lens on the end of an optical fibre.

By way of background, the preferred embodiment relates to an area of machining know generally as micro- machining. Micro-machining essentially involves machining of minute objects, such as optical fibre.

As can be seen in figure 1, the system 1 has an obtaining means 3 which includes a camera 3a and a terminal 3b. The camera 3a is used to obtain an image of an edge 5 of a shaping tool 7 (which in this embodiment is in the form of a circular grinding wheel) , the edge 5 being used to machine a lens 9 onto the end of an optical fibre 11 (as shown in figure 2) . To ensure the image is of high quality, the camera 3a has a resolution of about 176nm. As is discussed later in this description, the image of the edge 5 of the shaping tool 7 is used to determine a profile T U03/00232

11

thereof. To ensure the image is accurate (so that profile of the edge 5 can be accurately determined) , the obtaining means 3 includes a mount (not shown in the figures) for the camera that enables the camera 3a to be moved. The movement of the camera 3a allows the camera 3a to be moved so that the axis of the camera 3a tracks the tool 7 profile - thereby minimizing parallax error and spherical aberration, which would otherwise result in a profile which is not entirely accurate.

The terminal 3b (in the form of a computer terminal) allows a user of the system 1 to input a function which defines a desired profile for the end of an optical fibre 11, which in effect is the profile of the required lens 9.

The system 1 also includes a processor 13 which is in the form of computer hardware and software running thereon. The processor 13 is such that it is connected via a link 15 to the camera 3a so that the image of the edge 5 of the shaping tool 7 can be transferred from the camera 3a to the processor 13. The processor 13 also being connected to the terminal 3b via another link 17 so that the function defining the desired profile of the lens 9 of the optical fibre 11 can be transferred from the terminal 3b to the processor 13.

The processor 13 is such that it processes the image of the edge 5 of the shaping tool 7 and the function defining the desired profile of the lens 9 of the optical fibre 11 in order to determine a path for the end of the optical fibre 11 and/or edge 5 of the shaping tool 7 to be moved along. Effectively, the path is used to machine the end of the optical fibre 11 into the desired lens 9 profile.

In order to determine the path, the processor 13 first determines a profile 19 of the edge 5 of the shaping tool 7 (as shown in figure 3) . This is done by applying an optimal linear operator for step edge detection to the image of the edge 5 of the shaping tool 7. Effectively this allows the processor 13 to determine the edge 5 of the shaping tool 7, and in turn determine the profile 19 thereof. The optimal linear operator is used in the preferred embodiment because it allows sub-pixel edge detection and thereby accurately determine the profile 19 of the edge 5 of the shaping tool 7. Being able to accurately determine the profile 19 is particularly important for machining a lens 9 on the end of an optical fibre 11 because it allows the lens 9 to be accurately machined. It is, however, possible to use other techniques to determine the edge 5 of the shaping tool 7.

A good description on the optimal linear operator for step edge detection can be found in the article by J. Shen and S . Castan, entitled "An Optimal Linear Operator for Step Edge Detection" and which appeared in Graph.

Models and Image Process., Vol. 54, no. 2, pp. 112-133 March 1992. A copy of this article is included in appendix A of this document.

The next step carried out by the processor 13 in determining the path, is to use the function defining the desired profile of the lens 9 of the optical fibre 11 (which was input into the terminal 3b) in order to select a number of parts along the desired profile of the end of the optical fibre 11 (the parts being marked A and B in figure

3) . For each of the parts, the processor 13 examines the profile 19 of the edge of the shaping tool 7 in order to identify a location on the edge 5 of the shaping tool 7 which corresponds to the spaced apart location on the desired profile of the optical fibre 11. This involves, in this embodiment, determining a tangent for each of the parts along the desired profile of the optical fibre 11 and for each spaced apart location along the desired profile of the optical fibre 11 using the profile 19 of the edge 5 of the shaping tool 7 to identify a location (which are marked A' and B' in figure 3) thereon that the same tangent as the location along the desired profile of the optical figure 11.

As can be seen in figure 3, location A of the desired lens 9 profile has the same tangent as the location A⁷ on the profile 19 of the edge 5. The same can also be said for locations B and B .

The processor 13 then superimposes the parts along the desired lens 9 profile of the optical fibre 11, onto the optical fibre 11. At this stage, the processor 13 calculates the path such that each one of the number of parts superimposed onto the optical fibre 11 can come into contact with the corresponding locations on the edge 5 of the shaping tool 7 when the end of the optical fibre 11 and/or edge 5 of the shaping tool 7 is moved along the path. In effect, the path is such that locations A and A' will come together, after which locations B and B' will come together.

The system 1 also includes an assembly 21 for providing relative movement between the optical fibre 11 and the edge 5 of the shaping tool 7 such that a portion of the optical fibre 11 is removed when the optical fibre 11 and edge 5 of the shaping tool 7 come into contact with each other.

As shown in figure 4, the assembly 21 includes drive motor 23 coupled to a member 25 which is capable of retaining the shaping tool 7. The drive motor 23 is in the form of an air driven motor which is controlled by an air regulator, whilst the member 25 is in the form of a chuck. The drive motor 23 and member 25 being such that they cooperate to rotate the shaping tool 7 at about 10,000rpm, thereby providing the relative movement between the edge 5 of the shaping tool 7 and the optical fibre 11. The drive motor 23 is coupled to the member 25 via a flexible coupling 27. The member 25 is also mounted to an air bearing 29 which is fed with filtered compressed air at about 90psi from a compressor 31. The flexible coupling 27 and the air bearing 29 ensuring that eccentricities in the shaping tool 7 (particularly at the edge 5) are kept to about lOnm whilst being rotated.

The system 1 also includes an apparatus 33 which is capable of moving the optical fibre 11 and/or edge 5 of the optical fibre 11 along the path, previously determined by the processor 13, whilst the assembly 21 is providing the relative movement between the edge 5 of the shaping tool 7 and the optical fibre 11. This movement effecting the machining of the end of the optical fibre 11 into the desired lens profile entered into the terminal 3b.

As illustrated in figure 5, the apparatus 33 includes a chuck 35 which is capable of holding the optical fibre 11, and an actuation means which is coupled to the chuck 35. The actuation means 37 is such that it can move the chuck 35 in order to effect movement of the end of the optical fibre 11 along the path whilst being held by the chuck 35.

As shown in figure 5, the actuation means 37 also includes a feedback circuit (not illustrated) which produces a signal that is used by the actuation means 37 in order to determine the location of the chuck 35, which in turn is used to ascertain the location of the end of the optical fibre 11. The feed back circuit is in the form of a capacitive transducer which is associated with the chuck 11 and a fixed structure. The actuation means 37 being such that it can move, either incrementally and/or transitionally, the chuck a distance up to 200 microns in at least x and z axis. The actuation means 37 is in the form of a number of piezoelectric actuators that are appropriately positioned about the chuck 35 to provide movement in the x and z axis.

The apparatus 33 also includes a drive motor 39 and an adjustable device 41 (illustrated in figure 6) , both of which are coupled together. The adjustable device 41 is coupled to the chuck 35, whilst the drive motor 39, which is in the form of an air driven motor, cooperates with the device 41 in order to effect rotation of the chuck 35, which in turn rotates the optical fibre 11.

As depicted in figure 6, the device 41 includes, a shaft 43 coupled to the chuck 35, a housing 45 in which the shaft resides, and a plurality of adjustable supports 47 that support the shaft 43 in the housing 45. The adjustable supports 47, which include picomotors 49 that can adjust the supports 47, to effect movement of the shaft 43. The device 41 also includes an air bearing 51 to which the shaft 43 is coupled. The air bearing is such that it ensures that eccentricities in the fibre 11 are kept to a minimum whilst being rotated. The air bearing 51 is supported with filtered compressed air at around 90psi from the compressor 31. The movement provided by the device 21 allows the chuck 35 to be moved, and in turn, the optical fibre 11 to be moved. The movement provided by the adjustable device 41 is such that it essentially allows a rotational axis of the optical fibre 11 and a rotational axis of the chuck 35 to coincide.

The system 1 also includes a control circuit (not illustrated) which is capable of detecting whether the rotational axis of the optical fibre 11 and the rotational axis of the chuck 35 coincide. To determine this, the control circuit includes a camera (which could be camera 3a) which captures an image of the end of the optical fibre 11 whilst it is being rotated by the apparatus and includes a sensor which is used to detect the rotational axis of the chuck 35. The control circuit also includes an image processor which is connected to the camera so that a captured image of the end of the optical fibre 11 rotating can be transferred from the camera to the image processor.

The image processor is in the form of hardware/software which analzises the images of the rotating fibre 11 in order to determine whether there is an angular and/or positional error in the edge of the optical fibre 11. Essentially, an angular difference will exist if there is an angular difference between the rotational axis of the fibre 11 and the rotational axis of the chuck 35, as shown in figure 7. It can be seen from figure 7 that there exists an angular difference between the two axis because the axis of the fibre 11 is inclined relative to the axis of the chuck 35. A positional error will be considered to exist if the two axis are apart from each other. An angular and/or positional error indicates that the rotational axis of the optical fibre 11 and the rotational axis of the chuck 35 do not coincide. Using the angular and/or positional error, the control circuit generates an electrical control signal which is proportional to the angular and/or positional error. The electrical control signal is fed to the picomotors 49 of the adjustable supports 47 of the device 41, which effects adjustment of the supports 47 such that the rotational axis thereof and the rotational axis of the device 41 coincide.

It will be appreciated that the present invention is not limited to machining a lens on the end of an optical fibre. It could also be use to, for example, machine threads onto minute working parts of a machine. Those skilled in the art will appreciate that the invention described herein is susceptible to variations and modifications other than those specifically described. It should be understood that the invention includes all such variations and modifications which fall within the spirit and scope of the invention.

APPENDIX A

An Optimal Linear Operator for Step Edge Detection

JUN SHEN*

Institut de Geodynamique, Bat. Geologie, Uniυersii Bordeaux-Ill, Avenue des Fa dtέs, 334U5 Taumm, France

AND

SERGE CASTAN

IRIT, UA CNRS, Universile Paul Sabatier, 118 Route de Narbonne, 31062 Toulouse, France Received July 26, 1990; accepted June 20, 199!

Laplacian operator [1-8]. Note that these differential op¬

Step edge detection is an important subject in image processing erators are in general of small size. Because the noise in and computer vision and many methods, including some optimal images is generally random, it is difficult to efficiently filters, have been proposed. In. this paper, we propose an optimal remove it from the image data merely in a small window. linear operator of an infinite window size for step edge detection. This operator is at first derived from the well-known mono-step On the contrary, the attributes extracted from a great edge model by use of a signal/noise ratio adapted to edge detecnumber of pixels would be less sensitive to noise, which tion. Because of the infinite window size of the operator, we procan explain partially the success of the edge detection pose then a statistic muitiedge model and analyze the optimal theory of Marr and Hildreth, who proposed the use of the operator by spectral analysis. It is shown that the Infinite Symwell-known Gaussian filters of large window sizes [9]. metric Exponential Filter (ISEF) is optimal for both mono- and Filtering an image by Gaussian filters of large windows muitiedge detection. Recursive realization of ISEF and the derivaneeds much computation, and fast algorithms have been tives is presented and generalized to multidimensional cases also. proposed for their realization [10-13]. The Gaussian disThe performance of ISEF is analyzed and compared with that of tribution is a nonzero function from ~∞ to +∞, but in Gaussian and Canny filters, and it is shown that ISEF has a better practice, when one processes an image by Gaussian filperformance in precision of edge localization, insensibility to ters, the Gaussian function is considered to be nonzero noise, and computational complexity. Edge detection based on the optimal filter ISEF is thus presented and the essential difference only in a finite interval [-w/2, +w/2]; i.e., a mask of a between ISEF and some other optimal edge detectors is shown. finite size is used. The success of Marr's theory of edge The experimental results for computer-generated and real images, detection showed that images filtered by a low-pass mask which confirm our theoretical analysis, are reported. © 1992 A«- of large size are less sensitive to noise, which inspired us demic Press, Inc. to hypothesize that if we use a filter kernel of an infinite size, still better results would be obtained. Moreover, the limited kernel -size constraint will introduce a cutoff ef¬

1. INTRODUCTION fect in the filtered image and produces in turn some Dirac

Edge detection has been an important subject in image distributions in the derivatives at the boundaries of the processing because the edges correspond in general to finite window, which is equivalent to an introduction of important changes in physical or geometrical properties noise in the low-pass and derivative images. With the use of objects in the scene. The derivatives express well the of filters of infinite window size, the cutoff effect problem changes in the gray value function; edges can therefore can be avoided. ' be detected by the maxima of the gradient or the zero The essential problem for a Gaussian filter is the concrossings of the second derivatives including the Laplac- tradiction between the noise-suppressing effect and the iaπ, calculated by some differential operators. . edge localization precision. As is well known, the larger

As differential operators are sensitive to noise, a prethe standard deviation of a Gaussian kernel, the less senprocessing such as smoothing is often necessary to resitive to noise the Gaussian filter. However, on the other duce the noise. Many methods for edge detection in noisy hand, the larger the Gaussian kernel, the more planar the images have been proposed, such as the Robert gradient, Gaussian function at the center and the worse the preciSobel operators, Prewitt operators, facet model, and sion for edge localization. Moreover this loss of localization precision can in turn introduce difficulties for edge

* On leave from Southeast University, 210018 Nanjing, China. point verification afterward; for example, the gradient magnitude will be less important at the edge position decess. By use of the theory of linear filtering of stationary tected because this position deviates from the true edge. processes, we show that the symmetric exponential filter Given the above considerations, we propose that in of an infinite window size is also optimal as a preparation order to obtain better results, it would be interesting to for muitiedge detection. Note that the optimal kernel we use a filter that is of an infinite size for efficiently reducing develop in the present paper is deduced from mono- and he noise on the one hand and is sharper at the center muitiedge models by use of the criteria determined from than the Gaussian filters for improving the precision for the point of view of signal processing rather than as an edge localization on the other hand. To the knowledge of approximation or simplification of Canny's filter. Bethe authors, the first results of edge detection by filters of cause an ideal differentiation operator implies an infinite infinite window sizes can be found in [14], where a symenergy in the frequency domain, it is impossible to realize metric exponential filter of an infinite size was used, realexactly such operators, and if we use an approximation, ized by a recursive algorithm. additional errors will be introduced. To overcome this

Canny was the first to analyze the optimal filter for difficulty, in Section 4 we present how to calculate the edge detection by variational calculus [15]. By use of the band-limited first- and second-order derivatives of the mono-step edge model and the finite kernel size coninput image filtered by the optimal symmetric exponenstraint, he proposed an optimal filter, which can be aptial filter. In Section 5, we present the recursive realizaproximated by a Gaussian filter. A related work was that tion of the optimal filter and its first- and second-order of Deriche, who proposed a simplified version of Canny's derivatives, taking into account the computational comfilter by applying directly Canny's results obtained under plexity, the finite word length effect, and the adaptation finite kernel constraints to the case of infinite window to parallel processing. In Section 6, we generalize these size [16]. We think that it is not sufficiently reasonable to results to bi- and multidimensional cases by use of magnianalyze a filter of an infinite window size merely on the tude and maximum value distances, and the calculation basis of the mono-step edge model, because when the of the aplacian and partial Laplacian is introduced. window size is infinite, we cannot suppose that there Edge detection based on the optimal filter is presented in exists only one edge in the window. In practice, only in Section 7 and the adaptive gradient is introduced also. To rare cases does one edge exist all over an image. verify the performance of the optimal filter, Section 8

In the present paper, we analyze the problem of the we present the experimental results of our method and optimal filter for step edge detection, giving a summary of compare them with results of some other optimal filters

, our research on step edge detection since 1985 [14, 17- for edge detection such as the Gaussian and Canny fil20]. As mentioned above, to obtain better results, we ters. These results show that the optimal filter proposed prefer to use filters of infinite window size. In Section 2, in the present paper exhibits a better performance in infollowing the traditional mono-step edge model with an sensibility to noise, precision of edge localization, comadditive independent white noise, we model the problem putational complexity, and adaptation to parallel proof edge detection in noisy images as a smoothing filter for cessing, which confirms the theoretical analysis. The removing the noise followed by a differential operator for paper is terminated with some concluding remarks and detecting the change. The analysis is done from the point two tables for clarifying the essential relation and differof view of signal processing. Because the smoothing filter ence between some other current filters for edge detecworks as a preparation for step edge detection, the ention and our optimal filter ISEF. ergy of its response to the step edge and to the noise as well as to the differential noise is considered in our crite2. OPTIMAL FILTER BASED ON MONO-STEP rion. Note that the localization error is in fact implied in EDGE MODEL this criterion because this error is caused by the noise energy in the differential output which has been taken 2.1. Optimal Kernel for Monoedge Model into account. By use of variational analysis and in the In this section, we deduce the optimal filter for edge

. case of infinite filter window size, we conclude that an detection based on the monoedge model. An edge detecoptimal low-pass filter as a preparation for edge detection tion operator is considered as the cascade of a low-pass is a symmetric-exponential filter of an infinite size (ISEF, smoothing filter for removing the noise in the input image Infinite Symmetric Exponential Filter). The performance and a differential block for detecting the important gray of the filter is analyzed and compared with that of Gausvalue changes to localize the edge points (Fig. 1). The sian-like filters and it is sho n that our optimal filter has a advantage of this model is that it clarifies the role that signal/noise ratio and a precision of localization better each part plays and shows that the essential characteristhan those of Gaussian-like filters. Because we use a filter tic of edge detectors in noisy images is the implied equivof infinite size, an analysis based on only the mono-step alent smoothing kernel. Once we have the optimal edge would not be sufficient. In Section 3, we propose a smoothing kernel, its derivatives of first or second order, muitiedge model described by a stationary stochastic proincluding the Laplacian, can be used to detect edges. Input image Differeniial (dldx)S_n(x) = (dldx){S_i(x) * f(x)}

I(x) output

Low-pass filter Differential block = (dldx){S(x) *f(x)} + (d/dx){N(x) */(*)} ► to remove the noise to delect the grey value change Because

FIG. 1. An edge detector considered as a smoothing filter followed by a differential block. idldx){Six) * fix)} = {idldx)Six)} * {fix)} and idldx)Six) = A ^• δtø, (2.4)

Let f{x) be the low-pass smoothing filter kernel for removing the noise that we want to find that gives the where δ(x) is the Dirac distribution, we have best results for step edge detection, The input noisy image S\{x) is modeled as (Fig. 2) idldx){Six) *fix)} = A ' δix) *fix)

(2.5)

Six) = Six) + Nix), (2.1) = A - fix), where Six) is the step edge free of noise, which gives that the first-derivative response corresponding to the step edge in the output of the filter will be

A, for x > 0, A -/(0) at the edge position x = 0, and the energy of this response can therefore be measured by E_s with

Six) = A/2, for * = 0, 0, for x < 0, Es = A> -r-iϋ). (2.6) and Nix) is the white noise independent of Six) with Another part that corresponds to the noise in the first derivative of the filter output is idldx){Nix) *f(x)}, whose

E{Nix)} = 0 and E{N²ix)} = n energy can be measured by EN¹ with

E_tf = E{[idldx)Nix) *fix)f} where E{-} indicates the expectation.

The output S_ϋix) of the low-pass smoothing filter will = E [[j^ Nix - ) - fiu) - du be

S^βtø = S,(x) */tø As the noise Nix) is white, we have

(2.2)

= Six) * fix) + Nix) * fix),

En- = E {f_^ NHκ - u) - f u) - du where * denotes the convolution. (2.7)

On the right-hand side of Eq. (2.2), the first term is the = n^{2 •}

^■ dx. response of the filter to the step edge free of noise and the second, that to the noise. Because the edges are in general detected by the maxi¬

The energy of the noise in S₀M can be measured by mum of the first derivative or the zero crossing of the EN, with

second derivative, the optimal low-pass filter fix) as a Because f ix) is odd and /₂(x) is even, we have

0, 0,

C = V£,v E_tf'IE's. and

Ignoring the coefficients representing the amplitudes of f\\x) ^■ dx ≥ 0. the step edge and of the noise in the criterion C, we obtain the normalized criterion C,v, which expresses the performance of the filter fix) for mono-step edge detecThis means the odd part f (x) produces no other effect tion and is dependent only on the filter kernel, than to increase the effect of noise, so in order to minimize C_fi, we should take/i(x) = 0; i.e.,/(x) should be an even function. And Cn can be rewritten as

Cfl - fix) ^■ dx I^' f' x. dx\ //-(0), (2.8) and the kernel fix) should minimize . = 4 ^• j_o r-ix) dx - r-ix) ^■ dx m. (2.1D

Before determining the optimal kernel fix) from the criterion C,v, we extract some properties that fix) should Now we determine fix) that minimizes C,v- For the satisfy. convenience of analysis, we first analyze the filter kernel

From Eq. (2.5), we see that the first derivative of the of a finite window size 2W and then take the limit case filtered step edge is A ^• fix). Because the edge position W → +∞ to find the optimal solution. corresponds to the maximum of the first derivative, A ^• Consider a filter fix) with a limited window size W; fix) should have its maximum at the edge position x = 0, we have from Eq. (2.11) and in order to avoid the introduction of additional maxima at the positions other than the edge position, this C² _N = 4 ^• J_β fix) ^■ dx ^■ fix ^■ dx/PiO). (2.12) maximum should be unique; i.e., the filter kernel fix) should satisfy

In order to find the optimal function fix), x 6 [0, W], max /(x) = /(0) (2.9) that minimizes CV, we create

Φ(/,/") = /'² + λ ' /², and > fixι) for |JΓ,| < | ₂|. where λ is a constant, λ > 0.

The optimal function fix) defined in the interval [0, W]

Second, because all functions can be decomposed into that satisfies the conditions of continuity and differentiathe sum of the odd and even parts, fix) can be rewritten tion for variational calculus will be the solution of Euler's as equation W = ιW +ΛW, (2-¹⁰) idtdx)Φ_r - Φ = 0; i.e., 2 - fix) - 2 ^• λ - fix) = 0,

the window, which is in fact equivalent to introducing a And for the Gaussian filter G(x, σ) = [l/(2τr)^{l 2 •} σ] noise effect in the differential output. So for our optimal exp(-x²/2σ²), filter for edge detection, we do not limit the window size; i.e., the limit case W→ +∞ should be considered. Taking C_N = 4 - G x, σ) - dx the case W→ +∞, from Eq. (2.13), we see that in order to obtain a convergent filter, fix) should satisfy the boundary condition

= (π/2)"² lim fix) = 0, » 1.253. which gives

We see that when the energy of the noise in the output of the filter and in its first derivative is considered with

C, = 0; i.e., fix) = C₂ ^• exp(-p ^• x), x e= [0, +∞). respect to the energy of the response of the filter to the step edge, the optimal filter ISEF shows a performance

Because /(x) is an even function as mentioned above, the better than that of the Gaussian filters; the difference optimal filter fix) is between them is to a factor of 25% or so. fix) = Cι ^■ exp(-p ^• |x|) with p > 0, x £ (-∞ +∞). 2.2.2. Precision for Edge Localization. Because the edge points are detected by the maximum of the gradient

As is well, known, filters /(x) and C -fix), where C is a or the zero crossing of the second derivative, the preciconstant, have essentially the same performance in resion of edge localization can be analyzed from the sign moving the noise, We would like to always take a normalchange and the slope of the second derivative of the filized filter kernel with the amplitude gain 1, i.e., tered image.

Consider the second derivative of the image filtered by fix) ^■ dx = 1, the kernel fix), tøWtøtø */(*)} ^< id²ldx²){Six) * fix)} ₍2 17) which gives + id²lώ ²){Nix) *fix)}. fix) = ip/2) ^■ expi-p ^■ |x|; Obviously, if there is no noise, i.e., Nix) = 0, we have rewritten as id ldx²){S,-ix) */(*)} = id ldx²){Six) *fix)} fix) = a ^■ fcM, (2.14) = Six) * id ldx^l)fix) (2.18)

= idldx)Six) * (dldx)f(x). where a = (-In b)l2 and 0 < b < 1. In the discrete case, we have a = (1 - b)l{l + b) and 0 < b < 1, which implies For the optimal filter fix) = a ^■ IJM, we have 0 < α < 1.

We conclude that on the basis of the mono-step edge [a ^■ In b ^■ b* for x > 0, model, the optimal low-pass linear filter as a preparation id/dx)fix) (2.19) for edge detection for removing the noise is a symmetric -a ^■ In b ^■ b^~x for x < 0, exponential filter of an infinite window size, briefly called the infinite size symmetric exponential filter. And edges which gives, from Eqs. (2.4) and (2.18), can be detected from the differentiation of the output of this filter, such as by maxima of the first derivative or A ^• a ^■ In b ^■ V- for x > 0, zero crossings of the second derivative or Laplacian. id²ldx²){Six) *fix)} = -A - a - h b - b-' for x < 0,

2.2. Performance of the Optimal Filter (2.20)

2.2.1. Noisel Signal Ratio iNSR). From Eq. (2.1 1), We see from Eqs. (2.18) and (2.20) that x = 0 is a zero we can calculate the NSR for the optimal filter: crossing of the second derivative of the image filtered by fix). That is, if there is no noise, the second derivative of

C_N = {4 ^• a^{2 •} b²-' ^• dx ^■ ia ^■ In b)^{2 ■} b²* ^■ dx] '/a² fix) localizes the zero crossing at the edge position with no error. Figure 3 shows the symmetric exponential filter

= 1. (2.15) of infinite size and its first derivative.

with

Now we analyze the case where the noise exists. As in > 0. (2.23)

(2.24)

id ldx²){Siix) * fix)} > 0 for x = xt The error for edge localization can therefore be measured by and e = £{|x_e|}, id ldx²){Siix) */(x)} < 0 for x = x;, which gives from Eq. (2.24) or id ldx ){Sιix) *fix)} < 0 for x = x x\ = E{\Nix) *f( \} lim - fe/2 fix) ^■ dx and

E{\Nix)\} id²ldx²){S;ix) */(x)} > 0 for x = x_e ^' - "\ |/"(x)| ^■ dx

Because |/"(x)| •

id²/dx-){Siix) *fix)} = id ldx²){Six) */«} + R~ (2.21) with R„ = id²!dx²){Nix) * fix)}, the zero crossing of id lέx²){Siix) */(x)} will be the level crossing of id²/ dx¹) (d^,/ x^z) S(x)*OW = (d <lx) GW {Six) * /(x)} passing through the level -R_a. From Eq. (2.20), we see that id²ldx²){Six) */(x)} has a jump from -A ^■ a ^• In b ^• b^~x to A ^• a ^■ In b ^■ If when x changes from 0" to 0^" (Fig. 4), so the error introduced by the noise R„ in the second derivative of the filtered image produces no effect on zero-crossing localization; i.e., the zero cross

ing will still be detected at x = 0 and no error in edge FIG. 4. Precision of zero crossing localization, (a) Symmetrical exlocalization will be introduced even when there is noise. ponential filter, (b) Gaussian filter. Removing the coefficients corresponding to the ampliduced by the noise R„ in the second derivative causes no tudes of the step edge and of the noise, we obtain the error for zero-crossing localization. Comparing Eq, measure of the edge localization error L_e : (2.28) with Eq. (2.31), we conclude that our optimal filter ISEF localizes the step edge with a precision better than that of the Gaussian filters. It should be noted that in the lim \f"ix)\ .- dx ISEF method, the output is the convolution of the input

-ε/2 l/"WI ^• dx image with the second derivative of a symmetric expo¬

(2.25) nential filter of infinite window size which includes a Dirac distribution at the center of the kernel (see Eq.

For our optimal filter fix) = a ^■ b^ we have from Eq. (2.27)). ItTs this Dirac distribution for the ISEF filter that (2.19) that fix) is the distribution ensures a very good precision for edge localization.

Note that in noisy cases, because of the remaining

[a ^■ i\n b)^{2 ■} b^ for ^ O, noise, we have, in general, another zero crossing farther fix) = (2.26) from the real edge position than the one detected, regardt(2α ^• In b) ^■ Six) for x = 0; less of which smoothing filter is used (Fig. 4). This false zero crossing will be removed by gradient thresholding i.e. and/or the "derivative sign correspondence principle," which is presented in Section 7.1. fix) - a - la b - [(In b) ^• *W + 2 ^• Six)]. (2.27)

Substituting Eq. (2.27) in Eq. (2.25), we have for the 3. OPTIMAL FILTER FOR MULTIEDGE DETECTION optimal filter

In Section 2, we have analyzed the optimal filter by use

0. (2.28) of the mono-step edge model. As is well known, the mono-step edge model is widely used in edge detection

For the Gaussian filter G(x, σ) = [l/(2ιτ)^{lβ •} σ] studies such as [15]. When Canny deduced his optimal exp(-x²/2σ²), we have filter for step edge detection, he used the mono-step edge model and analyzed the optimal filter with the constraint that the filter kernel is of a finite window size. It is realim (2 ^• ιr)^υ (2.29) sonable to use the mono-step edge model in the case of

|G"(x)| ^■ dx filters of limited window sizes because we can suppose that there exists only one step edge within the filter winand dow when the window size is sufficiently small. But, in our case, because we prefer to use filters of infinite window size to remove efficiently the noise and to avoid the

|G"(x)| ^• dx = 4 ^• (e)"^{2 •} σ~ (2.30) cutoff effect and because the optimal filter we found is a symmetric exponential filter of an infinite window size,

Substituting Eqs. (2.29) and (2.30) in Eq. (2.25), we have an analysis based uniquely on the mono-step edge model for the Gaussian filter evidently will be insufficient. In practical cases, we always have many edge points in an image and the case = ^• (2 ^• e ^■ v) 1/2 (2.31) where there exists only one unique edge point on all the image is rarely encountered. So we think that for the

We see that the edge localization error of a Gaussian filters of an infinite window size, an analysis based on filter is proportional to the standard deviation of the muitiedge models would be absolutely necessary. Gaussian kernel; the larger the standard deviation, the In this section, we analyze the optimal filter problem more important the edge localization error. In fact, when for edge detection based on muitiedge models. the noisy step edge is filtered by a Gaussian filter of a Our task is to find an optimal filter that can eliminate large standard deviation, the slope of the second derivathe noise and preserve as well as possible the step edges tive of the filtered image will be decreased, and the reso that its derivatives can be used for edge detection. sponse to the noise in the second derivative, i.e., R„, will Because the edge positions in images change from one to cause a more important localization error (Fig. 4). As for another, our analysis is statistical. the symmetric exponential filter of an infinite size, beWhat we are interested in is the step edges rather than cause the second derivative of the filtered step edge the DC or low-frequency components, so we model the changes its value from -A ^• a In b • b^~x to A • a In b • b^x step edge sequences by the following stationary stochaswhen x changes from 0^~ to 0⁺ (Fig. 4), the error introtic processes. The muitiedge model SM(x) is shown in Fig. 5. SM(x) equal to -A², so we have the autocorrelation of the muitakes values -A or A with A > 0, and a jump of SM(x) tiedge sequence SM(x) as follows, from -A to A (respectively from A to -A) coσesponds to step edge. Suppose SM(x) is a stationary stochastic Rsir) = £{SM(x) ^■ SM(x - r)} process satisfying the following conditions:

(1) Stability. The probability that we have a step edge = A² ∑ Puir) - ∑ r) (3.3) in an arbitrary interval (x₀, x₀ + Δx) is independent of x₀. 1=0 t=0

(2) Orthogonality. The number of step edges in an in= A^{2 •} exp(-2 ^■ λ ^• |τ|), terval (xo , xι) is independent of that in another interval (x₂ , x₃) if (xo , xι) n (x₂ , x ) = 0, which is an assumption which gives the spectral density of SM(x), much used in image analysis [1],

(3) Finite edge density. The probability that there exist r+∞ Psiω) = illiή ^■ J Rsir) ^• cos(ω ^■ T) ^■ dr two or more step edges in an interval (x, x + Δx) is very small when Δx → 0; i.e., ,P₂(Δx)/Δx → 0 when Δx → 0, (3.4)

= (2/τr) ^• (A^{2 •} λ)/(ω² + 4 ^• λ²). where Λ(Δx) denotes the probability that we have two edge points in the interval (x, x + Δx). Because Nix) is the white noise, its spectral density

The noisy step edge sequence SM|(x) is will be constant:

SMι(x) = SM(x) + Nix), p_Niω) = (2/τr) ^• B² (£ > 0). (3.5) where SM(x) is the step edge sequence free of noise as We therefore have from Eqs. (3.4) and (3.5) that the described above and JV(x) is the white noise independent spectral density of the noisy step edge sequence SMj(x) is of SM(x) with E{Nix)} = 0 and E{N²ix)} = n¹.

From the model SM(x) above, the probability that _PsNiω) = (2/ιr) ^• [(A^{2 ■} λ)/(ω² + 4 ^• λ²) + B²]. (3.6) there exist k step edges in an interval Δx is

The optimal linear filter /(x) removing the noise in the (Δx) = exp(-λ ^• Δx) ^• (λ ^• Δx)%! (3.2) input noisy image should give an optimal estimation Mix) of SM(x) from the noisy image SMj(x) such that the differwith λ > 0, the average density of edge points in the ence between the estimation Mix) and SM(x) is image, and kl = k - ik - I) 1; Pt(Δx) is therefore a

Poisson distribution.

Obviously, when there exists an even number of edge points in the interval (x - T, X), SM(x) ^• SM(x - T) will be in [M(x) - SM(x)]^{2 •} dx = Minimum

equal to A², and when there exists an odd number of edge According to the theory of linear filtering of stationary points in the interval (x - T, x), SM(x) ^• SM(x - T) will be stochastic processes [21 , 22], the spectral characteristic of the optimal filter fix) should satisfy

= 0 . The

(3.7) (3.8)

FIG. 5. Muitiedge model. The optimal filter kernel (x) can therefore be found by fix) = ®- { M(<O)} with α = (-In b)/2 and 0 < b < 1;

(3.9)

= [(A^{2 •} λ)/(2 ^• B^{2 ■} a)] ^■ exp(-α ^• |χ|). [a ^• In b ^■ b^x for x > 0, Ax) = idldx)fix) = (4.2)

We conclude that based on the muitiedge model that {-a - ln b - b^~x for .r < 0; we proposed in this section, the optimal linear smoothing filter is still a symmetric exponential filter of an infinite and window size, i.e., the same as that found based on the mono-step edge model. fix) = id²ldx^l)fix) = a ^■ In b ^• [(In b) ^■ &M + 2 ^• δ(x)J.

Note that in Eqs. (3,8) and (3.9), λ is the density of (4.3) edge points, i.e., the average number of edge points in an interval of unit length. So when λ is increased, i.e., the Consider the Fourier transform of/(x) in Eq. (4.1); we average distance between neighboring edge pixels is dehave creased, a will be increased and the optimal filter will be sharper to reduce the influence between neighboring edge 9\fix)} = 4 ^• α /(ω² + 4 ^• a²) (4.4) pixels. And, on the other hand, when the signal/noise ratio of the noisy image is decreased, the optimal filter and Eq. i4.4) can be rewritten as will become planar to effectively remove the noise. But in all cases, we have always a ≥ λ, otherwise the filter 9{fix)} = [2 ^• alii ^■ a + j - ω)] ^■ [2 ^■ β/(2 ^■ a - j - ω)], kernel fix) would be too planar to avoid an important (4.5) influence of neighboring edges. where j = V-ϊ. Taking the inverse Fourier transform, we have

4. DERIVATIVE COMPUTATION FOR EDGE DETECTION f{x) = fdx) *Mx), (4.6)

We have shown that based on the mono-step edge and where multi-step edge models, the optimal low-pass smoothing filter for removing the noise as a preparation for edge 2 ^■ a ^■ exp(-2 ^■ x) = 2 ^• a ^■ b^x for x ≥ 0, detection is a symmetric exponential filter of an infinite fiix) window size. Edges can therefore be detected by use of 0, for x < 0; the first or second derivatives of the smoothed image. (4.7) The scheme in Fig. 1 is convenient for the analysis of the optimal filter for edge detection, but for practical realizaand tion, it is not necessary to realize the low-pass filtering and the differentiating block separately. One reason is for x > 0, that it is almost impossible to exactly realize a differential /*(*) = operator without smoothing; for example, the first-order exp(2 ^• a ^■ x) ~ 2 • a ^■ b^~ for x ≤ 0. differentiation will have j • ω as the Fourier transform, (4.8) which has infinitely important high-frequency components and is not convergent, so it cannot be realized Comparing Eqs. (4.7) and (4.8) with Eq. (4.2), we have physically. If we use an approximation of a finite window size to realize the differential operator, errors may be introduced by the approximation and an operator of a /'( ) = « ^• [/R(*) - / «] (4.9) limited window size will in general produce the cutoff effect, so the advantage of using an infinite window filter and another representation for/'(x) is will be lost. Another reason is from the point of view of computational complexity. It will be desirable to realize, ix) if possible, the low-pass filtering and the differential opera ^■ In b ^• b^x = In b ^■ [2 • b^x], forx > 0, ations together by a simple algorithm, which is important for practical applications. -a ^■ In b ^■ b^~x - -In b b^~x forx < 0,

Rewrite the symmetric exponential filter and its derivatives as which gives fix) = a ^■ b\^χ] (4.1) /'(x) = Hn έ) - [/(χ) -/_LW]. (4.10) From Eqs. (4.3), (4.7), and (4.8), we have where /_L(ι) is the equivalent linear filter with f ix)

From Eqs. (4.-2), i4.9), (4.10), and (4.11), with the coefBy comparing Eq. (4.7) with Eq. (5.3) and letting ficients neglected, the low-pass output of an input image Iix) filtered by the symmetric exponential filter fix) of an o = 2 ^• a infinite window size and the band-limited first- and sec(5.4) a_\ - b - exρ(-2 ^• a), ond-order derivatives can be calculated as the one-side filter /_L(x) in Eq. (4.7) is realized.

Low-pass filtered image: 7(x) */(x) = /(x) */_Ltø *Mx)

(4.12) Similarly, another one-side exponential filter /_R(x) defined in Eq. (4.8) can be realized by the algorithm

First-order derivative: rι(0 = flo (i) + ^βι rι(« + l). i = N, 1, (5.5) idldxMx) * fix)] ~ Iix) * Mx) - Iix) * /_L(x) (4.13) where Y is the output, X is the input, αo = 2 ^• a, and a, = idldx)[I(x) */«] ~ Hx)-* fix) - Hx) *Λ(x) (4.14) b = exp(-2 • a), as is mentioned above.

Normalizing the filters Ki) and Mi) to have an ampli¬

Second-order derivative: id l dx²)[Iix) */(x)]

(4.15) tude gain 1 in the discrete cases, we take ~/(x) */(x) - /(x): α₀ + α\ = 1 with 0 < flo < 1. (5.6) where left side ~ right side means that the left side can be calculated by the right side to a factor, so they are equivSubstituting Eq. (5.6) in Eqs. (5.1) and (5.5), we obtain alent as the absolute amplitude is not concerned. the following recursive algorithms to realize the one-side

We see that on the basis of the one-side exponential exponential filters /_L(i) and/_R(ι): filters fix) and Λ(x), the low-pass image and the band- limited first- and second-order derivatives can be easily ΛO^"): rι(0 = «o ' [Xi - W " DJ + W ~ D, calculated. (5.7) i = 1, . . . , N,

5. RECURSIVE REALIZATION and

From the preceding section, we conclude that given an input noisy image, the low-pass and the band-limited Aii): Yiif) = α_Q ^■ [Xii) - Y,U + 1)] + 7,(i + 1),

(5.8) first- and second-order derivative images filtered by the i = JV 1. optimal filter, i.e. , the symmetric exponential filter of infinite window size, can be calculated by the essential one- By use of the recursive filters in Eqs. (5.1) and (5.5) or side exponential filters /_L(x) and/_R(x). Evidently, to realin Eqs. (5.7) and (5.8), the symmetric exponential filter of ize these infinite impulse response filters, recursive infinite size and its first and second derivatives can be algorithms should be used. In this section, we present realized by their cascading or parallel combination, as is how to realize them by a very simple recursive algorithm. mentioned in Section 4.

5.1. Recursive Realization of One-Side 5.2. Recursive Realization Taking Account of Finite Exponential Filters Word Length Effect

Consider a recursive filter as Different combinations based on the above filters for realizing the ISEF filters and their derivatives have been r,(i) = αoX(θ + β,y, - i), / = ι, N, (5.1) proposed [16-20, 29]. Note that these algorithms were proposed with no special consideration of the errors inwhere Y_\ is the output, X is the input, and α₀ and a\ are troduced by finite word length effect. Here we present a the coefficients, with a₀, #ι > 0. new algorithm for realizing the ISEF filters and their deBy use of Z-transform [23], we have rivatives that shows less complexity of computation. And because the images are stored in general in 8-bit memo¬

WO = / (Ø * *(«). (5.2) ries, i.e., each pixel has only 8 bits, special consideration of reducing the error propagation of the finite word length it is easy to show that D_\ and D₂ are the first- and second- effect is also taken into account in our new algorithm. order derivatives of the image filtered by the symmetric

We present first the realization of the ISEF filter. Let exponential filter divided by a factor p ^• e' l and p- e'"/ α₀ in Eq. (5.1) be (1 - i)/(l + b) and α₀ in Eq. (5.5) be b ^■ 2, respectively. (1 - _>)/(! + b) i.e., we use the recursive filters Compared with the algorithms proposed in [20], the new algorithms proposed above need no more bits of

Y ii) = [(I - b)H\ + b)] ^■ XO) + * ^• Y ii ~ 1), memory but the amplitudes of the calculated derivatives

(5.9) are amplified lib times. Considering the finite word

N, length-effect [24], the errors introduced by the finite bits of memory are thus reduced by a factor b. Note that b < and 1 , and for many practical images, b is in general less than 0.5; a double precision is gained because of the new algo¬

1W9 = [-> ^• (. - b)Hl + b)] ^■ XO) + b ^■ Y_RsU + 1), rithm and no more memory is needed. Note that the one- side exponential filters used in the implementation are

(5.10) / = N, . . . . 1. first-order recursive filters, whose limit-cycle behavior has been analyzed -in the literature [24]. If the coefficients

By taking the sum of the filters are floating-point numbers and a floatingpoint multiplier is used, floating-point operations can

therefore be used in the calculation of the recursive filtering but the resulting image is stored in 8-bit memory; the it is easy to show that 7(0 is the output corresponding to dead band effect is avoided. the input image filtered by the discrete normalized sym6. GENERALIZATION IN MULTIDIMENSIONAL CASES metric exponential filter /(i) with

As shown in the preceding sections, based on the

/(0 = [(1 - b)lil + b)] ^• ill, mono- and multi-step edge models, the optimal filter as a preparation for edge detection is a symmetric exponential i = -», . . . , - 1, 0, 1 +00, filter of an infinite window size: and Σ /(i) = 1. fix) a ^■ e ^■r\x\ (p > 0).

Note that the algorithm above is well adapted to realization in parallel: 7_Ls and 7_Rs can be calculated in paralIn this section, we generalize the result obtained for lel, the structure and access to memory is simple, and the monodimensional cases to multidimensional cases, espeoverall time to calculate an output pixel is only one multicially to bidimensional cases, which are the most used in plication and two additions. image processing.

The scheme above is well adapted for calculating the Evidently, we have two possibilities for this generalsmoothed image, but it is not convenient for calculating ization. One is to use directional filters in different directhe derivatives with precision by this scheme. In the foltions and the filtered directional .derivatives (first or seclowing, we present a scheme for the computation of the ond order) can then be calculated for edge detection. derivatives. Discussions on directional derivative operators can be

Consider the filters in Eqs. (5.1) and (5.5) as found in the literature [1, 6]. Another possibility is to use nondirectional operators such as the band-limited Laplacian [9]. The advantage of such operators is that calcuL-(0 = (1 - 6) ^• X + b ^■ Yuii - 1), i = 1 N,

(5.12) lations in different directions are not needed and less computational complexity can therefore be achieved. A and discussion on nondirectional operators can be found in [25]. In both these two cases, the essential problem is the determination of the corresponding bidimensional low-

Yω(Q = il - b) - XiD- + b - Y_Mii + l), i = N, pass filter and its realization, because the band-limited

(5.13) derivatives can be easily calculated from the smoothed image, as we see later.

By taking the differences

6.1. Using Euclidean Distance ?

0,(0 = y_Bd(/ + 1) - Y i - D (5.14)

A direct generalization of a one-dimensional filter fix)

D U) = Yuϋ + 1) + Yuii - 1) - XH) - XO), (5.15) to bidimensional cases is to take (D(x, y)). where Dix, y) is a distance measure defined for (x, y) such that £>(x, y) nential filters. Obviously, the filter /( , j) is separable; degenerates to x in one-dimensional cases. and/(D(x, y)) i.e.. will be rotationally invariant for D(x, y); i.e., all (x, y) having the same distance value will correspond to the /(U) =/(0 */O). (6.6) same filter coefficient. For the symmetric exponential filter, we have the bidimensional filter (D(x, y)) as where /(<) and f[j) are the ID symmetric exponential filters respectively in the dimensions i and j. fiDix, y)) = a - _e-^p-°M _P > 0). (6.1) The directional derivatives can be calculated as

A natural choice of £⁾(x, y) is to take the Euclidean First derivative in dimension i: distance ... ., ,, .. ,, .. fiiι, j) = f(j) *fi(0 (6.7)

Dsi , y) = V(x² + y²), (6.2)

Second derivative in dimension i: which gives fiiii,ϊ) = f{j) * (6.8) fix, y) ~ a ^• exp[-/7 ^• V(x² + y²)]. (6.3) First derivative in dimension j: i6.9)

The filter /_e(x, y) is circularly symmetric for Euclidean /X/, ⁽0 *J5C/⁾ distance, as is desirable in most cases. The inconve¬

Second derivative in dimension j: nience of using/_e(x, y) is that no fast realizations have yet been found, so one must take an approximate filter mask fϊii,j) = fU) * fjj(f), (6.10) of a finite size to realize it, which will introduce a cutoff eσor, and a large mask size implies an important compuwhere fii,j) and /„( ,;^') mean the first- and second-order tational complexity. It is for this reason that we propose partial derivatives in the dimension x, respectively. in the following the use of some other distance measures The filter fi, j) and its derivatives can therefore be to deduce the bidimensional symmetric exponential fildecomposed into the cascade of the one-dimensional ters of which a fast realization is possible. symmetric exponential filters respectively in the dimensions i and;^'; each can be realized in turn by the combina¬

6.2. Magnitude Distance Symmetric tion of two one-side exponential filters realized by first- Exponential Filter order recursive algorithms in inverse directions, as is mentioned above. Because the lines (respectively the

One of the most utilized distance measures in image columns) are independent of each other for horizontal processing is known as the magnitude distance [1], de(vertical) operations, this decomposed realization can be fined by easily implemented by parallel line processors, for example, in a parallel system SYMPATI [32].

D_mix, y)

+ \y\ . (6.4)

6.3. Maximum Value Distance Symmetric

Substituting Eq. (6.4) in Eq. (6.1), we have Exponential Filter fix, y) = a - β- tøl⁺M⁾. (6.5) Another kind of distance measure in image processing is the maximum value distance D_xix, y) [1], defined by

Of course, an inconvenience of the use of the magnitude distance is that the 2D filter will no longer be iso- Z),(x, y) = Max(|x|, |y|), (6. H) tropic for Euclidean distance measure (the isovalue line of. the filter will be a square rather than a circle). But which gives the Maximum Value Symmetric Exponential considering the important advantage that a very simple filter (MVSE filter) as ■ implementation can be obtained and the filter will be iso- tropic for the magnitude distance measure, we propose f x, y) = a. , ^■ e -p,-Maι (6.12) using the magnitude distance measure to generalize the symmetric exponential filter from ID to 2D and -dimen- By use of the Z-transform, it is easy to see that a maxisional iM > 2) cases. mum distance filter is in fact a rotation of τr/4 of a magni¬

Given a 2D filter/(;,/) to realize, we show now how to tude distance filter [19]. So the separated realization of calculate the smoothed image and the first- aVid second- the maximum distance filter will be similar to that of the order derivatives of a 2D image by ID symmetric expo- magnitude distance filter except that one-dimensional fil— teriπg should be executed in two diagonal directions re/₂(x, y) - Iix, y) = ( 14a¹) ^• Iix, y) * Δ/₂(x, y). spectively rather than in line and column directions, and (6.15) we must take the sampling interval 2 pixels in two diagonal directions because of the rectangularly sampled imEquation (6.15) indicates that the difference between age. the input and the output of a 2D symmetric exponential filter is a measure of the band-limited Laplacian of the

6.4. Laplacian Calculation in Bidimensional Cases input image; we call this method the DRF (Difference of Recursive Filters) method for the Laplacian of ISEF.

As for the Laplacian, we have

Let all the pixels of a positive Laplacian take the value mi,J = ,UJ) +Mi,J), (6.13) 1 and the others take the value 0; we obtain a Binary Laplacian Image (BLI), which is a feature image of the which can easily be calculated from the partial derivaoriginal image [30, 31] and can serve edge detection also. tives. 6.5. M-Dimensional Cases

Another possibility for calculating the Laplacian is to use the difference between the bidimensionally smoothed It is direct to generalize the algorithms above to M- image and the original image. dimensional (M > 2) cases.

Let

6.5.1. Low-Pass Image and Derivatives. Given an image in the M-dimensional space 5_M (xι , . . . , x._w), the (^χ, y) = li^χ, y) * M^χ, y), magnitude distance in the space SM is defined by where Iix, y) and Λ(x, y) are respectively the input and

D_m = bidimensionally smoothed output images, and ₂(x, y) is = \x\\ + 1**1, the bidimensional symmetric exponential filter with where X = (xι, . . . , x.w), and an M-dimensional symmetric exponential filter is

which is separable; i.e., ix, y) - Iix, y) = /,(x, y) - /(x, y) */(x) + /(x, y) *fix) - Iix, y), (6.14) UX) = [a^{m ■} exp(-/? ^• |x,|)

* . . . * [a^{m •} exp(-p • |x_w|)]. (6.16) where fix) is the one-dimensional symmetric exponential filter in dimension x From Eqs. (6.14) hix, y) - Iix, y) =

=

= (l/4α²) - /(x, y) * {(a²/dy²) /₂(x, y) * [(a/ax_;)/_m(xj)], (6.17)

=

=

Neglecting the higher-order derivatives, we obtain FIG. 6. Isovalue surface of a t dimeπsional ISEF. in which

* ^{• • ■} * e_w, because the first which can be easily realized by the cascade of ID expopartial derivative idldx_j)f_m ixj) in the dimension X_j and the nential filters. low-pass exponential filters in the other dimensions can Edges can be detected by the zero crossings of the be realized by the cascade of oπe-dtmeπsioπal filters. The partial Laplacian. The advantage of the use of partial multidimensional derivative filter is thus realized by the Laplacian is that we can detect and obtain closed edge cascade of them. The calculation of higher-order partial supersurfaces in different subspaces adapted to the proband mixed derivatives is similar by use of the convolution lem concerned. Moreover, the low-pass filtering in the theorem, which is trivial. complement of the subspace smooths the M-dimensional

6.5.2. Laplacian and Partial Laplacian. Laplacian image to suppress the noise but does not blur the edges in of an M-dimensional image I(X) filtered by an M-dimen- the subspace in which we are interested, if the complesional exponential filter /_m(X) is defined by ment and the subspace are orthogonal to each other, as is the case in many practical images.

Δrø * /(X)] = [(3²/Sx?) /_m(X) * /(X)],

7. EDGE DETECTION BY ISEF AND ADAPTIVE GRADIENT which can be calculated by the sum of the second-order With the optimal linear filter for edge detection and its partial derivatives in each dimension. realization proposed above, edges can be detected by

Similar to 2D cases, it is easy to show that we can maxima of gradient, zero crossings of directional second calculate the M-dimensional Laplacian by the difference derivatives, or zero crossings of Laplacian, always filbetween the output and the input of the M-dimensional tered by the optimal symmetric exponential filter. The exponential filter; i.e., edge candidates thus obtained are then verified by gradient thresholding with or without hysterisis. Note that the Laplacian will be more sensitive to noise than directional i[/m(X) * I(X)] ~ /m(X) * /(X) - (X). (6- 18) derivatives if the linear variation condition is not satisfied, as was mentioned by Haralick and Marr [6, 9], so

Note that in M-dimensional cases, edge detection by the use of directional derivatives can give better results zero crossings of Laplacian shows more advantages than than the Laplacian. For ISEF, this is also the case [20]. 2D cases. First, using the M-dimensional Laplacian, the On the other hand, use of the Laplacian shows the folcomputational complexity is much more reduced than lowing advantages: (a) it is rotationally symmetric to corthat of directional derivatives as the dimension number is responding distance measure (for example, magnitude increased. Second, the zero crossings of the M-dimendistance or maximum value distance); (b) derivative calsional Laplacian form closed supersurfaces in the M-diculation in different directions ^"is not needed and a less mensional space, which will be difficult to obtain if only important computational complexity can therefore be directional derivatives were used. Of course, the shortachieved; and (c) the edges detected will be closed and of coming of the use of the Laplacian for edge detection is 1 pixel width. that it is more sensitive to noise if the linear variation condition is not satisfied [9]. 7.1, False Zero-Crossing Suppression

It is interesting to introduce the partial Laplacian in by Sign Correspondence multidimensional cases defined by

As is well known, step edges can be detected by the zero crossings of the second derivatives. In Fig. 7 the two

Δ(A > ^)[ -,(X) * /(X)] = (3²/3χ- ) _ra(X) different cases of smoothed step edges and the first- and

* J(X) + ^{• • •} + (3²/flxL)/_m(X) * i(X) (6.19) second-order derivatives are shown. We see that in case 1, the first derivative is positive and the second derivative will change its sign from positive to negative when x is withxt,- £ { ι , ^{■ • •} , x_M}, foπ^' = l, . . . , n. Evidently, this partial Laplacian can be calculated by the sum of the increased to pass through the zero crossing. For simplicity, we call such a zero crossing a positive zero crossing. directional second-order derivatives in the dimensions concerned, but it can also be calculated by the difference In case 2, the first derivative is negative, which correbetween the output and input of multidimensional filters sponds to a negative zero crossing (i.e., when x is increased to pass through the zero crossing, the second derivative changes its sign from negative to positive). We

Δtøti , • • • ■ kM ) * IGQ1 ^• /m(X) * /(X) conclude that for a step edge, if it has a positive first

- fmiXki , ^{■ ■} , xt,,) * /(X), derivative, it must correspond to a positive zero crossing;

Case 1 otherwise if it has a negative first derivative, it must corStep edge f respond to a negative zero crossing. We call this correspondence between the signs of the first and second deV rivatives of a step edge the "zero-crossing sign correspondence principle." And a zero crossing violating this principle cannot be a step edge.

Because of the existence of noise and the rounding or truncation errors, in practice when we calculate the derivatives of the input image to detect step edges, the results are in general not as simple as those in Fig. 7. For the step edges of case 1 and case 2, we obtain some false zero crossings as shown in Fig. 8.

Sometimes it is difficult to remove these false zero crossings by gradient thresholding, because they are -close to the true edge and can therefore correspond to a large gradient value (note that the calculated gradient is a smoothed one). A high threshold will cause a considerable loss of true edges- and a low threshold will not be sufficient for false zero-crossing suppression. But if we examine the sign of the derivatives, we see that these false zero crossings violate the sign correspondence principle and can therefore be easily removed even when they correspond to a high gradient value. That is why we propose removing them with the use of the sign correspondence for zero crossings before the gradient thresholding. According to our experiments, we find that the use of sign correspondence helps much in suppressing false zero crossings that are sometimes difficult to

remove by gradient thresholding. . FIG. 8. False zero crossings close to a step edge. Because zero crossings of the band-limited Laplacian and are verified by gradient thresholding, the quality of the gradient estimate plays an important role in edge detecE{[^"gτiP) - gr( )]²} = Minimum, (7.6) tion. Obviously, a shift-invariant band-limited gradient operator will smooth together the two regions of different where gr(P) is the gradient magnitude at P free of noise; gray values separated by the edge pixels and give therei.e. , fore an estimate smaller than the real gradient magnitude as the edges in a 2D image can have arbitrary forms. In gr(/>) = g; - g/. (7.7) this manner difficulties for edge pixel verification arise. We show here how to give an estimate that best approxiSubstituting Eqs. (7.1), (7.4), and (7.7) in Eq. (7.5), we. mates the real gradient magnitude with the help of BLI. have Our deduction shows that in 2D cases, this estimator is not shift-invariant.

Let W be a window centered at the pixel P which is a Mix, y) ^■ Nix, y) zero crossing of the band-limited Laplacian. Suppose

that W is composed of N regions Ri (i = I, . . . , N) of = 8ι - gj. (7.8) different gray values g, (i = I N) with an additive white noise Nix, y) Of mean 0 and R_r, Rj, the two regions We have from Eqs. (7.2) and (7.8) separated by P (Fig. 9). We have

Mix, y) = 0

Gix, y) = g, + Nix, y), (x, y) 6 R, e W, for (x, y) i = l, . . . , N, (7.1) eΛ, (/ = ι, N and i Φ I, J) and ∑ Mix, y) = 1 (7.9)

(-yjefl/

E[N[x, y)] = 0 (7.2) ∑ (M(x, y) = - 1 .yteRj ε *, y⁾] = «o (7.3)

Because JV(x, y) is white, substituting Eq. (7.9) in Eq. where G(x, y) is the gray value of the pixel (x, y) and £[^■] (7.6), we have indicates the expectation.

A linear gradient estimator Mix, y) gives the estimated £{[ ) - gr(P)]²}

gradient ^*gr( ) for the pixel P by ix. Σytex, , y)

W) = ∑ ∑ Mix, y) ^■ Gix, y) (7.4) (7.10)

+ ∑ M\^χ, y)

[x.y)SR, and ^*gr(P) should satisfy for all possible gray value distriminimum. butions in W,

From Eqs. (7.9) and (7.10), the optimal gradient estima¬

£{W)} = piP) (7.5) tor is

Edge image free of noise Original noisy image Edges detected

FIG. 10. Experimental results of ISEF filter for ID noisy edge signals.

We see that the optimal gradient estimate is the different regions separately by use of BLI. In general, the ence between the averages of gray values of the regions adaptive gradient gives a stronger response to edge R[ and R , which is evidently not shift-invariant. And the points; i.e., it shows the advantage of being sensitive to more pixels of regions Λ, and R contained in window W, weak edges. On the other hand, because of the small the less important the estimate error. As for practical window size used for adaptive gradient calculation, it applications, to make the calculation simple, we take may be more sensitive to noise than the first derivatives window size 5 x 5 or 7 x 7 and experiments show that calculated by ISEF, especially for very noisy images. Of the result is satisfactory. In this case, window W contains course, the adaptive gradient can be improved by use of a in general only two regions separated by the zero crosslarger window, but in this case this will imply a greater ings (Fig. 9). These two regions will correspond respeccomputational complexity. tively to zones of values 1 and 0 in W of the binary band- limited Laplacian image (BLI), because the ISEF method 8. EXPERIMENTAL RESULTS detects zero crossings with a good precision [17]. So the optimal gradient estimate can be calculated by the differOur optimal edge detection filter is deduced from one- ence between the gray value averages in the original imand muitiedge models. Obviously, real images will be still age corresponding to zones 1 and 0 in window W of the more complicated than these models. To examine the BLI. This estimate is known as the adaptive gradient robustness of the models and of the theoretical results, proposed by Shen and Castan [17]. the methods of edge detection by ISEF have been imple¬

It should be noted that the adaptive gradient can be mented by the recursive algorithms presented and tested used only when the possible edge distribution in window for different types of images, including computer-generW is known. This is why we propose using it for gradient ated and real ones. The experimental results are very thresholding after the detection of zero crossings. And it satisfactory. In Fig. 10 experimental results for ID noisy is the use of the knowledge of zero crossings obtained edge signals are given. Some examples for 2D images are before the gradient calculation that makes the adaptive given in Fig. 11 and Fig. 12. To compare the performance gradient give an estimate better than that of those that do of different filtering techniques for edge detection, we not make use of this already obtained knowledge. Comfirst use the noisy images created by computer. Because pared with the shift-invariant operators, the adaptive grain the artificial image, we know exactly where we do and dient operator separately smoothes the regions of differdo not have the edges with this kind of image, it is conent gray values to remove the noise but at the same time venient and confirmatory to examine and compare the no blurring effect is produced because it processes differperformance of different techniques, such as the sensibil-

FIG. 11. Experimental results for a computer-generated noisy image, (a) Original image; (b) edges detected by ISEF filter; (c) best result for edges detected by DOG; (d) best result for edges detected by Canny-Gaussian filter; (e) best result for edges detected by simplified version of Canny by Deriche.

ity to noise, the precision of localization, and the ability with that of the well-known Gaussian-like filters. We to detect real edges. An example is given in Fig. 11. We show that the optimal filter proposed in the present paper see that with the optimal filter presented in this paper, the has a better performance in terms of insensitivity to noise result is excellent: all the edge points are detected with and precision of edge localization. no errors of localization, and there exist no false edge Because the ISEF filter is of an infinite window size points caused by noise. With Gaussian filters or the simand many edge points always exist in a real image, an plified version of Canny's filters, we have never obtained analysis based on only the mono-step edge model is not such a result, though many different parameters were sufficient. Thus we propose a noisy muitiedge model that tested. In Fig. 12 some examples for real images are is described by a stationary stochastic process with an given. The experimental results show the good perforadditive white noise. On the basis of the spectral analymance of the optimal filter proposed for edge detection, sis, we show that the optimal linear filter for muitiedge which confirms the theoretical analysis. detection is still an ISEF filter.

We propose also the recursive realization of the ISEF and its first and second derivatives, taking into account

9. CONCLUSION the computational complexity, the finite word length effect, the facility for hardware implementation, and the

An essential difficulty- with Gaussian filters for edge adaptation to parallel processing. detection is the contradiction between the insensibility to These algorithms are generalized to bidimensional and noise and the precision of edge localization. I the multidimensional cases. It is shown that the ISEF filters present paper, we deduce the optimal linear operator for and their derivatives are separable for magnitude and edge detection from the point of view of signal processmaximum value distances, the two most used distances ing. A linear edge detector is considered as a low-pass in image processing. So a multidimensional ISEF and the smoothing filter to remove the noise followed by a differderivatives, including the Laplacian, can be easily calcuential element to detect the changes. The optimal operalated from the cascade of one-dimensional ISEF filters. tor is at first deduced from the well-known monostep The use of the Laplacian exhibits the advantage of exedge model, based on a combined signal/noise ratio critetremely little computational complexity and gives closed rion adapted to edge detection, i.e., maximizing the reedge supersurfaces even in high-dimensional cases. We sponse to the step edge and minimizing those to the noise introduced also the partial Laplacian for high-dimenand to the derivative of the noise. It is shown that an sional cases, which shows the advantages of obtaining optimal low-pass filter as a preparation for edge detection closed edge surfaces in some subspaces of interest, and is the infinite symmetric exponential filter. The perforthe smoothing of the complement of the subspace sup- mance of this optimal filter is analyzed and compared presses the noise without blurring the subspace edges.

FIG. 12 Edges detected by ISEF methods Top (a) original mage, (b) edges detected by ISEF (gradient threshold 5) Bottom (a) oπgiπal image, (b) edges detected by the DRF method (gradient thresholds 5 and 8), (c) edges detected by Laplacian of Gaussian filter (gradient thresholds 5 and 8) With the algorithms presented, edges can therefore be ISEF filters, which confirms the theoretical analysis of detected by the zero crossings of the second derivatives optimization and performance. or Laplacian, or by the maxima of the gradient, always It may be interesting to explain qualitatively the optifiltered by the ISEF filter. A technique using the sign mal operator for edge detection as the following: correspondence between the first and second derivatives In a strict mathematical sense, the differentiation at a is presented to remove some false zero crossings. The pixel should concern only its very close neighbors; the edge candidates thus detected are then verified by gradicloser a pixel to the pixel of interest the more important ent thresholding with or without hysteresis. This gradient the role that it should play in differential operation. Becan be calculated from the ISEF filter or from the adapcause of. the existence of noise, an additional smoothing tive gradient if non-shift invariant operators are considwill be necessary, but we should always keep in mind the ered. Note that the adaptive gradient can be used only principle that closer pixels should play a more important after the BLI is obtained. The advantage of the adaptive role in the differentiation operation. Obviously, in examgradient is that it is more sensitive to weak edges because ining the first derivative of the symmetric exponential regions of different gray values are smoothed separately. filters, we see that this principle is always respected no

The optimal linear operator for edge detection promatter how great the smoothing factor b for removing the posed is implemented and tested for computer-generated noise (Fig. 3). So even when one adjusts the smoothing and real images and compared with some other optimal factor b to adapt it to different noisy images, this propoperators such as the Gaussian filter, Canny's filter, and erty is preserved and always gives a Dirac distribution in its simplified version by Deriche. The experimental the second derivative of the filter kernel at the center of results show a significantly better performance by the the kernel, which assures that on the one hand the more

TABLE 1 Comparison between ISEF and Canny Filter

Canny filter ISEF filter

Model Mono-step edge Mono-step edge

.Muitiedge

Type First deriv. Low pass followed by differentiation

(first and/or second order)

Criterion S/N ratio Point of view of signal processing 1. Monoedge Precision of local. Max. resp. to step edge Max. unique Min. resp. to noise

Min. resp. to noise in deriv. Max. unique

2. Muitiedge Optimal estimation Analysis method Variational calculus Variational calculus

Spectral analysis for stochastic process

Constraint Limited window size, continuous deriv. Window size noπlimited Discontinuous deriv. permitted

Problem For limited size, mono-step edge model, For infinite window size, muitiedge reasonable model analysis necessary, (never only one edge)

Optimal filter obtained fix) = Hie" sin ωx Low pass: /(.r) = cε^"°^w, first and second cos ωx deriv.

-t-iW™ cos ωx + c

Separability Nonseparable Separable for distances of four and of eight neighbors Realization Approximated by first deriv. of Gaus- Strictly optimal and realized by recursians sive algorithms (first and/or second deriv., Laplacian)

Particularity Continuous second deriv. δ distribution in second deriv. at the center

Jump in first deriv. at the center

Edge localization error 4 - (2 - < Γ)"^{1 •} σ (zero error impos- 0 sible)

SIN ratio (Shen-Castan's criterion: (2/τr)"^: < 1 1 EsHE_f, ^• £ir)^ln) TABLE 2 ters,^' this problem is avoided. It should be noted that

Comparison between Gaussian, Canny, there exist other edge detection operators of first-derivaand Deriche Filters and ISEF tive type having a Dirac distribution in its derivative at

1. Edge localization error x_c [17] the center, such as the box difference filters (average difference filters) [28]. But because first the form of this

Gaussian filter ISEF filter kind of kernel is not optimal for edge detection in complix_tG = 4 ^■ (2 ^• e ^■ τr)^ιala jr_eE = 0

Canny filter cated noisy images and second there also exist Dirac dis.ϊ_eC = 0.81/α tributions in the derivatives at the boundary of these ker¬

Deriche filter nels thatrserve to introduce a noise effect in the second x_c0 = 4 ^■ e^~]/a = 1.47/a derivatives, they show a performance worse than that of

2. Canny's signal/noise ratio at the edge point detected [20]" the ISEF filters.

Gaussian filter ISEF filter In order to show more clearly the relation and the es¬

SNRo = 2 ^■ a ^■ _e-^H»/τ^lfl SNR_E = Ha sential difference between our optimal ISEF filter for

Canny filter edge detection and the Gaussian and^' Canny filters, we list SNRc = 0.39/α ^' them in Table 1 as a summary. A comparison between the

Deriche filter Gaussian, Canny, and Deriche filters and the ISEF is SNR_D = 2 ^• (1 + 4/e)^{3 •} e-»"la = 0.64/α given in Table 2. We conclude that the optimal ISEF filter for edge detection shows a better performance in insensi¬

3. Computational complexity for one pixel in a 2D image* bility to noise, precision of edge localization, and re¬

Deriche filter ISEF filter duced computational complexity and is well adapted to Low-pass filtering Low-pass filtering

16X and 14+ 4X and 9+ parallel processing. First-derivative operator First-derivative operator 1 13X and 13+ 4x and 10+ Second-derivative operator Second-derivative operator REFERENCES

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" SNR = p( _cyjlχ fⁿ{x) dx (r,, estimation of edge position detected). 3. J. Prewitt, Object enhancement and extraction, in Picture Processk We only give a comparison between the exponential filter and the ing and Ps chopictories (B. Lipkiπ and A. Rosenfeld, Eds.), pp. Deriche filter [16], because they are implemented by recursive algo75-149, Academic Press, New York, 1970. rithms needing less computation. 4. R. Haralick, Edge and region analysis for digital image data, Corn- put. Vision Graphics Image Process, 12, 1980, 60-73.

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Claims

THE CLAIMS DEFINING THE INVENTION ARE AS FOLLOWS:

1. A system for machining an object, the system including: an obtaining means for obtaining a desired profile of the object, and information about an edge of a shaping tool; a processor for processing the desired profile of the object and the information about the edge in order to determine a path for the object and/or edge to be moved along.

2. The system as claimed in claim 1, further including an assembly for providing a relative movement between the object and the edge such that a portion of the object is removed when the object and edge come into contact with each other.

3. The system as claimed in claim 1 or 2, further including an apparatus for moving the object and/or edge along the path whilst the relative movement is being provided, to thereby effect..machining of the object into the desired profile.

4. The system as claimed in any one of the preceding claims, wherein the obtaining means includes a camera for obtaining an image of the edge, the image being the information about the edge.

5. The system as claim in claim 4, wherein the camera has a resolution of about 176nm.

6. The system as claim in any one of the preceding claims, wherein the obtaining means includes a terminal for allowing a user of the system to enter a function which defines the desired profile.

7. The system as claim in any one of the preceding claims, wherein in order to determine the path the processor: processes the information about the edge in order to determine a profile thereof; selects a number of parts along the desired profile of the object; for each one of the number of parts, selects a location on the profile of the edge which corresponds to the one of the number of parts; and superimposes the number of parts along the desired profile onto the object, wherein the path is such that when the apparatus moves the object and/or edge along the path, each one of the number of parts superimposed onto the object comes into contact with the location on the profile of the edge.

8. The system as claim in claim 7, wherein in order to determine the location on the profile of the edge, the processor determines a tangent for each one of the number of parts along the desired profile of the object, the processor selecting the location on the profile of the edge on the basis that it has a tangent that is the same as the tangent for the one of the number of parts along the desired profile.

9. The system as claim in claim 7 or 8, wherein the processor includes an image-processing device for determining the outer limits of the edge, the processor using the outer limits to create the profile of the edge.

10. The system as claim in claim 9, wherein the processor determines the outer limits using an optimal linear operator for step edge detection.

11. The system as claim in any one of the preceding claims, wherein the assembly includes a drive motor which is coupled to a member for retaining the shaping tool, the drive motor and member being such that they are capable of rotating the shaping tool in order to provide the relative movement between the object and the edge.

12. The system as claimed in claim 11, wherein the drive motor is coupled to the member via a flexible coupling and the member is mounted to an air bearing such that eccentricities are kept to an acceptable level whilst the shaping tool is being rotated.

13. The system as claim in claim 12, wherein the acceptable level of eccentricities is about lOnm.

14. The system as claim in any one of claims 11

- 13, wherein the drive motor is such that the shaping tool can be rotated at about 10,000 rpm.

15. The system as claimed in any one of claims

11 - 14, wherein the drive motor is driven by compressed air.

16. The system as claimed in any one of the preceding claims, wherein the apparatus includes a chuck for holding the object, and an actuation means coupled to the chuck, the actuation means being capable of moving the chuck in order to effect movement of the object along the path whilst being held by the chuck.

17. The system as claimed in claim 16, wherein the actuation means includes a feedback circuit capable of producing a signal that can be used by the actuation means to determine a location of the chuck.

18. The system as claimed in claim 17, wherein the feedback circuit includes a capacitive transducer.

19. The system as claimed in claim 16 or 18, wherein the actuation means is capable of providing incremental and/or translational movement of the chuck in at least x and z axes .

20. The system as claimed in any one of claims 15 - 18, wherein the actuation means is capable of moving the chuck a distance of about 200 microns

21. The system as claimed in any one of claims 15 - 19, wherein the actuation means includes a piezoelectric actuator.

22. The system as claimed in any one of the preceding claims, wherein the apparatus includes a drive motor coupled to an adjustable device capable of holding the object, the drive motor effecting rotation of the adjustable device such that the object can be rotated when held by the device.

23. The system as claimed in claim 22, wherein the device is capable of being adjusted so that the fibre can be moved into a position such that a rotational axis of the object and a rotational axis of the device coincide.

24. The system as claimed in claim 23, further including a control circuit- capable of detecting whether the rotational axis of the object and the rotational axis of the device coincide, and upon detecting that the rotational axis of the object and the rotational axis of the device do not coincide, adjusting the supports such that the rotational axis of the object and the rotational axis of the device coincide.

25. The system as claimed in claim 24, wherein the control circuit includes a camera for capturing an image of the object whilst being rotated by the apparatus, and a processor for processing the image and in order to determine whether the rotational axis of the object coincides with the rotational axis of the device, the processing also being capable of adjusting the supports.

26. The system as claimed in any one of claims 22 - 25, wherein the device includes a chuck for holding the object, a shaft coupled to the chuck, a housing in which the shaft resides, and a plurality of adjustable supports for supporting the shaft in the housing, whereby adjusting the supports allows the fibre to be moved into the position.

27. A method for machining an object, the method including the steps of: obtaining a desired profile of the object; obtaining information about an image of an edge of a shaping tool; and processing the desired profile of the object and the information about the edge in order to determine a path for the object and/or edge to be moved along.

28. The method as claimed in claim 27, further including the step of providing a relative movement between the object and the edge such that a portion of the object is removed when the object and edge come into contact with each other.

29. The method as claimed in claim 28, further including the step of moving the object and/or edge along the path whilst the relative movement is being provided, to thereby effect machining of the object into the desired profile.

30. The method as claimed in any one of claims 27 - 29, wherein the obtaining step includes using a camera to obtaining an image of the edge, the image being the information about the edge.

31. The method as claimed in any one of claims

27 - 30, wherein the obtaining step includes using a terminal to enter a function which defines the desired profile.

32. The method as claim in any one of claims 27

- 31, wherein the processing step includes: processing the information about the edge in order to determine a profile thereof; selecting a number of parts along the desired profile of the object; for each one of the number of parts, selecting a location on the profile of . the edge which corresponds to the one of the number of parts; and superimposing the number of parts along the desired profile onto the object, wherein the path is such that when the object and/or edge move along the path, each one of the number of parts superimposed onto the object comes into contact with the location on the profile of the edge.

33. The method as claim in claim 32, wherein selecting the location on the profile of the edge includes determining a tangent for each one of the number of parts along the desired profile of the object, the location on the profile of the edge being selected on the basis that it has a tangent that is the same as the tangent for the one of the number of parts along the desired profile.

34. The method as claim in claim 32 or 33, wherein processing the image of the edge includes determining the outer limits of the edge, the step of using the desired profile using outer limits to create the profile of the edge.

35. The method as claim in claim 34, wherein the outer limits are determined using an optimal linear operator for step edge detection.

36. The method as claim in any one of claims 27 - 35, further including the step of rotating the shaping tool in order to provide the relative movement between the object and the edge.

37. The method as claimed in claims 36, wherein the shaping tool is rotated at about 10,000 rp .

38. The method as claimed in any one of claims

27 - 37, further including the step of moving the object along the path.

39. The method as claimed in claim 38, wherein the object is moved incremental and/or translational in at least x and z axes.

40. The method as claimed claim 38 or 39, wherein the movement is about 200 microns

41. The method as claimed in any one of claims 27 - 40, further including the step of rotating the object.

42. A system for determining a machining path for motion of a shaping tool relative to an object, for machining the object, the system including: an obtaining means for obtaining a desired profile of the object, and information about an edge of a shaping tool; a processor for processing the desired profile of the object and the information about the edge in order to determine a path for the object and/or edge to be moved along .

43. A method for determining a machining path for motion of a shaping tool relative to an object, for machining the object, the method including the steps of: obtaining a desired profile of the object, and information about an edge of a shaping tool; processing the desired profile of the object and the information about the edge in order to determine a path for the object and/or edge to be moved along.

44. A computer program including instructions for controlling a computing device to implement the steps of claim 43.

45. A computer readable medium, providing a computer program in accordance with claim 44.

46. A micro-machining apparatus, including: a mounting device being adapted to provide mounting for a workpiece to^" be machined; and a machining head including a grinding element having a peripheral edge portion, the machine head being operatively coupled to the mounting device to permit relative movement between the machining head and the workpiece to provide contact therebetween, the grinding element being rotated and contacted with the workpiece whereby the peripheral edge portion effects machining of the workpiece in a predetermined profile.

47. A micro-machining head including a grinding element having a peripheral edge portion, the grinding element being rotatable about a rotational axis and its peripheral edge portion being configured to contact a workpiece which on rotation of the grinding element is machined in a predetermined^" profile.

48. A method of micro-machining a work-piece, the method including the steps of:

Providing a mounting device to which the workpiece is mounted, and a machining head having a grinding element and being operatively coupled to the mounting device;

Rotating the grinding element about a rotational axis; and

Providing relative movement between the machining head and the workpiece to effect contact between a peripheral edge portion of the grinding element and the workpiece to provide machining of said workpiece in a predetermined profile.