WO1997024692A1

WO1997024692A1 - Locating shapes in two-dimensional space curves

Info

Publication number: WO1997024692A1
Application number: PCT/US1996/013026
Authority: WO
Inventors: David Michael
Original assignee: Cognex Corporation
Priority date: 1995-12-31
Filing date: 1996-08-09
Publication date: 1997-07-10
Also published as: US5751853A; EP0871940A4; EP0871940A1; JP3927239B2; JP2000502824A

Abstract

A method and apparatus is provided for finding a feature of an object by finding at least one instance of a model two-dimensional space curve that represents the feature within a target two-dimensional space curve that represents at least a portion of the object. Unlike previous methods for finding a two-dimensional space curve within another two-dimensional space curve, the method of the invention is able to find any shape at any orientation or position in the two-dimensional space curve. The model (12) and target two-dimensional space curves (16) each include a plurality of points. Each point is characterized by a position in two dimensions, where the position is a function of distance S along the two-dimensional space curve. For each of the plurality of points of the two-dimensional space curve, the position in two dimensions as a function of distance S is converted into an angle as a function of distance S to provide an angle-based space curve (18). Then, at runtime, the angle-based model space curve is used to find at least one position along the angle-based target space curve that results in a match metric value that exceeds a threshold.

Description

LOCATING SHAPES IN TWO-DIMENSIONAL SPACE CURVES

FIELD OF THE INVENTION

This invention relates generally to computer vision, and relates particularly to image

analysis of two-dimensional space curves.

BACKGROUND OF THE INVENTION

In computer vision, images are sometimes obtained that contain two-dimensional space

curves. For example, when an image of an object is analyzed using an edge detector, the

boundary of the object can be represented by a series of linked edge points that together form a

two-dimensional space curve. The object can be, for example, a semiconductor wafer.

Other ways to produce a space curve given an image of an object include using a boundary

tracker based on a setable threshold, such as the method disclosed in co-pending patent

application Serial No. 08/458,908, filed 6/2/95, or a boundary tracker that follows the "peak" of

an edge, such as the method disclosed in copending patent application with Attorney Docket No.

C95-043, filed 12/6/95, both assigned to Cognex Coφoration.

A space curve can also be generated by a position sensor that detects a stream of two-

dimensional space points and produces a time-parameterized signal. For example, the curve

could represent the two-dimensional position over time of a truck on a road. In this case, a particular sub-curve to be detected might be a particular traffic pattern. Alternatively, if a

thermometer or seismograph is used as the sensor, the curve can represent the temperature at a

location as a function of time, or the excursion of a seismograph's indicator as a function of time,

respectively.

In a space curve that represents the boundary of an object, a particular shape in that

space curve may correspond to a feature of the object. For example, a concave angular curve

segment (two legs of a triangle) found in the boundary of a semiconductor wafer may represent

the notch that is typically found in the perimeter of a semiconductor wafer.

It is sometimes desirable to find one or more instances of a particular curve segment in a

larger two-dimensional space curve. There are many approaches to finding two-dimensional

shapes in two-dimensional images, but there are no known general techniques for finding rotation-

invariant and translation- invariant curve segments of arbitrary shape in two-dimensional space-

curves.

Presently, applicant is aware of a method for finding only rectangular curve segments

within a larger two-dimensional space curve. For example, if the curve segment is a rectangle

made up of straight lines, each straight line will be characterized by one of four different angles.

In this case, an angle histogram is used to find the angular position of the rectangular curve

segment, and consequently find the angular orientation of an object having the rectangular curve

segment as a boundary. This technique is described in co-pending patent application Serial No.

08/330,223, filed 10/26/94, assigned to Cognex Coφoration, which is a continuation of

application Serial No. 07/979,848, filed 11/23/92, now abandoned. If there were a general technique for finding space curve segments of arbitrary shape in a

two-dimensional space curve, where the space curve represents the boundary of an object, it

would be possible to find the position and/or orientation of a feature of an object by finding the

particular segment of the space curve that corresponds to the feature. For example, if the space

curve segment that represents the notch of a semiconductor wafer.(two legs of a triangle) could be

found in the boundary of the image of a semiconductor wafer, the angular orientation of the

notch, and therefore the angular orientation of the wafer, could be found.

PROBLEMS TO BE SOLVED BY THE INVENTION

It is a goal of the invention to provide a method and apparatus for finding instances of a

shape in a space curve.

It is a further goal of the invention to provide a method and apparatus for finding a two-

dimensional space curve of any shape within another larger two-dimensional space curve.

It is a further goal of the present invention to provide a method and apparatus that can

find a model two-dimensional space curve within another larger target two-dimensional space

curve, regardless of the orientation, translation, or scaling of the model two-dimensional space curve.

It is another goal of the invention to provide a method and apparatus for finding instances

of a shape in a space curve that is not restricted to finding rectangular-shaped space curves. It is a further goal of the invention to provide a method and apparatus for finding

instances of a shape of a model space curve in a larger target space curve that exploits the entire

model space curve to be located.

It is an additional goal of the invention to provide a method and apparatus for finding

instances of a shape of a model space curve in a target space curve that does not require feature

detection to be performed on the model space curve to be located.

SUMMARY OF THE INVENTION

A method and apparatus is provided for finding a feature of an object by finding at least

one instance of a model two-dimensional space curve that represents the feature within a target

two-dimensional space curve that represents at least a portion of the object. The method

includes the steps of, at train-time, acquiring the model two-dimensional space curve, where the

model two-dimensional space curve includes a plurality of points. Each point is characterized by

a position in two dimensions, where the position is a function of distance S along the model two-

dimensional space curve. Then, for each of the plurality of points of the model two-dimensional

space curve, the position in two dimensions as a function of distance S is converted into an angle

as a function of distance S to provide an angle-based model space curve.

Then, at run-time, the target two-dimensional space curve is acquired. The target two-

dimensional space curve includes a plurality of points, each point being characterized by a

position in two dimensions, where the position is a function of distance S along the target two-

dimensional space curve. Next, for each of the plurality of points of the target two-dimensional space curve, the position in two dimensions as a function of distance S is converted into an angle

as a function of distance S to provide an angle-based target space curve. Then, the angle-based

model space curve is used to find at least one position along the angle-based target space curve

that results in a match metric value that exceeds a match threshold.

It is preferred to, after the step of converting the model two-dimensional space curve into

the angle-based model space curve, reduce the dynamic range of, and/or remove discontinuities in

the angle-based model space curve. It is also preferred to, after the step of converting the target

two-dimensional space curve into the angle-based target space curve, reduce the dynamic range

of, and/or remove discontinuities in the angle-based target space curve.

To provide invariance with respect to rotation, translation, and scaling in the angle-based

space curves, it is preferred to use normalized correlation as the match metric. Alternatively, if

rotation invariance is not required for the particular application, a match metric such as sum of

absolute differences or correlation can be used.

The method and apparatus of the invention can be used to find the notch or flat on the

perimeter of a semiconductor wafer. For example, the model space curve that represents the

notch can be synthetically generated based on a mathematical description of the notch or flat.

Alternatively, a composite model two-dimensional space curve can be statistically compiled from an ensemble of exemplar two-dimensional space curves.

Both the model space curve and the target space curve can be extracted from images of an

object to be analyzed using either edge detection on an image that represents at least a portion of

the object, or boundary tracking on an image that represents at least a portion of the object. 1 The technique for converting the position in two dimensions as a function of distance S

2 into an angle as a function of distance S involves performing a first derivative operation with

3 respect to the distance S upon each dimension of the position in two dimensions as a function of

4 distance S to provide a quotient of first derivatives, the arctangent of which provides

5 the angle at the distance S along the space curve. Advantageously, a change in a first dimension

6 per unit change in S and a change in a second dimension per unit change in S can be used as

7 indices into a look-up table having an array of angle values to provide an angle value at each

8 particular distance S along the space curve.

9 Unlike previous methods for finding a two-dimensional space curve within another two-

I 0 dimensional space curve, the method of the invention is able to find any shape at any orientation

I I or position in the two-dimensional space curve. In addition, the method is invariant to scale

1 2 changes in the angle representation of the two-dimensional space curve. The method of the

1 3 invention is not restricted to finding rectangular- shaped space curves. The method of the

1 4 invention exploits the entire space curve to be located, and does not require feature detection

1 5 within the space curve to be located.

1 6

1 7

1 8 BRIEF DESCRIPTION OF THE DRAWINGS

1 9 The invention will be more fully understood from the following detailed description, in

20 conjunction with the accompanying figures, wherein: Fig. 1 shows a target space curve and a model space curve that can be found in the target

space curve; Fig. 2 shows the model space curve of Fig. 1, next to a plot of angle as a function of the

distance along the model space curve, as well as a rotated instance of the model space curve of

Fig. 1 , next to a plot of angle as a function of the distance along the rotated model space curve;

Fig. 3 shows the model space curve of Fig. 1, next to a plot of angle as a function of the

distance along the model space curve, as well as a scaled instance of the model space curve of Fig.

1, next to a plot of angle as a function of the distance along the scaled model space curve;

Fig. 4A is a flow chart of the train-time phase of the method of the invention; and

Fig. 4B is a flow chart of the run-time phase of the method of the invention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The invention provides a method for finding one or more instances of a model two-

dimensional space curve of arbitrary shape, rotation, and translation, anywhere in a target rwo-

dimensional space curve.

The method includes a training phase to generate a model two-dimensional space curve,

and a run-time phase that uses the model two-dimensional space curve to find its position

somewhere along a target two-dimensional space curve. During the training phase, the pattern to

be located, in the form of a model two-dimensional space curve, is converted from a two-

dimensional space curve, where the two dimensions specify the (X and Y) position of each point

disposed at a distance S along the space curve, into a one-dimensional space curve, where the single dimension is the angle of the tangent to the space curve at each point disposed at a distance

S along the space curve. In other words, the pattern to be located is converted from a space curve

which is expressed as position in a two-dimensional coordinate system as a function of distance S

along the space curve, into a space curve which is expressed as an angle as a function of distance

S along the space curve.

Any discontinuities in the angle representation of the space curve are removed using

properties of geometry and/or by using more than one copy of the model or target space curve, as

will be explained below.

At run-time, the space curve to be searched, i.e., the target space curve, is transformed

from the two-dimensional position representation to the one-dimensional angle representation, as

was the model space curve at train-time. Next, a search is performed for one or more positions S of the model space curve in the angle representation along the target space curve in the angle

representation where a match metric attains a value that exceeds a preselected match threshold.

In a preferred embodiment, the match metric is the normalized correlation between the

two space curves in the angle representation. Other match metrics can be used, such as

correlation or sum of absolute differences, but these metrics will have different properties with

respect to rotation, translation, and scaling of the space curves. For example, the sum of absolute

differences metric would not behave the same as normalized correlation with respect to rotation

invariance. As a final result, the one or more positions S where the match metric attains a value that

exceeds a preselected match threshold are the found locations of the model space curve in the

target space curve.

We first discuss some properties of two-dimensional space curves. Referring to Fig. 1, a

target space curve 10 and a model space curve 12 are shown. The target space curve 10 includes

one instance 14 of the model space curve 12, although that instance 14 is rotated about 45° and

translated with respect to the model space curve 12, as shown in Fig. 1. Nevertheless, the

rotated and translated instance 14 of the model space curve 12 can be found along the target space

curve 10 using a particular embodiment of the method of the invention, e.g., the embodiment that

employs a normalized correlation match metric, to be discussed below.

Fig. 2 shows a second instance 16 of the model space curve 12, next to a plot 18 of angle

'A' as a function of the distance 'S' along the curve 16, as well as a third instance 20 of the model

space curve 12, next to a plot 22 of angle 'A' as a function of the distance 'S' along the curve 20.

Note that the third instance 20 is just the second instance 16 rotated by 90°. Also note that the

plot 22 is just the plot 18, but shifted down by a constant value. Thus, rotating a space curve

has the effect of shifting along the A axis the plot of angle A versus distance S.

Fig. 3 shows the second instance 16 of the model space curve 12, next to the plot 18 of

angle 'A' as a function of the distance 'S' along the curve 16, as well as a fourth instance 24 of

the model space curve 12, next to a plot 26 of angle 'A' as a function of the distance 'S' along the

curve 20. Note that the fourth instance 24 is just the second instance 16 compressed by about

30%. Also note that the plot 26 is just the plot 18, but expansively scaled by a constant value. Thus, compressing a space curve has the effect of expansively scaling along the A axis the plot of

angle A versus distance S.

It should be noted that match metrics with the properties of normalized correlation will

find a substantially perfect match between plots 18 and 22, 22 and 26, and 18 and 26. That is

because match metrics like normalized correlation are not sensitive to linear transformations such

as scaling of a plot, as in the plot 26 with respect to plot 18, and/or adding a constant to a plot,

as in the plot 22 with respect to plot 18. This is why, when the method of the invention

employs a match metric like normalized correlation, the method is invariant with respect to

rotation, translation, and a type of scaling of either the model or the target angle-based space

curve.

Referring to Fig. 4A, in the first step (30) of the training phase of the method of the

invention, a two-dimensional model space curve is obtained, where the space curve is described

by the two dimensions X(S) and Y(S). Here, X(S) represents the x-coordinate each point on the

space curve, Y(S) represents the y-coordinate of each point on the space curve, and S is a

parameter that represents a corresponding distance along the space curve for each point on the

space curve represented by the X(S), Y(S) coordinate pair.

As explained in the Background section above, using an edge detector to analyze an image

of an object can provide the boundary of the object as represented by a series of linked edge

points X(S), Y(S) that together form a two-dimensional model space curve. The object can be,

for example, the notch of a semiconductor wafer. The edge points can be linked by iteratively

testing each edge point as to whether it has exactly one neighbor that is already on the space curve, and then marking it as being on the space curve too. Alternatively, the edge points can be

linked by recursively testing all of the neighbors of the last edge point added to the space curve.

tracker based on a setable threshold, such as the Cognex Boundary Tracker, disclosed in co-

pending patent application Serial No. 08/458,908, filed 6/2/95, or a boundary tracker that follows

the "peak" of an edge, such as the method disclosed in copending patent application with

Attorney Docket No. C95-043, filed 12/6/95, both assigned to Cognex Coφoration.

An additional way to produce a model space curve is to synthetically generate a model

two-dimensional space curve based on a mathematical specification of the space curve X(S),

Y(S). For example, if a triangular shape model space curve is desired, the equation for a particular

triangle can be used to generate the particular corresponding two-dimensional space curve that is

represented by X(S), Y(S).

Yet another way to produce a model space curve is to statistically compile a composite

two-dimensional model space curve using an ensemble of exemplar two-dimensional space

curves.

Next, the two-dimensional model space curve X(S), Y(S) is converted (32) into a one-

dimensional, angle-based space curve A(S). Note that the curve A(S) is still parameterized by the

distance S along the curve. To perform the conversion, essentially, the method of the invention

finds the angle of a tangent to the space curve X(S), Y(S) at each point S along the space curve,

thereby providing the space curve A(S). Thus, first a tangent to the space curve X(S), Y(S) at

each point S along the space curve must be computed, and then the angle of the tangent at each point S must be determined. In a preferred embodiment that employs a look-up table, to be

discussed below, the steps of finding a tangent and determining its angle can be combined into a

single table look-up operation. To find the tangent to the space curve X(S), Y(S), a quotient of partial first derivatives is

computed at each point S along the curve. This is accomplished by taking the partial derivative

of X(S) with respect to S at each point S along the curve, and dividing the partial derivative of

X(S) with respect to S by the partial derivative of Y(S) with respect to S at each point S along

the curve.

To determine the angle of the tangent to the space curve X(S), Y(S) at each point S, the

arctangent of the quotient of partial first derivatives as computed above is determined. This is

best accomplished using table look-up.

One way to accomplish the steps of finding the tangent to the curve, and the angle of the

tangent for a plurality of points along the curve parameterized by S, is to apply the Cognex Angle Finder to the curve. The Cognex Angle Finder examines consecutive points along a space

curve, perhaps separated by several points, and finds the angle for that curve segment by table

look-up. In the look-up table, the partial derivatives explained above are approximated as first

differences, i.e., a first difference in X ("delta X", i.e., the change in X per unit change in S), and a

first difference in Y ("delta Y", i.e., the change in Y per unit change in S). Here, delta X and delta

Y serve as indices into the look-up table, where for each possible combination of delta X and

delta Y, there is a value of the arctangent of the quotient of that particular delta X and delta Y stored at the location indexed by that delta X and delta Y. The value of the arctangent is the angle

of the tangent represented by the quotient of delta X and delta Y.

Discontinuities may occur along the space curve A(S) in its angle representation. This

can occur anywhere the space curve A(S) changes direction abruptly, such as at a comer. For

example, a left-hand turn of 90 degrees would result in a 90 degree discontinuity in the space

curve A(S). Furthermore, this discontinuity is essentially identical to a right-hand rum of 270

degrees which would be represented as a -270 degree discontinuity in the space curve A(S).

Thus, the two discontinuities appear to be different, when in fact they are equivalent. This can

result in failure to find a true instance of the model space curve in the target space curve.

One way to make the discontinuities equal is to add 360 degrees to each negative

discontinuity encountered, since this is equivalent to adding 0 degrees. Altematively, the rule can

be to always reduce the largest absolute discontinuity by either adding or subtracting 360 degrees,

whichever reduces the absolute magnitude of the largest absolute discontinuity. Thus, in the

above example, -270 would be reduced by adding 360 degrees to obtain 90 degrees. Regardless of

which rule is applied, the number of apparently unequal discontinuities is reduced (34), and the

resulting angle-based space curve is then better suited for use as a model space curve.

When normalized correlation is the match metric used, the dynamic range of the model space curve can be reduced, and angular offset of the model space curve can be adjusted by the

following technique. To reduce the dynamic range of the model space curve, the entire angle-

based version of the model space curve is multiplied by a constant angle value. Then, to ensure

that all the angle values of the angle-based model space curve are positive, another constant angle value is added to the entire angle-based version of the model space curve. The result is a model

space curve with a small positive dynamic range. In addition, for some applications, it is

advantageous to use the above technique to provide a target space curve with a small positive

dynamic range It is possible to adapt the present invention for finding an instance of a model space curve

in a target space curve to a multi-scale approach. To accomplish this, a multi-scale version of the

model space curve can be generated. The simplest multiscale representation can be generated by

low-pass filtering the model space curve, and then sub-sampling it. Each scale of the multi-scale representation has associated with it a model space curve that is about one fourth as long if the

sub-sampling and low-pass filtering were one octave in the X and Y dimensions.

Referring to Fig. 4B, in the first step (36) of the run-time phase of the method of the invention, a two-dimensional target space curve is obtained, where the target space curve is

described by the two dimensions X(S) and Y(S). The target space curve is the space curve within

which one or more instances of the model space curve is sought. Here, as before in the case of

the model space curve, X(S) represents the x-coordinate each point on the target space curve,

Y(S) represents the y-coordinate of each point on the target space curve, and S is a parameter that represents a corresponding distance along the target space curve for each point on the target

space curve represented by the X(S), Y(S) coordinate pair.

points X(S), Y(S) that together form a two-dimensional target space curve. The object can be, for example, a semiconductor wafer. The edge points can be linked by iteratively testing each edge

point as to whether it has exactly one neighbor that is already on the space curve, and then

marking it as being on the space curve too. Altematively, the edge points can be linked by

recursively testing all of the neighbors of the last edge point added to the target space curve.

Other ways to produce a target space curve given an image of an object include using a

boundary tracker based on a setable threshold, such as the Cognex Boundary Tracker, disclosed

in co-pending patent application Serial No. 08/458,908, filed 6/2/95, or a boundary tracker that

follows the "peak" of an edge, such as the method disclosed in copending patent application with

Attorney Docket No. C95-043, filed 12/6/95, both assigned to Cognex Corporation.

Next, the two-dimensional target space curve X(S), Y(S) is converted (38) into a one-

dimensional, angle-based target space curve A(S) using the same method as describe above in step

(32).

Just as in the case of the model space curve A(S), discontinuities may occur along the

target space curve A(S) in its angle representation, and must somehow be compensated (40). In

particular, it is important to attempt to remove discontinuities in places along the space curve

where the angle along the target space curve A(S) crosses 0 degrees. If such discontinuities

cannot be removed, then multiple versions of the target space curve A(S) must be generated

where each version has the discontinuity at a different location. Then, each version of the target

space curve A(S), A'(S), etc, must be searched with the model space curve.

Creating multiple versions of the target space curve can also advantageously reduce the

dynamic range of the angle of the target space curve. Unlike the case of the model space curve A(S), the value of the angle can increase to an extent that can begin to exceed the capacity of fixed

point arithmetic hardware and software. For example, if all of the turns of the target space curve

are +90 degrees, then the value of the angle will increase with each turn. If -360 degrees is added

to the angle at some positions along the target space curve A(S), the dynamic range of the angle value can be limited. Limiting the dynamic range of the angle value in this way can provide

greater accuracy when fixed point arithmetic hardware and software are used.

Next, the target space curve A(S) is searched for instances of the model space curve A(S)

(42). A match metric is calculated at each candidate position of the model space curve A(S) along

the target space curve A(S) to produce a sequence of match metric values. The candidate

position(s) resulting in a match metric value that exceeds an empirically determined threshold

indicate the existence of one or more instances of the shape of the model space curve in the target space curve. The setting of the threshold depends on the amount of noise in the model space

curve and the target space curve, and on the discrimination capability of the model. If no

candidate position of the model space curve results in a match metric value that exceeds the

threshold, then report that no instance of the model space curve was found. The exact position of each instance of the shape of the model space curve in the target

space curve is best determined using inteφolation, such as parabolic inteφolation. Altematively,

the position of the greatest match metric value for a particular instance of the model space curve

in the target space curve can represent the position of the shape of the model space curve in the

target space curve. 1 The preferred match metric is normalized correlation, although other match metrics can be

2 used, such as sum of absolute differences and correlation. Normalized correlation is preferred

3 when translation invariance, rotation invariance , and scale invariance in the angle-based space

4 curve is desired. Normalized correlation allows angle-based space curves to be matched that are

5 related by a linear transformation, i.e., some combination of scaling and offset. For example, if

6 one angle-based space curve is twice as large as another (scaling), or is shifted up in angle by 100

7 units (offset), the matching properties of the scaled and/or offset angle-based space curves are

8 unaffected. See Figs. 2 and 3 to observe how offset and scaling of an angle-based space curve

9 correspond to rotation and scaling of the corresponding position-based space curves.

I 0 Normalized correlation can be advantageously accelerated by using the Cognex VC-l/VC-

I I 3 integrated circuitry, or using the method for performing fast normalized correlation disclosed in

1 2 copending patent application Serial No. 08/203,812, filed 3/1/94, assigned to Cognex

1 3 Coφoration.

1 4 Using the multi-scale model space curve described above, it may be desirable to use a

1 5 multi-scale approach to searching for instances of the model space curve in the target space curve.

1 6 Other modifications and implementations will occur to those skilled in the art without

1 7 departing from the spirit and the scope of the invention as claimed. Accordingly, the above

1 8 description is not intended to limit the invention except as indicated in the following claims.

Claims

1 CLAIMS

2

1 What is claimed is:

1 1. A method for finding a feature of an object by finding at least one instance of a model two-

2 dimensional space curve that represents the feature within a target two-dimensional space curve

3 in an image that represents at least a portion of the object, the method comprising the steps of:

4 at train-time:

5 acquiring said model two-dimensional space curve, said model two-dimensional space

6 curve including a plurality of points, each point being characterized by a position in two

7 dimensions, said position being a function of distance S along said model two-dimensional space

8 curve;

9 for each of said plurality of points of said model two-dimensional space curve, converting

I 0 said position in two dimensions as a function of distance S into an angle as a function of distance

I I S to provide an angle-based model space curve; and

1 2 at run-time:

1 3 acquiring said target two-dimensional space curve, said target two-dimensional space

1 4 curve including a plurality of points, each point being characterized by a position in two

1 5 dimensions, said position being a function of distance S along said target two-dimensional space

1 6 curve; for each of said plurality of points of said target two-dimensional space curve, converting

said position in two dimensions as a function of distance S into an angle as a function of distance

S to provide an angle-based target space curve; and

using said angle-based model space curve to find at least one position along said angle-

based target space curve that results in a match metric value that exceeds a match threshold.

2. The method of claim 1, further including the step of:

after the step of converting said model two-dimensional space curve into said angle-based

model space curve, reducing the dynamic range of said angle-based model space curve.

3. The method of claim 1, further including the step of:

after the step of converting said target two-dimensional space curve into said angle-based

target space curve, reducing the dynamic range of said angle-based target space curve.

4. The method of claim 1, wherein said match metric is normalized correlation.

5. The method of claim 1, wherein said match metric is sum of absolute differences.

6. The method of claim 1, wherein said match metric is correlation.

7. The method of claim 1, wherein said object is a semiconductor wafer, and said feature is a

notch in the perimeter of said semiconductor wafer.

8. The method of claim 1, wherein the step of acquiring said model two-dimensional space curve

includes the step of:

synthetically generating a model two-dimensional space curve.

9. The method of claim 1 , wherein the step of acquiring said model two-dimensional space curve

includes the step of: statistically compiling a composite model two-dimensional space curve using an ensemble

of exemplar two-dimensional space curves.

10. The method of claim 1 , wherein the step of acquiring said model two-dimensional space

curve includes the step of:

performing edge detection on an image that represents at least a portion of said object to provide a two-dimensional space curve.

11. The method of claim 1, wherein the step of acquiring said model two-dimensional space

curve includes the step of:

performing boundary tracking on an image that represents at least a portion of said object

to provide a two-dimensional space curve.

12. The method of claim 1, wherein the step of acquiring said target two-dimensional space curve

includes the step of:

performing edge detection on an image that represents at least a portion of said object to

provide a two-dimensional space curve.

13. The method of claim 1, wherein the step of acquiring said target two-dimensional space curve

includes the step of:

to provide a two-dimensional space curve.

14. The method of claim 1, wherein each step of converting said position in two dimensions as a

function of distance S into an angle as a function of distance S includes the steps of:

performing a first derivative operation with respect to the distance S upon each dimension

of said position in two dimensions as a function of distance S to provide a pair of first

derivatives;

dividing a first member of said pair by a second member of said pair to provide a quotient;

and

providing the arctangent of said quotient as said angle at said distance S.

15. The method of claim 1 , wherein each step of converting said position in two dimensions as a function of distance S into an angle as a function of distance S includes the steps of:

for each pair of points along said two-dimensional space curve, computing a change in a first dimension and a change in a second dimension; and

using said change in said first dimension and said change in said second dimension as

indices into a look-up table having an array of angle values to provide an angle value that

corresponds to said change in said first dimension and said change in said second dimension.

16. The method of claim 2, wherein the step of removing discontinuities includes the step of:

adding 360° at a point along said angle-based model space curve.

17. The method of claim 1, further including the step of:

generating a multi-scale representation of said angle-based model and target space curves.

18. An apparatus for finding a feature of an object by finding at least one instance of a model

two-dimensional space curve that represents the feature within a target two-dimensional space curve in an image that represents at least a portion of the object, the apparatus comprising:

means for acquiring at train time said model two-dimensional space curve, said model

two-dimensional space curve including a plurality of points, each point being characterized by a

position in two dimensions, said position being a function of distance S along said model two-

dimensional space curve; first means for converting, cooperative with said means for acquiring at train time, for

each of said plurality of points of said model two-dimensional space curve, said position in two

dimensions as a function of distance S into an angle as a function of distance S to provide an

angle-based model space curve; and

means for acquiring at run time, cooperative with said first means for converting, said

target two-dimensional space curve, said target two-dimensional space curve including a plurality

of points, each point being characterized by a position in two dimensions, said position being a

function of distance S along said target two-dimensional space curve;

second means for converting, cooperative with said means for acquiring at n time, for

each of said plurality of points of said target two-dimensional space curve, said position in two

angle-based target space curve; and

means, cooperative with said first means for converting and said second means for

converting, for using said angle-based model space curve to find at least one position along said

angle-based target space curve that results in a match metric value that exceeds a match threshold.