Abstract:
A method and apparatus are provided for rapidly refining a given approximate location of a pattern to produce a more accurate location. The invention employs a multi-dimensional space that includes translation, orientation, and scale. The invention can serve as a replacement for the fine resolution phase of any coarse-fine system for pattern location. Patterns and images are represented by a feature-based description that can be translated, rotated, and scaled to arbitrary precision much faster than digital image re-sampling, and without pixel grid quantization errors. Thus, accuracy is not limited by the ability of a grid to represent small changes in position, orientation, or size (or other degrees of freedom). The invention determines an accurate object pose from an approximate starting pose in a small, fixed number of increments that is independent of the number of dimensions of the space, and independent of the distance between the starting and final poses, provided that the starting pose is within the “capture range” of the true pose. Thus, accuracy need not be sacrificed to keep execution time acceptable for practical applications. Specifying locations in four or more dimensions will often result in better matches between the pattern and image than two-dimensional location systems, thereby improving accuracy. Accuracy is not degraded if some portion of the object is missing or occluded, or if unexpected extra features are present.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
   This application is a continuation of U.S. patent application Ser. No. 08/979,588, filed Nov. 26, 1997 now abandoned. 

   FIELD OF THE INVENTION 
   This invention relates to machine vision, and particularly to systems for pattern localization in an image. 
   BACKGROUND OF THE INVENTION 
   Digital images are formed by many devices and used for many practical purposes. Devices include TV cameras operating on visible or infrared light, line-scan sensors, flying spot scanners, electron microscopes, X-ray devices including CT scanners, magnetic resonance imagers, and other devices known to those skilled in the art. Practical applications are found in industrial automation, medical diagnosis, satellite imaging for a variety of military, civilian, and scientific purposes, photographic processing, surveillance and traffic monitoring, document processing, and many others. 
   To serve these applications the images formed by the various devices are analyzed by digital devices to extract appropriate information. One form of analysis that is of considerable practical importance is determining the position, orientation, and size of patterns in an image that correspond to objects in the field of view of the imaging device. Pattern location methods are of particular importance in industrial automation, where they are used to guide robots and other automation equipment in semiconductor manufacturing, electronics assembly, pharmaceuticals, food processing, consumer goods manufacturing, and many others. 
   Another form of digital image analysis of practical importance is identifying differences between an image of an object and a stored pattern that represents the “ideal” appearance of the object. Methods for identifying these differences are generally referred to as pattern inspection methods, and are used in industrial automation for assembly, packaging, quality control, and many other purposes. 
   One early, widely-used method for pattern location and inspection is known as blob analysis. In this method, the pixels of a digital image are classified as “object” or “background” by some means, typically by comparing pixel gray-levels to a threshold. Pixels classified as object are grouped into blobs using the rule that two object pixels are part of the same blob if they are neighbors; this is known as connectivity analysis. For each such blob we determine properties such as area, perimeter, center of mass, principal moments of inertia, and principal axes of inertia. The position, orientation, and size of a blob is taken to be its center of mass, angle of first principal axis of inertia, and area, respectively. These and the other blob properties can be compared against a known ideal for proposes of inspection. 
   Blob analysis is relatively inexpensive to compute, allowing for fast operation on inexpensive hardware. It is reasonably accurate under ideal conditions, and well-suited to objects whose orientation and size are subject to change. One limitation is that accuracy can be severely degraded if some of the object is missing or occluded, or if unexpected extra features are present. Another limitation is that the values available for inspection purposes represent coarse features of the object, and cannot be used to detect fine variations. The most severe limitation, however, is that except under limited and well-controlled conditions there is in general no reliable method for classifying pixels as object or background. These limitations forced developers to seek other methods for pattern location and inspection. 
   Another method that achieved early widespread use is binary template matching. In this method a training image is used that contains an example of the pattern to be located. The subset of the training image containing the example is thresholded to produce a binary pattern and then stored in a memory. At run-time, images are presented that contain the object to be found. The stored pattern is compared with like-sized subsets of the run-time image at all or selected positions, and the position that best matches the stored pattern is considered the position of the object. Degree of match at a given position of the pattern is simply the fraction of pattern pixels that match their corresponding image pixel, thereby providing pattern inspection information. 
   Binary template matching does not depend on classifying image pixels as object or background, and so it can be applied to a much wider variety of problems than blob analysis. It also is much better able to tolerate missing or extra pattern features without severe loss of accuracy, and it is able to detect finer differences between the pattern and the object. One limitation, however, is that a binarization threshold is needed, which can be difficult to choose reliably in practice, particularly under conditions of poor signal-to-noise ratio or when illumination intensity or object contrast is subject to variation. Accuracy is typically limited to about one whole pixel due to the substantial loss of information associated with thresholding. Even more serious, however, is that binary template matching cannot measure object orientation and size. Furthermore, accuracy degrades rapidly with small variations in orientation and/or size, and if larger variations are expected the method cannot be used at all. 
   A significant improvement over binary template matching came with the advent of relatively inexpensive methods for the use of gray-level normalized correlation for pattern location and inspection. The methods are similar, except that no threshold is used so that the full range of image gray-levels are considered, and the degree of match becomes the correlation coefficient between the stored pattern and the image subset at a given position. 
   Since no binarization threshold is needed, and given the fundamental noise immunity of correlation, performance is not significantly compromised under conditions of poor signal-to-noise ratio or when illumination intensity or object contrast is subject to variation. Furthermore, since there is no loss of information due to thresholding, position accuracy down to about ¼ pixel is practical using well-known interpolation methods. The situation regarding orientation and size, however, is not much improved with respect to binary template matching. Another limitation is that in some applications, contrast can vary locally across an image of an object, resulting in poor correlation with the stored pattern, and consequent failure to correctly locate it. 
   More recently, improvements to gray-level correlation have been developed that allow it to be used in applications where significant variation in orientation and/or size is expected. In these methods, the stored pattern is rotated and/or scaled by digital image re-sampling methods before being matched against the image. By matching over a range of angles, sizes, and x-y positions, one can locate an object in the corresponding multidimensional space. Note that such methods would not work well with binary template matching, due the much more severe pixel quantization errors associated with binary images. 
   One problem with these methods is the severe computational cost, both of digital re-sampling and of searching a space with more than 2 dimensions. To manage this cost, the search methods break up the problem into two or more phases. The earliest phase uses a coarse, subsampled version of the pattern to cover the entire search space quickly and identify possible object locations in the n-dimensional space. Subsequent phases use finer versions of the pattern to refine the locations determined at earlier phases, and eliminate locations that the finer resolution reveals are not well correlated with the pattern. Note that variations of these coarse-fine methods have also been used with binary template matching and the original 2-dimensional correlation, but are even more important with the higher-dimensional search space. 
   The location accuracy of these methods is limited both by how finely the multidimensional space is searched, and by the ability of the discrete pixel grid to represent small changes in position, orientation, and scale. The fineness of the search can be chosen to suit a given application, but computational cost grows so rapidly with resolution and number of dimensions that practical applications often cannot tolerate the cost or time needed to achieve high accuracy. The limitations of the discrete pixel grid are more fundamental—no matter how finely the space is searched, for typical patterns one cannot expect position accuracy to be much better than about ¼ pixel, orientation better than a degree or so, and scale better than a percent or so. 
   A similar situation holds when gray-level pixel-grid-based methods are used for pattern inspection. Once the object has been located in the multidimensional space, pixels in the pattern can be compared to each corresponding pixel in the image to identify differences. Some differences, however, will result from the re-sampling process itself, because again the pixel grid cannot accurately represent small variations in orientation and scale. These differences are particularly severe in regions where image gray levels are changing rapidly, such as along object boundaries. Often these are the most important regions of an object to inspect. Since in general, differences due to re-sampling cannot be distinguished from those due to object defects, inspection performance is compromised. 
   Another pattern location method in common use is known as the Generalized Hough Transform (GHT). This method traces its origins to U.S. Pat. No. 3,069,654 [Hough, P. V. C., 1962], which describes a method for locating parameterized curves such as lines or conic sections. Subsequently the method was generalized to be able to locate essentially arbitrary patterns. As with the above template matching and correlation methods, the method is based on a trained pattern. Instead of using gray levels directly, however, the GHT method identifies points along object boundaries using well-known methods of edge detection. A large array of accumulators, called Hough space, is constructed, with one such accumulator for each position in the multidimensional space to be searched. Each edge point in the image corresponds to a surface of possible pattern positions in Hough space. For each such edge point, the accumulators along the corresponding surface are incremented. After all image edge points have been processed, the accumulator with the highest count is considered to be the multidimensional location of the pattern. 
   The general performance characteristics of GHT are very similar to correlation. Computational cost rises very rapidly with number of dimensions, and accuracy is limited both by fineness of the Hough space and grid quantization effects. Coarse-fine methods have been developed to improve performance of GHT, but are computationally expensive at high accuracy. The edge detection module generally eliminates problems due to local variations in object contrast, but increases susceptibility to noise. 
   SUMMARY OF THE INVENTION 
   In one general aspect, the invention is a method and apparatus for refining a given approximate location of a pattern to produce a more accurate location. This process of refinement is called localization, and occurs within a multidimensional space that can include, but is not limited to, x-y position (also called translation), orientation, and size. The localization method is fast and extremely accurate. In another general aspect, the invention is a method for identifying differences between a stored pattern and a matching image subset, where variations in pattern position, orientation, and size do not give rise to false differences. The process of identifying differences is called inspection. 
   In another general aspect, the invention is a method for determining a precise n-dimensional position of a model pattern within an object image. The method includes extracting pattern features from the model pattern to represent a pattern boundary. Then, a vector-valued function is generated using the pattern features to provide a pattern field. Also, image features are extracted from the object image. A better n-dimensional transformation is determined that provides an improved correspondence between the pattern features and the image features by using both the pattern field and an initial n-dimensional transformation that relates the image features with the pattern features. Using the better n-dimensional transformation, the precise n-dimensional position (i.e., location) of the trained image pattern within the object image is provided. 
   To avoid ambiguity we will call the location of a pattern in a multidimensional space its pose. More precisely, a pose is a coordinate transform that maps points in an image to corresponding points in a stored pattern. In a preferred embodiment, a pose is a general six degree of freedom linear coordinate transform. The six degrees of freedom can be represented by the four elements of a 2×2 matrix, plus the two elements of a vector corresponding to the two translational degrees of freedom. Alternatively and equivalently, the four non-translational degrees of freedom can be represented in other ways, such as orientation, scale, aspect ratio, and skew, or x-scale, y-scale, x-axis-angle, and y-axis-angle. 
   The invention can serve as a replacement for the fine resolution phase of any coarse-fine method for pattern location and inspection, such as the prior art methods of correlation or GHT. In combination with the coarse location phases of any such method, the invention results in an overall method for pattern location and inspection that is faster and more accurate than any known prior art method. The invention can also work with any other method for producing approximate object poses, including blob analysis, mechanical dead reckoning, and manual human positioning. 
   In a preferred embodiment, PatQuick™ tool, sold by Cognex Corporation, Natick Mass., is used for producing an approximate object pose. 
   The invention uses a stored pattern that represents an ideal example of the object to be found. The pattern can be created from a training image or synthesized from a geometric description. According to the invention, patterns and images are represented by a feature-based description that can be translated, rotated, and scaled to arbitrary precision much faster than digital image re-sampling and without pixel grid quantization errors. Thus accuracy is not limited by the ability of a grid to represent small changes in position, orientation, or size (or other degrees of freedom). Furthermore, pixel quantization errors due to digital re-sampling will not cause false differences between the pattern and image that can limit inspection performance, since no re-sampling is done. 
   Accuracy is also not limited by the fineness with which the space is searched, because the invention does not test discrete positions within the space to determine the pose with the highest degree of match. Instead the invention determines an accurate object pose from an approximate starting pose in a small, fixed number of increments that is independent of the number of dimensions of the space (i.e. degrees of freedom) and independent of the distance between the starting and final poses, as long as the starting pose is within some “capture range” of the true pose. Thus one does not need to sacrifice accuracy in order to keep execution time within the bounds allowed by practical applications. 
   Unlike prior art methods where execution time grows rapidly with number of degrees of freedom, with the method of the invention execution time grows at worst very slowly, and in some embodiments not at all. Thus one need not sacrifice degree of freedom measurements in order to keep execution time within practical bounds. Furthermore, allowing four or more degrees of freedom to be refined will often result in better matches between the pattern and image, and thereby improved accuracy. 
   The invention processes images with a feature detector to generate a description that is not tied to a pixel grid. The description is a list of elements called dipoles that represent points along object boundaries. A dipole includes the coordinates of a point along an object boundary and a direction pointing substantially normal to the boundary at that point. In a preferred embodiment, object boundaries are defined as places where image gradient (a vector describing rate and direction of gray-level change at each point in an image) reaches a local maximum. In another preferred embodiment, gradient is estimated at an adjustable spatial resolution. In another preferred embodiment, the dipole direction is the gradient direction. In another preferred embodiment, a dipole contains additional information as further described in the drawings. In yet another preferred embodiment, dipoles are generated not from an image but from a geometric description of an object, such as might be found in a CAD system. 
   The stored model pattern to be used by the invention for localization and inspection is the basis for generating a dipole list that describes the objects to be found by representing object boundaries. The dipole list derived from the model pattern is called the field dipole list. It can be generated from a model training image containing an example object using a feature detector, or it can be synthesized from a geometric description. The field dipole list is used to generate a 2-dimensional vector-valued function called afield. For each point within the region of the stored model pattern, the field gives a vector that indicates the distance and direction to the nearest point along a model object boundary. The vector is called the force at the specified point within the stored model pattern. 
   Note that the nearest point along a model object boundary is not necessarily one of the model object boundary points represented by the field dipoles, but in general may lie between field dipole positions. Note further that the point within the stored model pattern is not necessarily an integer grid position, but is in general a real-valued position, known to within the limits of precision of the apparatus used to perform the calculations. Note that since the force vector points to the nearest boundary point, it must be normal to the boundary (except at discontinuities). 
   In a preferred embodiment, if no model object boundary point lies within a certain range of a field position, then a special code is given instead of a force vector. In another preferred embodiment, the identity of the nearest field dipole is given in addition to the force. In another preferred embodiment, one additional bit of information is given that indicates whether the gradient direction at the boundary pointed to by the force is the same or 180° opposite from the force direction (both are normal to the boundary). In another preferred embodiment, additional information is given as further described in the drawings. In another embodiment, the field takes a direction in addition to a position within the pattern, and the force returned is the distance and direction to the nearest model object boundary point in approximately the given direction. 
   The stored model pattern used by the invention includes the field dipole list, the field, and a set of operating parameters as appropriate to a given embodiment, and further described throughout the specification. 
   Given an object image and an approximate starting pose, pattern localization proceeds as follows. The object image is processed by a feature detector to produce a dipole list, called the image dipole list. The starting pose is refined in a sequence of incremental improvements called attraction steps. Each such step results in a significantly more accurate pose in all of the degrees of freedom that are allowed to vary. The sequence can be terminated after a fixed number of steps, and/or when no significant change in pose results from the last step, or based on any reasonable criteria. In a preferred embodiment, the sequence is terminated after four steps. 
   For each attraction step, the image dipoles are processed in any convenient order. The position and direction of each image dipole is mapped by the current pose transformation to convert image coordinates to model pattern (field) coordinates. The field is used to determine the force at the point to which the image dipole was mapped. Since each image dipole is presumed to be located on an object boundary, and the force gives the distance and direction to the nearest model object boundary of the stored model pattern, the existence of the image dipole at the mapped position is taken as evidence that the pose should be modified so that the image dipole moves in the force direction by an amount equal to the force distance. 
   It is important to note that object boundaries generally provide position information in a direction normal to the boundary, which as noted above is the force direction, but no information in a direction along the boundary. Thus the evidence provided by an image dipole constrains a single degree of freedom only, specifically position along the line of force, and provides no evidence in the direction normal to the force. 
   If the current pose is a fair approximation to the true object position, then many image dipoles will provide good evidence as to how the pose should be modified to bring the image boundaries into maximum agreement with the boundaries of the stored model pattern. For a variety of reasons, however, many other image dipoles may provide false or misleading evidence. Thus, it is important to evaluate the evidence provided by each image dipole, and assign a weighting factor to each image dipole to indicate the relative reliability of the evidence. 
   In one embodiment, the direction (as mapped to the pattern coordinate system) of an image dipole is compared with the force direction, and the result, modulo 180°, is used to determine the weight of the image dipole. If the directions agree to within some specified parameter, the dipole is given a high weight; if they disagree beyond some other specified parameter, the dipole is given zero weight; if the direction difference falls between the two parameters, intermediate weights are assigned. 
   In another embodiment, the image dipole direction is compared to the gradient direction of the model pattern boundary to which the force points. A parameter is used to choose between making the comparison modulo 180°, in which case gradient polarity is effectively ignored, or making it modulo 360°, in which case gradient polarity is considered. In a preferred embodiment, the field itself indicates at each point within the stored model pattern whether to ignore polarity, consider polarity, or defer the decision to a global parameter. 
   In one embodiment, the force distance is used to determine the dipole weight. In a preferred embodiment, if the force distance is larger than some specified parameter, the dipole is given zero weight, on the assumption that the dipole is too far away to represent valid evidence. If the force distance is smaller than some other specified parameter, the dipole is given a high weight, and if it falls between the two parameters, intermediate weights are assigned. 
   In a preferred embodiment, the parameters specifying the weight factor as a function of force distance are adjusted for each attraction step to account for the fact that the pose is becoming more accurate, and therefore that one should expect image dipoles representing valid evidence to be closer to the pattern boundaries. 
   In one embodiment, the gradient magnitude of the image dipoles is used to determine the dipole weight. In a preferred embodiment, a combination of dipole direction, force distance, and gradient magnitude is used to determine the weight. 
   For each attraction step, the invention determines a new pose that best accounts for the evidence contributed by each image dipole, and taking into account the dipole&#39;s weight. In a preferred embodiment, a least-squares method is used to determine the new pose. 
   The evaluation of each image dipole to produce a weight can also provide information for inspection purposes. It is desirable to look for two distinct kinds of errors: missing features, which are pattern features for which no corresponding image feature can be found, and extra features, sometimes called “clutter”, which are image features that correspond to no pattern feature. In one embodiment, image dipoles with low weights are considered to be clutter. In a preferred embodiment, a specific clutter value is computed for each image dipole, as further described in the drawings below. 
   In an embodiment of the invention that can identify missing pattern features, the field at each point gives identity of the nearest field dipole, if any, in addition to the force vector. Each field dipole contains an evaluation, which is initialized to zero. Each image dipole transfers its evaluation (weight) to that of the nearest field dipole as indicated by the field. Since in general the correspondence between image and field dipoles is not one-to-one, some field dipoles may receive evaluations from more than one image dipole, and others may receive evaluations from none. Those field dipoles that receive no evaluation may represent truly missing features, or may simply represent gaps in the transfer due to quantization effects. 
   When more than one evaluation is transferred to a given field dipole, the evaluations can be combined by any reasonable means. In a preferred embodiment, the largest such evaluation is used and the others are discarded. Gaps in the transfer can be closed by considering neighboring field dipoles. In one embodiment, methods known in the art as gray-level mathematical morphology are used to close the gaps. In the case of the invention, one-dimensional versions of morphological operations are used, since field dipoles lie along one-dimensional boundaries. In a preferred embodiment, a morphological dilation operation is used. 
   If the starting pose is too far away from the true pose, there may be insufficient good evidence from the image dipoles to move the pose in the right direction. The set of starting poses that result in attraction to the true pose defines the capture range of the pattern. The capture range depends on the specific pattern in use, and determines the accuracy needed from whatever method is used to determine the starting pose. 
   In a preferred embodiment, the feature detector that is used to generate dipoles is tunable over a wide range of spatial frequencies. If the feature detector is set to detect very fine features at a relatively high resolution, the accuracy will be high but the capture range will be relatively small. If on the other hand the feature detector is set to detect coarse features at a relatively lower resolution, the accuracy will be lower but the capture range will be relatively large. This suggests a multi-resolution method where a coarse, low resolution step is followed by a fine, high resolution step. With this method, the capture range is determined by the coarse step and is relatively large, while the accuracy is determined by the fine step and is high. 

   
     BRIEF DESCRIPTION OF THE DRAWING 
     The invention will be more fully understood from the following detailed description, in conjunction with the following figures, wherein: 
       FIG. 1  is a high-level block diagram of an embodiment of the invention; 
       FIG. 1A  is an illustration of a pixel array having a 2:1 aspect ratio; 
       FIG. 2  is a block diagram of the training block of  FIG. 1 ; 
       FIG. 3  is a block diagram of the feature detection block of  FIG. 2 ; 
       FIG. 4  is a diagram showing bit assignments of a 32-bit word, and optional ‘nearest dipole’ bits; 
       FIG. 5  is an illustration of a field element array, including a border of field elements; 
       FIG. 6  illustrates field seeding, showing some of the field elements of  FIG. 5 , including a plurality of straight line segments of a pattern boundary, and the associated field dipoles; 
       FIG. 7A  illustrates field dipole connecting, showing some of the field elements of  FIG. 6 , including a plurality of straight line segments of a pattern boundary, and a plurality of associated right and left links; 
       FIGS. 7B and 7C  are diagrams illustrating the order in which neighboring field elements are examined; 
       FIG. 8  illustrates chain segmentation of  FIG. 2 , showing some of the field elements of  FIG. 5 , including a plurality of straight line segments, and a plurality of left and right links; 
       FIGS. 9A ,  9 B, and  9 C illustrate part of the analysis that is performed by the propagate phase of the field generation module of  FIG. 2 ; 
       FIG. 10  shows further details of the propagate phase of the field generation module of  FIG. 2 ; 
       FIG. 11  shows the same portion of the field array that was shown after seeding in  FIG. 6 , but with new force vectors resulting from one propagation; 
       FIG. 12  shows the same portion of the field array as  FIG. 11 , after two propagations; 
       FIG. 13  is a block diagram of the run-time module of the preferred embodiment of  FIG. 1 ; 
       FIG. 14  is a diagram illustrating a least squares method for determining a pose that best accounts for the evidence of the image dipoles at each attraction step; 
       FIG. 15  is a block diagram of the attraction module of  FIG. 13 ; 
       FIG. 16  is a block diagram of the map module of  FIG. 15 ; 
       FIG. 17  is a block diagram of the field module of  FIG. 15 ; 
       FIG. 18  is a block diagram of the rotate module of  FIG. 15 ; 
       FIGS. 19A ,  19 B, and  19 C show output as a function of input for three fuzzy logic processing elements; 
       FIG. 20A  is a schematic diagram of the portion of the evaluate module of  FIG. 15 , showing a preferred system for calculating ‘weight’ and ‘eval’; 
       FIG. 20B  is a schematic diagram of the portion of the evaluate module of  FIG. 15 , showing a preferred system for calculating ‘clutter’; 
       FIG. 21  is a schematic diagram of the sum module of  FIG. 15 ; 
       FIGS. 22A-D  are block diagram of the solve module of  FIG. 15 , showing the equations and inputs for providing ‘motion’ and ‘rms error’ for various degrees of freedom; 
       FIG. 23  is a block diagram of the equations of the compose module of  FIG. 15 ; 
       FIG. 24  is a block diagram of the equations of the Normal Tensor module of  FIG. 15 ; 
       FIG. 25  is a graphical illustration of a plurality of image dipoles and a plurality of connected field dipoles, showing field dipole evaluation; 
       FIG. 26  is a high-level block diagram of a multi-resolution embodiment of the invention; 
       FIGS. 27A and 27B  are flow diagrams illustrating the sequence of operations performed by the modules of  FIG. 15 ; and 
       FIG. 28  is a flow diagram illustrating a multi-resolution mode of operation of the invention. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   In the following figures, “modules” can be implemented as software, firmware, or hardware. Moreover, each module may include sub-modules, or “steps”, each of which can be implemented as either hardware, software, or some combination thereof.  FIG. 1  is a high-level block diagram of one embodiment of the invention. A training (model) image  100  containing an example of a pattern  105  to be used for localization and/or inspection is presented. A training module  110  analyzes the training image and produces a stored model pattern  120  for subsequent use. At least one run-time image  130  is presented, each such image containing zero or more instances of patterns  135  similar in shape, but possibly different in size and orientation, to the training (model) pattern  105 . 
   Each run-time image  130  has an associated client map  131 , chosen by a user for a particular application. A client map is a coordinate transformation that maps, i.e., associates points in an orthonormal but otherwise arbitrary coordinate system to points in the image. An orthonormal coordinate system has perpendicular axes, each axis having a unit scale. The client map provides an orthonormal reference that is necessary to properly handle the orientation degree of freedom, as well as the skew, scale, and aspect ratio degrees of freedom. In practical applications, the images themselves are almost never orthonormal, since practical image sensors almost never have perfectly square pixels. For example, pixels having an aspect ration of 2:1 are possible, as shown in FIG.  1 A. In this case, the client map would be a 2×2 matrix:
         0.5 0.0   0.0 1.0       

   If the pixels are square, the client map is the identity transform, i.e., each diagonal entry in the transform matrix is 1.0, and each off-diagonal element is 0.0. 
   Furthermore, it is sometimes useful to have a significantly non-orthonormal field. For example, a field generated from a square pattern can be used to localize and inspect a rectangular or even parallelogram-shaped instance of the pattern by using an appropriate starting pose. These cases can only be handled if an orthonormal reference is available. 
   For each run-time image, a starting pose  132  is determined by any suitable method, such as coarse gray-level correlation with orientation and size re-sampling, a coarse generalized Hough transform, or the Cognex PatQuick™ tool. The starting pose  132  is a six-degree-of-freedom coordinate transformation that maps points in the pattern  105  to approximately corresponding points  135  in the run-time image  130 . A run-time module  140  analyzes the image  130 , using the stored pattern  120 , the starting pose  132 , and the client map  131 . As a result of the analysis, the run-time module  140  produces a pose  134  that maps pattern points  105  to accurately corresponding image points  135 . 
   The run-time module  140  produces an rms error value  136  that is a measure of the degree of match between the pattern  105  and the image  130 . The rms error value  136  is the root-mean-square error from the least squares solution, or other error minimization solution, that is used to determine a pose that best fits the evidence of the image dipoles, to be described in more detail below. A value of zero represents a perfect fit, while higher values represent poorer fits. 
   The run-time module  140  produces a coverage value  138  that is a measure of the fraction of the pattern  105  to which corresponding image features have been found. The coverage value  138  is computed by summing the field dipole evaluations and dividing by the number of field dipoles, to be described in more detail below. 
   The run-time module  140  produces a clutter value that is a measure of extra features found in the image that do not correspond to pattern features. In a preferred embodiment, the clutter value is computed by summing the individual image dipole clutter values and dividing by the number of field dipoles. 
   The run-time module  140  produces an evaluated image dipole list  150  and an evaluated field dipole list  160 . The evaluated image dipole list  150  identifies features in the image  130  not present in the pattern  105 , and the evaluated field dipole list  160  identifies features in the pattern  105  not present in the image  130 . The differences between the image and pattern so identified can be used for inspection purposes. 
     FIG. 2  shows a block diagram of the training module  10  and pattern-related data  120 . A training image  100  containing an example of a pattern  105  to be localized and/or inspected is presented to training module  110 , which analyzes the training image  100  and produces a stored pattern  120  for subsequent use. The training module  110  consists of two modules, a feature detection module  200  and a field generation module  210 . Both modules use various parameters  220  to control operation as appropriate for the application. These parameters, as well as those needed for subsequent run-time modules, are pattern-dependent and therefore are collected and stored in the pattern  120  as shown. 
   The feature detection module  200  processes the training image  100 , and using the parameters  220 , to produce a field dipole set  230 , which is stored in the pattern  120  as shown. The field generation module  210  uses the field dipole set  230  and parameters  220  to produce a field  240 , stored in the pattern  120  as shown. 
   As described in the summary above, the field  240  produces information as a function of theoretically real-valued position within the region of the pattern  105 . In practice, since all the field values cannot be computed analytically and stored, a 2-dimensional array is used that stores field values at discrete points on a grid. Given a real-valued position (to some precision determined by the particular details of the embodiment), an interpolation method, such as the method shown in  FIG. 17 , is used to compute field values at intermediate points between grid elements. Since the field grid is never translated, rotated, or scaled, no re-sampling is needed and grid quantization effects are small. Instead, the image dipoles, which are not grid-based, are mapped to the fixed field coordinates in accordance with the map of FIG.  16 . 
   Thus the purpose of the field generation module  210  is to compute the elements of the 2-dimensional array that encodes the field  240 . The field generation module  210  is itself composed of many steps or submodules, as shown in FIG.  2 . Each of these steps modifies the field in some way, generally based on the results of the previous steps. Some of the steps also add information to the field dipole set  230 . The specific sequence of steps shown in  FIG. 2  corresponds to a preferred embodiment; many other variations are possible within the spirit of the invention; the essential requirement is that the stored pattern  120  be able to provide certain information as a function of position within the region of the pattern  105 . 
   In a preferred embodiment as shown in  FIG. 2 , the field generation module  210  consists of the following steps. An initialization step  250  loads predefined codes into the field array elements  520 ,  540 , shown in  FIG. 5. A  seed step  252  sets up field array elements at positions corresponding to the dipoles in the field dipole set  230 . A connect step  254  uses the seeded field array to identify neighboring dipoles for each field dipole. Once identified, the field dipoles are connected to neighboring ones to form chains by storing the identity of left and right neighbors along pattern boundaries, if any. A chain step  256  scans the connected field dipoles to identify and catalog discrete chains. For each chain, the starting and ending dipoles, length, total gradient magnitude, and whether the chain is open or closed is determined and stored. 
   A filter step  258  removes weak chains from the pattern by removing the dipoles they contain from the field array (i.e. reversing the seeding step  252  for those dipoles). A variety of criteria can be used to identify weak chains. In a preferred embodiment, chains whose total gradient magnitude or average gradient magnitude are below some specified parameter are considered weak. 
   A segment step  260  divides chains into segments of low curvature, separated by zones of high curvature called corners. Corner dipoles are marked in the field array for use as described in subsequent figures. Curvature can be determined by a variety of methods; in a preferred embodiment, a dipole is considered a corner if its direction differs from that of either neighbor by more than some specified parameter, e.g., as further described in conjunction with FIG.  8 . 
   A sequence of zero or more propagate steps  262  extend the field out from the seeded positions. The result is that force vectors pointing to pattern boundaries, as well as other information needed by the run-time steps, can be obtained at some distance from the boundaries. The number of propagate steps  262  is controlled by a parameter and determines the distance from pattern boundaries that the field will contain valid force vectors, as well as the computation time needed for pattern training. In a preferred embodiment, four propagation steps are used. Field elements beyond the range of propagation will contain the code set during the initialization step  250 . 
     FIG. 3  shows a preferred embodiment of a feature detector to be used for practice of the invention. The feature detector processes a source image  300 , which can be either a training image or a run-time image. A low-pass filter  310  and image sub-sampler  320  are used to attenuate fine detail in the source image that for a variety of reasons we wish to ignore. For example we may wish to attenuate noise or fine texture, or we may wish to expand the capture range of the pattern by focusing on coarse pattern features. Also, we may wish to decrease the number of image dipoles, thereby reducing processing time. 
   The response of the filter  310  and sub-sampler  320  are controlled by parameters  220  stored in the pattern (not shown in this figure). One setting of the parameters effectively disables the filter and sub-sampler, allowing the source image  300  to pass without modification for maximum resolution. 
   Methods for low-pass filtering and sub-sampling of digital images are well known in the art. In a preferred embodiment, a constant-time second-order approximately Gaussian filter is used, as described in [U.S. patent pending “Efficient, Flexible Digital Filtering”]. 
   The filtered, sub-sampled image is processed by a gradient estimation module  330  to produce an estimate of the x (horizontal) and y (vertical) components of image gradient at each pixel. A Cartesian-to-polar conversion module  340  converts the x and y components of gradient to magnitude and direction. A peak detection module  350  identifies points where the gradient magnitude exceeds a noise threshold and is a local maximum along a 1-dimensional profile that lies in approximately the gradient direction, and produces a list of the grid coordinates (row and column number), gradient magnitude, and gradient direction for each such point. 
   A sub-pixel interpolation module  360  interpolates the position of maximum gradient magnitude along said 1-dimensional profile to determine real-valued (to some precision) coordinates (x i , y i ) of the point. The result is a list of points that lie along boundaries in the image, which includes the coordinates, direction, and magnitude of each point. This list can be used as the basis for either a field or image dipole list, to which additional information may be added as appropriate. 
   Methods for identifying points along image boundaries are well-known in the art. Any such method can be used for practice of the invention, whether based on gradient estimation or other techniques. Methods for gradient estimation, Cartesian-to-polar conversion, peak detection, and interpolation are also well-known. In a preferred embodiment, the methods described in [U.S. patent pending “Method and Apparatus for Fast, Inexpensive, Subpixel Edge Detection”] are used. 
   In a preferred embodiment, the source image has eight bits of gray-scale per pixel. The low-pass filter produces a 16-bit image, taking advantage of the inherent noise-reduction properties of a low-pass filter. The gradient estimation module uses the well-known Sobel kernels and operates on either a 16-bit filtered image, if the parameters are set so as to enable the filter  310 , or an 8-bit unfiltered image if the parameters are set so as to disable the filter  310 . The x and y components of gradient are always calculated to 16 bits to avoid loss of precision, and the gradient magnitude and direction are calculated to at least six bits preferably using the well-known CORDIC algorithm. 
   Several parameter values are needed for feature extraction, both in the training module  110  and in the run-time module  140 . Generally these parameters include those controlling the response of the low-pass filter  310 , the sub-sampling amount used by sub-sampler  320 , and the noise threshold used by peak detector  350 . Other values may be needed depending on the exact details of the feature extractor used to practice the invention. 
   Appropriate settings for the parameter values depend on the nature of the patterns and images to be analyzed. In a preferred embodiment certain defaults are used that work well in many cases, but in general no rules can be given that work well in all cases. Said preferred embodiment is further described below in conjunction with FIG.  26 . 
     FIG. 4  shows an element of the field array as used in a preferred embodiment of the invention. Information is packed into a 32-bit word  400 , both to conserve memory and to speed up access on conventional computers by maximizing the number of elements that will fit in data cache and using a word size that keeps all elements properly aligned on appropriate address boundaries. Fixed point representations are used for the force vector, both because they are more compact than floating point representations and to allow best use to be made of the signal processing capabilities of modern processors such as the Texas Instruments TMS320C80 and the Intel Pentium-MMX. 
   In a preferred embodiment, a field element stores a force vector that gives the distance and direction to the nearest point along a pattern boundary, and one bit that specifies whether the gradient direction at that boundary point is in the same, or 180° opposite, direction as the force vector. This is accomplished as shown in  FIG. 4  by storing a signed force magnitude  410  and a gradient direction  420 . If the force direction is the same as the gradient direction, the force magnitude  410  will be positive; if the force direction is opposite from the gradient direction, the force magnitude  410  will be negative. 
   The magnitude/direction representation for the force vector is preferred over an x-y component representation because it is necessary to be able to represent vectors that have zero length but a well-defined direction. Such vectors are called pseudo-null vectors. The equivalent x-y components can be calculated by the well-known formula 
         (           force   x               force   y           )     =     magnitude   ⁡     (           cos   ⁡     (   direction   )                 sin   ⁡     (   direction   )             )           
 
   Note that gradient direction can be used in the above formula, since the stored magnitude is negative if the force direction is opposite the gradient direction. 
   In the preferred embodiment shown, the force magnitude  410  is in units of field grid increments, using a two&#39;s complement representation of 16 total bits, of which the least significant 11 are to the right of the binary point and the most significant is the sign bit. Thus the maximum force vector length is just under 16 field grid units, and the resolution is {fraction (1/2048)} th  of a grid unit. 
   The gradient direction  420  is preferably represented as a 12-bit binary angle in the range 0° to 360°, with a resolution of 360°/4096=0.088°. In other embodiments, the bits of field element  400  are divided between force  410  and direction  420  to provide greater or lesser precision and range, as needed for each particular application. 
   A 4-bit flags element  430  is also stored in the field element  400 . An 2-bit eval code  440  determines how an image dipole is to be evaluated if the current pose maps it to a field position within the region covered by this element  400 . The don&#39;t care code specifies that the image dipole should be ignored. The expect blank code specifies that no features are expected in this region of the pattern, and so if any image dipoles map here they should be given a low evaluation for inspection purposes, should be given a high clutter rating, and should not be used as evidence for localization. The evaluate only code specifies that the image dipole should be evaluated by the usual criteria for inspection purposes, but should not contribute evidence for localization purposes. The attract code specifies that the image dipole should be evaluated and used both for localization and inspection. 
   If the eval code  440  is either “don&#39;t care” or “expect blank”, the force vector is undefined and is said to be invalid. If the eval code is “evaluate only” or “attract”, the force vector is said to be valid. 
   A 1-bit corner code  450  specifies whether or not the pattern boundary point pointed to by the force vector is in a high-curvature zone (“is corner”) or a low-curvature segment (“not corner”). If the force vector is invalid, the corner-code is set to “not corner”. 
   A 1-bit polarity code  460  specifies whether the image dipole evaluation should consider or ignore gradient direction, as described above in the summary section and further described below. A parameter is used to specify whether or not to override the polarity flags stored in the field element  400 , and if so, whether to force polarity to be considered or ignored. 
   In a preferred embodiment, the field element  400  also specifies the identity of the nearest field dipole  401  in addition to the force vector  400 . The identity  401  can be represented as a index into the field dipole list. In a preferred embodiment, a 16-bit index is used, which is stored in a separate array so as to satisfy data alignment guidelines of conventional computers. 
     FIG. 5  shows details of the initialization step  250  of the field generation step  210 . A 2-dimensional array  500  of field elements  400  is used. Any reasonable grid spacing can be used; in a preferred embodiment, the grid spacing is the same as that of the image that is input to the gradient estimation module  330  of the feature detector  200 . 
   Field elements  520 , (indicated as white in FIG.  5  and having the same structure as field element  400 ) cover the region of the training pattern  105 , together forming a “training region”. The field elements  520  are initialized so that the eval code  440  is set to “expect blank”. As described above, in this state the force vector is considered invalid and need not be initialized. In one embodiment, however, further described below in conjunction with  FIG. 20   b , the gradient direction field  420  of these field elements  520  are set equal to the corresponding gradient directions of the training image. A border of additional field elements  540 , (indicated as gray in FIG.  5  and having the same structure as field element  400 ) are initialized so that the eval code  440  is set to “don&#39;t care”. This reflects the fact that in general we don&#39;t know what features might lie beyond the bounds of the training region. These “don&#39;t care” values will be replicated inwards during each propagation step  262 , so that image features lying just outside the training pattern  105  don&#39;t attract to pattern features just inside. 
   A separate corresponding array of field dipole indices  401 , identical in size to the white-shaded field elements  520 , is also used, but need not be initialized. The values in this array are considered valid only if the force vector of the corresponding field element of array  500  is valid. 
     FIG. 6  shows details of the seed step  252  of the field generation module  210 . Shown is a subset of the field elements  520  of the field array  500 . Each field dipole is located within some field element. For example, the field dipole at point  600  falls within field element  620 , indicated as gray in FIG.  6 . Also shown is a small straight-line section of pattern boundary  660  corresponding to the example field dipole at point  600 . This section of boundary is shown primarily to aid in understanding the figure. Its orientation, and position along a line normal to its orientation, are significant, but its length is essentially arbitrary. 
   The field element  620  is set to have force vector  640 . The force vector points from the center of element  620  to a point on boundary section  660  and either in the direction, or opposite to the direction of the dipole (i.e. normal to the boundary), whichever is required to bring the head of the vector to the boundary  660 . In the example shown, the point on the boundary to which the vector points is coincident with the dipole position  600 , but in general it need not be.  FIG. 6  shows several other examples of seeded force vectors. 
   It also may happen that a field dipole&#39;s position falls exactly at the center of a field element, so that the length of the force vector is zero. In this case the force vector is pseudo-null—its direction is well-defined and must be set properly. 
   In a preferred embodiment, for each field element that receives a seed force vector, the eval code is set to “attract”, the corner code is set to “no corner”, and the polarity code is set to “consider polarity”. Other schemes may be devised within the spirit of the invention to suit specific applications. 
   For each field element that receives a seed force vector, the corresponding element  401  of the array of field dipole indices is set to identify the field dipole used to seed the field element. 
   In a preferred embodiment using the feature detector of  FIG. 3 , as further described in [U.S. patent pending “Method and Apparatus for Fast, Inexpensive, Subpixel Edge Detection”], and where the field grid has the same geometry as the image that is input to the gradient estimation module  330 , no more than one field dipole will fall within any given field element, and there will be no gaps in the boundary due to grid quantization effects. In a less preferred embodiment using different methods for feature detection, various schemes can be used to handle multiple dipoles that fall within a given field element, or gaps in the boundary due to quantization effects. The preferred method for multiple dipoles within one field element is to choose the one whose force vector is shortest, and discard the others. The preferred method for gaps in the boundary is to do nothing and let the propagation steps fill in the gaps. 
     FIG. 7  shows details of the connect step  254  of the field generation module  210 .  FIG. 7   a  shows the same subset of field elements  520  of the field array  500  as was shown in FIG.  6 . Also shown is the example field element  620 , indicated as light gray. 
   For every field dipole, the seeded field is examined to identify neighboring positions that contain dipoles to which the dipole should be connected. For the example field element  620 , the neighboring positions  700  are shown, shaded medium gray. The neighboring positions  700  are examined in two steps of four neighboring positions each, each step in a particular order, determined by the direction of the field dipole corresponding to field element  620 . 
   In one step, a left neighbor field element  710  is identified, and a left link  715  is stored in the field dipole corresponding to field element  620  identifying the field dipole corresponding to field element  710  as its left neighbor. In the other step, a right neighbor field element  720  is identified, and a right link  725  is stored to identify the field dipole&#39;s right neighbor. If a given neighbor cannot be found, a null link is stored. Note that “left” and “right” are defined arbitrarily but consistently by an imaginary observer looking along the dipole gradient direction. 
     FIG. 7   b  shows the order in which neighboring field elements are examined for a dipole whose direction falls between arrows  740  and  742 , corresponding to a pattern boundary that falls between dotted lines  744  and  746 . The sequence for identifying the left neighbor is +1, +2, +3, and +4. The first neighbor in said sequence that contains a dipole (seeded field element), if any, is the left neighbor. Similarly, the sequence for identifying the right neighbor is −1, −2, −3, and −4. 
     FIG. 7   c  shows another example, where the dipole direction falls between arrows  760  and  762 , corresponding to a pattern boundary that falls between dotted lines  764  and  766 . The sequences of neighbors are as shown. The sequences for all other dipole directions are simply rotations of the two cases of  FIGS. 7   b  and  7   c.    
   Note that the sequences given in  FIGS. 7   b  and  7   c  show a preference for orthogonal neighbors over diagonal neighbors, even when diagonal neighbors are “closer” to the direction of the pattern boundary. This preference insures that the chains will properly follow a stair-step pattern for boundaries not aligned with the grid axes. Clearly this preference is somewhat dependent on the specific details of how the feature detector chooses points along the boundary. 
   Once connections have been established for all field dipoles, a consistency check is performed. Specifically, the right neighbor of a dipole&#39;s left neighbor should be the dipole itself, and the left neighbor of a dipole&#39;s right neighbor should also be the dipole itself. If any links are found for which these conditions do not hold, the links are broken by replacing them with a null link. At the end of the connect step, only consistent chains remain. 
   Many alternate methods can be used to connect dipoles within the spirit and scope of the invention. In some embodiments, particularly where no inspection is to be performed, the connect  254  is omitted entirely. 
     FIG. 8  shows details of the segment step  260  of the field generation step  210 . Field elements  800  shaded medium gray are identified as “corners” because the dipole directions differs from that of their left and/or right neighbors by more than some specified parameter. In a preferred embodiment, the parameter is 16.875 degrees. For these elements the corner code  450  is set to “is corner”. Other field elements  820 , shaded light gray, lie along one chain segment, while field elements  840 , also shaded light gray, lie along another chain segment. For these elements it is not necessary to set the corner code, because it was set to “no corner” when the field was seeded. 
     FIG. 9  shows examples of part of the analysis that is performed by the propagate step  262  of the field generation step  210 . In the example of  FIG. 9   a , field element  900  initially does not have a valid force vector; its eval code is “expect blank”, as set by the initialization step  250 . Neighboring element  902  has a valid force vector  904 , which points to a segment of pattern boundary  906  that is assumed to be an approximately straight line. 
   A vector  908  is constructed from the center  916  of element  900  to the center of the neighbor  902 . The projection  910  of vector  908  onto force vector  904  is constructed. A new force vector  912  is constructed from the center  916  of field element  900  to the boundary  906  by adding the neighbor&#39;s force vector  904  to the projection  910 . An offset value is computed whose magnitude is equal to the length  914  of the difference between vector  908  and projection  910 , and whose sign is determined by the direction  918  by which vector  908  must be rotated to coincide with projection  910 , where anti-clockwise is positive as shown and clockwise is negative. The result of this analysis is the new force vector  912  and offset value of magnitude  914  and sign  918 . 
   A similar example but for a diagonal neighbor  932  of element  930  is shown in  FIG. 9   b . The projection  940  of vector  938  onto force vector  934  is constructed. A new force vector  942  is constructed from the center  946  of field element  930  to the boundary  936  by adding the neighbor&#39;s force vector  934  to the projection  940 . An offset value is computed whose magnitude is equal to the length  944  of the difference between vector  938  and projection  940 , and whose sign is negative since vector  938  must be rotated clockwise  948  to coincide with projection  940 . 
   Another example is shown in  FIG. 9   c , where in this case the boundary  966  passes between field element  960  and its neighbor  962 . The projection  970  of vector  968  onto force vector  964  is constructed. A new force vector  972  is constructed from the center  976  of field element  960  to the boundary  966  by adding the neighbor&#39;s force vector  964  to the projection  970 . An offset value is computed whose magnitude is equal to the length  974  of the difference between vector  968  and projection  970 , and whose sign is negative since vector  968  must be rotated clockwise  978  to coincide with projection  970 . 
   Further details of propagate step  262  of the field generation step  210  are shown in FIG.  10 . Each element of the field array is examined. Any element whose eval code  440  is “expect blank” is considered for possible propagation of the field to that element. All other field elements are already in a final state and are skipped. For each field element so considered, the eight neighbors are examined. If two or more adjacent neighbors have valid force vectors or have eval codes equal to “don&#39;t care”, the field will be propagated to the said field element; otherwise, the field element will be skipped and possibly considered again on a subsequent propagate step. 
   The rule specifying two or more adjacent neighbors is used to insure that there is sufficient information to be able to interpolate the field between neighbors. “Adjacent” means either sharing an edge, such as elements  1010  and  1012  of  FIG. 10 , or sharing a corner, such as elements  1010  and  1014 . 
   In  FIG. 10  element  1000  shaded light gray has eval code “expect blank”, and neighbors  1010 ,  1012 , and  1014 , shaded medium gray, have valid force vectors (the field has been seeded at or already propagated to the neighbors). Neighboring element  1010  has force vector  1030 , and following the method of  FIG. 9  new force vector  1032  and positive offset  1034  are computed. Neighboring element  1012  has force vector  1040 , and following the method of  FIG. 9  new force vector  1042  and negative offset  1044  are computed. Neighboring element  1014  has force vector  1050 , and following the method of  FIG. 9  new force vector  1052  and negative offset  1054  are computed. 
   The neighbors of field element  1000  are scanned anti-clockwise in sequence  1070 . The starting and ending points of sequence  1070  are arbitrary. If exactly one positive to negative offset transition between adjacent neighbors is found, the field is propagated to element  1000  by constructing a force vector  1080  by interpolating between new force vectors  1032  and  1042 . 
   In a preferred embodiment, the interpolation is a weighted average of vectors  1032  and  1042 . The vector  1032  is weighted by the magnitude of offset  1044 , and the vector  1042  is weighted by the magnitude of offset  1034 . The effect is that the weight of a vector is proportional to the other vector&#39;s offset and inversely proportional to its own offset, so that vectors are more heavily weighted if they pass closer to their corresponding neighbor&#39;s force vector. As shown in  FIG. 10 , the offset corresponding to vector  1032  has the smaller magnitude, so it is more heavily weighted and therefore force vector  1080  passes closer to vector  1032  than to vector  1042 . 
   One method for constructing a weighted average of vectors is to scale each vector by its corresponding weight, add the results, and then scale by the inverse of the sum of the weights. This is equivalent to an independent weighted average of the x and y components. In a preferred embodiment, an independent weighted average of the magnitude and direction is used. 
   If the vectors  1032  and  1042  participating in the interpolation are pointing to distant points along a pattern boundary, or to different boundaries, the field is considered indeterminate at element  1000  and the eval code is set to “don&#39;t care”. In a preferred embodiment, the vectors are considered to be pointing to distant points or different boundaries if either their magnitudes differ by more than 3 grid units or their directions differ by more than 135°. 
   In a preferred embodiment, a special case rule is used to propagate the field at the ends of an open chain. If a neighboring element with a valid force produces a small positive offset, and no anti-clockwise adjacent neighbor has a valid force, the field will propagate without interpolation by using the new force vector as constructed by the method of FIG.  9 . Similarly, if a neighboring element with a valid force produces a small negative offset, and no clockwise adjacent neighbor has a valid force, the field will propagate without interpolation by using the new force vector as constructed by the method of FIG.  9 . In a preferred embodiment, a small offset is one whose magnitude is less than {fraction (1/10)} th  of a grid unit. 
   If more than one positive to negative offset transition between adjacent neighbors, or application of the special case rule, is found, or if none are found, the field is considered indeterminate at element  1000  and the eval code is set to “don&#39;t care”. One reason that no such transitions might be found is that neighboring field elements are themselves set to “don&#39;t care”, for example the border elements  540  set by the initialization step  250 . 
   If a valid force is propagated to element  1000 , then corner code  450 , polarity code  460 , and the index of the nearest field dipole are also propagated by copying from whichever of the neighboring elements participating in the interpolation has the smallest offset magnitude (greatest weight). In the example of  FIG. 10 , the values would be copied from element  1010 . If the special case rule was applied, the values are copied from the neighbor with small offset used to construct the new force vector. 
   Many variations on the above rules can be used within the spirit of the invention to achieve similar results. Indeed any method that produces force vectors that point to the nearest point along a pattern boundary can be used to practice this invention. 
     FIG. 11  shows the same portion of the field array that was shown after seeding in  FIG. 6 , but with new force vectors resulting from one propagation step.  FIG. 12  shows the same portion after two propagation steps. Note in  FIG. 12  field element  1200  whose eval code is set to “don&#39;t care” because more than one positive to negative offset transition between adjacent neighbors was found. 
     FIG. 13  shows a block diagram of the run-time module  140  of a preferred embodiment. Run-time module  140  analyzes the image  130 , using the stored pattern  120 , the starting pose  132 , and the client map  131 . As a result of the analysis, the run-time module produces a pose  134  that maps pattern points to accurately corresponding image points. 
   The run-time module  140  produces an rms error value  136  that is a measure of the degree of match between the pattern and the image, a coverage value  138  that is a measure of the fraction of the pattern to which corresponding image features have been found, and a clutter value  137  that is a measure of extra features found in the image that do not correspond to pattern features. 
   The run-time module  140  produces an evaluated image dipole list  150  and an evaluated field dipole list  160 , e.g., as shown in FIG.  25 . The clutter values of the evaluated image dipole list  150  can be used to identify features in the image  130  not present in the pattern  105  (shown in FIG.  1 ). These probability values range from 0 to 1 and indicate the likelihood that the image feature is not present in the pattern. The “eval2” values ( FIG. 25 ) of the evaluated field dipole list  160  can be used to identify features in the pattern  105  not present in the image  130 . These probability values range from 0 to 1, and indicate the likelihood that the pattern feature was found in the image. 
   The run-time module  140  uses a feature detection module  200  to process the image  130  to produce an image dipole list  1300 . In a preferred embodiment, the feature detection module  200  is identical to that used by training module  110 , and is controlled by the same parameter settings stored in pattern parameters  220 , and is further described in conjunction with FIG.  26 . In other embodiments, different methods or different parameters setting are used as appropriate for a specific application. 
   At least one attraction module  1350  uses pattern parameters  220 , field dipole set  230 , field  240 , image dipole list  1300 , and the starting pose  132  and client map  131 , to refine the starting pose  132  and produce the other outputs  136 ,  138 ,  160 , and  150 . 
     FIG. 14  is a diagram that is used to derive the mathematical basis for a preferred embodiment that uses a least-squares method to determine a pose that best accounts for the evidence of the image dipoles at each attraction step. An image dipole, mapped by the current pose to field point  1400  and with direction  1402 , is considered. A small section of image boundary  1404  is also shown as an aid in understanding the diagram. 
   The field has force f  1430  at mapped image dipole point  1400 , pointing to the nearest point  1435  along pattern boundary  1410 . Note that the force  1430  is normal to the pattern boundary  1410  at point  1435 , and the mapped image dipole direction  1402  is similar but not equal, modulo 180°, to that of the force. 
   The mapped image dipole point  1400  has position vector p  1450  relative to the field origin  1420 . The existence of an image dipole at field point  1400 , with force  1430  and mapped dipole direction similar to the force direction (in a preferred embodiment, similar modulo 180°), is taken as evidence that the current pose should be modified so that the image dipole is mapped so as to lie somewhere along line  1440  tangent to pattern boundary  1410  at point  1435 . The dipole provides no evidence as to position normal to the force, that is along tangent  1440 . The position vector p′  1470  defines a point on tangent  1440 , and the difference vector p−p′  1460  indicates how the mapped dipole position might move as a result of the force. 
   Suppose that [C, t] is a six-degree-of-freedom coordinate transform that maps the current pose into a new, hopefully more accurate, pose. This transform is called the motion transform, because it tells how the image dipoles will move with respect to the field under the influence of the forces of the field. Here C is a 2×2 matrix and t is a translation vector. The evidence under consideration suggests that this transform should map p to p′:
 
 p′=Cp+t.   (1) 
 
   Let I be the identity matrix and define 
       f=|f|   (2)                f   ^     =     f        f                  (   3   )             
  Ċ=C−I   ( 4 ) 
   From the diagram of  FIG. 14  it can be seen that
 
 f= ( p′−p )· {circumflex over (f)}   (5a) 
 
 =( Cp+t−p )· {circumflex over (f)}   (5b)
 
=[( C−I ) p+t]·{circumflex over (f)}   (5c) 
 
=( Ċp+t )· {circumflex over (f)}   (5d) 
 
   Thus, given an image dipole that maps to field point p with force f=f{circumflex over (f)}, we have one equation in the six unknowns [C, t] that tells us how to map the current pose to get a new pose. With six dipoles we can solve for the six-degrees-of-freedom, but in practice the evidence obtained from only six dipoles is generally not sufficient to get an accurate or even meaningful solution. In practice we use many dipoles, typically anywhere from a few dozen to a few thousand, and some method for solving an over-determined set of equations. 
   In a preferred embodiment, a least-squares method is used. An error term for the i th  dipole can be defined as
 
 e   i =( Ċp   i   +t )· {circumflex over (f)}   i   −f   i   (6) 
 
   With this definition a least-squares problem can be set up and solved by methods well-known in the art. If weights w i  are determined for each dipole, we can write the sum squared error as 
             E   =       ∑   i     ⁢           ⁢         w   i     ⁡     [         (         C   .     ⁢     p   i       +   t     )     ·       f   ^     i       -     f   i       ]       2               (   7   )             
 
   In practice it is usually desirable to solve for fewer than six-degrees-of-freedom. Some patterns would result in a singular or unstable solutions if certain degrees of freedom are included. For example, circles cannot be solved for orientation and corners cannot be solved for size. In other cases, a solution would be possible, but some degrees of freedom, particularly aspect ratio and skew, are known not to vary and might cause problems if included. Perhaps the most serious such problem is that unreliable evidence, present to some degree in all images, will have a more serious effect when more degrees of freedom are allowed to vary. Another problem is that somewhat more computation is needed to solve for the additional degrees of freedom. 
   In a preferred embodiment the least-squares problem is set up in 4 degrees of freedom corresponding to x translation, y translation, orientation, and size. Sums needed for a least-squares solution in the 4 degrees of freedom are computed, and pattern parameters  220  specify which of the degrees of freedom will be solved for. 
   In an orthonormal coordinate system we can constrain the matrix Ċ to orientation and size variation by writing it as 
               C   .     =     (         p       q             -   q         p         )             (     8a     )                       ⁢     =     p1   +     q   ⁡     (         0       1             -   1         0         )                   (     8b     )             
 
   In practical applications, however, the images themselves are almost never orthonormal. CCD cameras, for example, typically have pixels that are non-square by a percent or so. For line scan cameras, the angle between the coordinate axes depends on mechanical alignment and so the coordinate axes may not be perfectly orthogonal. The variations from square are small, but easily detectable given the accuracy that can be achieved with the invention. Furthermore, it is sometimes useful to have a significantly non-orthonormal field. For example, a field generated from a square pattern can be used to localize and inspect a rectangular or even parallelogram-shaped instance of the pattern by using an appropriate starting pose. 
   In these cases we generally want the orientation degree of freedom defined by an orthonormal, real-world coordinate system rather than image or field coordinates. We re-write Ċ as
 
 Ċ=p 1 +qN   (9) 
 
where the elements of matrix N are the components of the normal tensor in the field coordinate system. The normal tensor is a mixed 2 nd -rank tensor, a vector-valued function of vectors that, informally, tells how to rotate a vector 90°. In an orthonormal coordinate system, of course, the components of the normal tensor are 
         (         0       1             -   1         0         )     .       
 
   The components of the normal tensor are computed from the current pose and from a coordinate transform called the client map that transforms points in an orthonormal but otherwise arbitrary reference coordinate system to points in the run-time image. 
   We can now re-write equation 7, the sum squared error, as 
             E   =       ∑   i     ⁢           ⁢         w   i     ⁡     [         [         (     p1   +   qN     )     ⁢     p   i       +   t     ]     ·       f   ^     i       -     f   i       ]       2               (     10a     )                       ⁢     =       ∑   i     ⁢           ⁢         w   i     ⁡     [         pp   i     ·       f   ^     i       +       qNp   i     ·       f   ^     i       +     t   ·       f   ^     i       -     f   i       ]       2                 (     10b     )             
 
   Now we can substitute 
               r   i     =       p   i     ·       f   ^     i               (   11   )                 s   i     =       Np   i     ·       f   ^     i               (   12   )               t   =     (         x           y         )             (   13   )                   f   ^     i     =     (           u   i               v   i           )             (   14   )             
 
into equation 10b and finally we have 
             E   =       ∑   i     ⁢       w   i     [       (       xu   i     +     yv   i     +     pr   i     +     qs   i     -     f   i       ]     2                 (   15   )             
 
   A least-squares problem based on equation 15 is easy to set up and solve for x, y, p, and q by well-known methods. The desired motion transform [C, t] is obtained from said solution using equations 4, 9, and 13. The current pose is composed with the motion transform to obtain the new pose. 
     FIG. 15  is a block diagram of the attraction module  1350  for a preferred embodiment based on a least-squares method of best accounting for the evidence of the image dipoles. In addition, to further clarify a preferred sequence of operation of the modules of  FIG. 15 , a flow chart is provided in  FIGS. 27A and 27B . Steps of the flow chart include reference numbers from  FIG. 15  in parentheses to help cross-correlate the figures. A current pose  1500  is initially set to the start pose  2702  and updated at the end of each attraction step  2724 . After the sum module  1535  is initialized to zero  2704 , the Normal tensor computation module  1510  uses the current pose and client map to compute the normal tensor N  2706  for the current attraction step. 
   Each image dipole  1515  of the image dipole list  1300  ( FIG. 13 ) is processed  2708 . The position and direction of dipole  1515  are mapped  2710  from image coordinates to field coordinates by map module  1520 , using the current pose  1500 . The mapped position is used by field module  1525  to determine the force, flags  430 , and index of nearest field dipole  2712 . The force, flags, and index are stored in the image dipole  1515  for later use. The normal tensor, force, and image dipole position in field coordinates are used by a rotate module  1530  to obtain the dipole&#39;s position in force coordinates (r, s)  2714  as specified by equations 11 and 12. 
   An evaluate module  1545  examines the force, flags, image dipole direction in field coordinates, and dipole gradient magnitude and computes a weight for attraction (localization) purposes and evaluation and clutter values for inspection purposes  2714 . In some embodiments, the evaluate module  1545  also considers the rms error from the previous attraction step in determining the weight and evaluation. The evaluation and clutter values are stored in the image dipole  1515  for later use. 
   A sum module  1535  uses the force, dipole position in force coordinates, and weight to compute sums needed for the least-squares solution  2716 . If there are no more dipoles  2718 , a solve module  1540  uses the sums and the normal tensor to solve for the motion transform and compute the rms error  2720 . A compose module  1505  composes the current pose with the motion transform to produce a new pose  2722 , which will be the current pose for the next attraction step, or the final pose if this is the last attraction step  2726 . 
   In a preferred embodiment where inspection is being performed, at the end of the last attraction step, the field dipole evaluation module  1550  evaluates the image dipole list  2728 , which has now been evaluated by evaluate module  1545 , and the field dipole set  230  stored in the pattern  120 , and produces an evaluated field dipole list, coverage rating, and clutter rating  2730 . 
     FIG. 16  gives details for the map module  1520  of FIG.  15 . Inputs are image dipole position in image coordinates  1600 , image dipole direction with respect to image coordinates  1610 , and the current pose  1620 . One output is the dipole position in field coordinates  1630 , computed as shown. 
   The other output is the dipole direction with respect to field coordinates  1640 , computed as shown. The formula for output dir(field)  1640  effectively does the following, reading the vector and matrix operations right to left:
         Construct a unit vector in the dipole direction, with respect to image coordinates, by computing the cosine and sine of the angle θ d .   Rotate the unit vector 90° to get a direction along the boundary that contains the dipole.   Map the rotated unit vector to field coordinates using C p  to get a boundary direction in field coordinates.   Rotate the mapped rotated unit vector −90° to get a direction normal to the boundary in field coordinates.   If the determinant of the pose matrix is negative, the transform changes the left- or right-handedness of the coordinate system, so rotate the vector 180° because the −90° of the previous step should have been +90°.   Compute the angle of the resulting vector using the well-known version of the arctangent function of two arguments whose result is in the range 0° to 360°.       

   Note as shown in output dir(field)  1640  that these calculations can be simplified considerably. In a preferred embodiment, the simplified formula is used at the beginning of each attraction step to compute a 256-element lookup table, indexed by an 8-bit binary angle, for use by the map block  1520 . This allows the direction mapping operation to be executed at high speed for each dipole. 
   When computing the lookup table, the symmetry of the formula requires us to compute only 128 elements of the table; the other elements are the negative of the computed ones. As a further improvement in computation time, 64 even-indexed elements are computed, and the odd-indexed elements are determined by interpolation from the even-indexed elements. Thus the formula need only be applied 64 times. The arctangent function is computed using the well-known CORDIC method. 
   Note that in computing output dir(field)  1640  we map the boundary direction instead of the dipole direction. This is because, in the embodiment described herein, directions are determined by a gradient estimation method  330  and Cartesian to polar conversion method  340  that assumes square pixels. This is not a problem except when mapping directions between non-orthonormal coordinate systems. In that case, the boundary direction must be used. 
     FIG. 17  shows a block diagram of the field module  1525  of  FIG. 15 , and a corresponding geometric diagram that illustrates the computation being performed. Image dipole position in field coordinates  1754  is input to the field block  1525 . The coordinates  1754  fall within field grid cell  1750 . 
   The coordinates are rounded to integer field grid position  1758  by integer rounding module  1700 . The integer field grid coordinates  1758  are used by address generation module  1704  to produce a memory address used to look up field element  1708 , corresponding to grid cell  1750 . In the embodiment shown, the index of the nearest field dipole is stored with the other field information, but in some embodiments, as described herein, the index is kept in a separate array. 
   The force direction θ f , flags, and index obtained from field element  1708  are direct outputs of field module  1525 , but the force magnitude  1774  is interpolated so that the force vector is a reasonably smooth function of real-valued position within the field. 
   Force interpolation is based on the assumption that the force stored in field element  1708 , corresponding to integer grid position  1758 , points to an approximately straight-line section of pattern boundary  1782 . This is a fast and accurate interpolation we can use with the information available. A more compute intensive interpolation could use neighboring field elements as well. 
   To interpolate force magnitude, the integer position  1758  is subtracted by module  1712  from the real-valued position  1754  to produce sub-grid position vector  1762 . A unit vector  1766  in the force direction is constructed by cosine/sine module  1716 , implemented as a lookup table in a preferred embodiment. The dot product  1770  of sub-grid position vector  1762  and unit vector  1766  is computed by dot product module  1720 . The dot product  1770  is subtracted from force magnitude  1774  by the subtraction module  1724  to produce interpolated force magnitude  1778 . 
   The interpolated force magnitude  1778 , unit vector in the force direction  1766 , and force direction angle stored in field element  1708 , are collected in output module  1728  and become part of the force vector  1526  produced by field module  1525 . 
   In another embodiment, not shown, at least one force vector is stored in each field element, pointing to the nearest points along at least one pattern boundary. The field module  1525  examines image dipole direction in addition to position, and uses the stored force vector that is closest to the dipole direction for interpolation and output to subsequent steps. 
     FIG. 18  gives details for the rotate module  1530  of FIG.  15 . Inputs are the normal tensor  1800 , force  1810 , and image dipole position in field coordinates  1820 . Output is image dipole position in force coordinates pos(force)  1830 , computed as shown, and as described above by equations 11 and 12. 
     FIG. 19  shows various preferred fuzzy logic processing modules that are used in evaluate module  1545  of a preferred embodiment illustrated in FIG.  20 . 
     FIG. 19   a  shows a fuzzy greater than module, which takes a real-valued input  1900  and a fuzzy threshold  1904 , and produces a fuzzy logic value  1912 . The fuzzy threshold  1904  is an ordered pair that specifies points along the x axis of graph  1908 . The graph  1908  shows the fuzzy logic output  1912  as a function of input  1900 . As can be seen, the fuzzy logic value falls within the range 0.0 to 1.0, inclusive. 
     FIG. 19   b  shows a fuzzy less than module, which takes a real-valued input  1930  and a fuzzy threshold  1934 , and produces a fuzzy logic value  1942 . The fuzzy threshold  1934  is an ordered pair that specifies points along the x axis of graph  1938 . The graph  1938  shows the fuzzy logic output  1942  as a function of input  1930 . As can be seen, the fuzzy logic value falls within the range 0.0 to 1.0, inclusive. 
     FIG. 19   c  shows a fuzzy not module. Fuzzy logic value input  1960  is inverted by subtracting it from 1 to produce fuzzy logic value output  1964 . 
     FIG. 20  is a block diagram of a preferred embodiment of the evaluate module  1545  of FIG.  15 .  FIG. 20   a  shows the portion responsible for computing the weight and eval values, and  FIG. 20   b  shows the portion responsible for computing the clutter value. 
   Referring to  FIG. 20   a , the computation of weight and eval is based on the force magnitude ‘f’, a comparison of the force direction θ f , and image dipole direction dir(field), and the image dipole&#39;s gradient magnitude ‘mag’. For each of these three factors in the evaluation, a fuzzy logic value is produced by fuzzy logic modules  2004 ,  2040 ,  2064 , respectively, that indicates confidence in the reliability of the evidence provided by the image dipole being evaluated. The three fuzzy confidence factors so-produced are combined into a single confidence score by a combination module  2080  in the range 0.0 to 1.0. The weight and eval outputs are obtained by using the eval code  440  ( FIG. 4 ) to select either the confidence score or the value 0.0. 
   Absolute value module  2002  computes the length of the force vector from force magnitude f, which in the preferred embodiment being described may be negative if the force and gradient directions differ. Fuzzy less than module  2004  compares the force length to a field strength threshold  2000 , to produce a strength confidence factor that indicates high confidence for force lengths “below” the field strength threshold. 
   The field strength threshold  2000  is set based on pattern parameters  220  for the first attraction step. In a preferred embodiment, the first attraction step uses field strength threshold values t zero =2.0 field grid units, and t one =3.0 field grid units. 
   In the embodiment shown in  FIG. 20 , the field strength threshold  2000  is modified after each attraction step based on the rms error from the previous step. The modification is accomplished by addition module  2012 , which adds the rms error to both the t zero  and t one  components of a field strength margin parameter  2008  to produce the new field strength threshold  2000 . As a result, the field strength threshold  2000  is matched to how well the particular run-time image being analyzed fits the stored pattern at each attraction step. 
   The method of adjusting the field strength threshold based on the rms error of the previous step is effective in some applications, but in other cases it has been observed to result in some oscillation of the attraction rather than convergence on one solution. In a preferred embodiment, not shown, the field strength threshold is reduced in equal steps after each attraction step. Thus, as the attraction converges to a solution, image dipoles must be closer to pattern boundaries to be given high confidence. 
   The image dipole direction dir(field) is compared with the pattern boundary gradient direction by subtract module  2024 . Recall that in the embodiment being described, the “force” direction θ f  reported by the field is actually boundary gradient direction, which is the same as or opposite of the true force direction. If pattern parameters  220  and polarity code  460  of flags  430  specify that gradient polarity is to be ignored, the angle difference from subtract module  2024  is constrained to the range −90° to +90° by mod 180° module  2028 ; otherwise, the angle θ f  is passed unmodified. The magnitude of the resulting angle difference is determined by absolute value module  2032 . 
   The angle difference magnitude is compared to one of two fuzzy thresholds by fuzzy less than module  2040  to produce a direction confidence factor. If corner code  450  of flags  430  indicates “no corner”, the field direction threshold  2044  is chosen by selection module  2036 . If the corner code indicates “is corner”, the field corner threshold  2048  is chosen. In a preferred embodiment, the field direction threshold  2044  has values t zero =11.25° and t one =22.5°, and the field corner threshold  2048  has values t zero =39.375° and t one =50.625°, reflecting the fact that a wider range of image dipole directions can reasonably correspond to a pattern boundary corner. In an alternate embodiment, a real-valued measure of curvature can be used instead of the binary “is corner” code, with multiple values of the field direction threshold possible. 
   The image dipole&#39;s gradient magnitude ‘mag’ is compared to a fuzzy magnitude threshold  2060  by fuzzy greater than module  2064  to produce a magnitude confidence factor. The magnitude threshold is intended to throw out very weak dipoles that are likely due to image noise or other artifacts, but the use of a fuzzy threshold gives more stable results than the more traditional hard threshold. In a preferred embodiment, the magnitude threshold  2060  uses the same value for t zero  as the noise threshold chosen for the peak detector  350 , and uses a value of t one  equal to twice the value of t zero . 
   The strength, direction, and magnitude confidence factors are combined by multiply module  2080  to produce an overall confidence score in the range 0 to 1. Based on eval code  440  of flags  430 , the selection module  2084  chooses a value for weight and the selection module  2088  chooses a value for eval. If the eval code is “attract”, the confidence score is chosen for the weight; otherwise the constant 0 is chosen so that the dipole is ignored for localization purposes. If the eval code is “attract” or “evaluate only”, the confidence score is chosen for eval; otherwise the constant 0 is chosen to indicate that the dipole does not correspond to any portion of the pattern. 
     FIG. 20   b  shows a preferred embodiment for the calculation of the clutter value. The direction confidence factor produced by fuzzy less than module  2040  is inverted by fuzzy not element  2042 . The image dipole&#39;s gradient magnitude is compared to a fuzzy clutter threshold  2070  by fuzzy greater than module  2074  to produce a clutter confidence factor  2075 . The clutter confidence factor  2075  is multiplied by the inverted direction confidence factor by multiplier  2090  to produce a tentative clutter value  2091 . If the eval code  440  is anything but “don&#39;t care”, selection module  2092  chooses this tentative clutter value  2091  as the dipole&#39;s clutter value; otherwise the constant 0 is chosen. 
   If the eval code  440  is “expect blank”, the force magnitude ‘mag’ is meaningless, but in a preferred embodiment, the force direction θ f  encodes the gradient direction from the training image as described above in conjunction with FIG.  5 . In this case, the computation of clutter uses this direction as it would the force direction θ f . This mode of operation is appropriate when it is desirable to minimize false alarms. Alternatively, if it is appropriate to minimize the chances of missing clutter, one can consider only the clutter confidence factor  275  when the eval code is “expect blank”. 
   In a preferred embodiment, the clutter threshold  2070  has values of t zero  and t one , each value being equal to 1.5 times the t zero  and t one  values used for the magnitude threshold  2060 . 
   In a preferred embodiment, the magnitude confidence factors and clutter confidence factors for all of the image dipoles are computed once and stored in the image dipole list  1300 , rather than being recomputed for each attract step. This can be done because these confidence factors are independent of the current pose  1500 . 
     FIG. 21  is a block diagram of sums module  1535  of FIG.  15 . This module accumulates the weighted sums needed for the solution of the least-squares problem of equation 15. Five multiply modules  2100  perform the weighting. Fifteen multiply-accumulate modules  2130  and one accumulate module  2160  compute and store the sums needed. The sixteen accumulators are set to zero at the beginning of each attraction step. This four degree-of-freedom case is an exemplary embodiment, other numbers of degrees of freedom being possible. 
     FIGS. 22   a-d  give details of the solve module  1540  of  FIG. 15 , which produces the motion transform and the rms error value. The formulas shown are based on the solution of the least-squares problem of equation 15. Pattern parameters  220  specify which degrees of freedom are to be determined. 
     FIG. 22   a  shows the solution for the 2 translation degrees of freedom only—size and orientation are as specified in the start pose.  FIG. 22   b  shows the solution for translation and orientation. This preferred solution is based on an approximation that assumes a small angle of rotation. If the assumption is violated, some size variation will be introduced.  FIG. 22   c  shows the solution for translation and size, holding orientation fixed, and  FIG. 22   d  shows the solution for all 4 degrees of freedom. 
     FIG. 23  gives details of the compose module  1505  of  FIG. 15 , which composes the current pose  2300  with the motion transform  2330  computed by the solve module  1540  to produce the new pose  2360 . 
     FIG. 24  gives details of the normal tensor module  1510  of  FIG. 15 , which computes the normal tensor  2460  from the current pose  2400  and the client map  2430 . 
     FIG. 25  shows an example of field dipole evaluation performed as part of field dipole evaluation module  1550  of FIG.  15 . In the example, a first image dipole  2500 , second image dipole  2510 , and third image dipole  2520  have received evaluations 0.85, 0.93, and 0.88 respectively. Four field dipoles labeled  2540 ,  2550 ,  2560 , and  2570  lie along a chain as determined by connect step  254  during training module  110 . The chain is defined by the left links  2580  and right links  2585 . 
   For image dipole  2500 , an index  2505  was determined by field module  1525  to identify the nearest field dipole  2540 . The evaluation 0.85 is transferred from image dipole  2500  to the “eval1” slot of field dipole  2540 . 
   No image dipole identified field dipole  2550  as nearest, so its “eval1” slot holds its initial value 0. 
   For image dipole  2510 , an index  2515  was determined to identify the nearest field dipole  2560 . For image dipole  2520 , the same index  2515  was determined to identify the nearest field dipole  2560 . The larger of image dipole  2510  evaluation 0.93 and image dipole  2520  evaluation 0.88 is transferred to the “eval1” slot of field dipole  2560 . 
   Field dipole  2570  has evaluation 0.90 transferred from some image dipole not shown. 
   To fill in the gap at field dipole  2550 , a dilation operation is performed, wherein all field dipoles receive an evaluation equal to the maximum of their own evaluation and that of their left and right neighbors. The dilated evaluations are shown in the “eval2” slot of each field dipole. Note that it is not actually necessary to store both “eval1” and “eval2” values;  FIG. 25  shows them for clarity. 
   Once the field dipoles have been evaluated, the coverage value produced by field dipole evaluation module  1550  is computed by averaging all of the field dipole evaluations. 
   In a preferred embodiment, the field dipoles are evaluated and coverage is computed only after the last attraction step, and only if pattern inspection is desired. 
     FIG. 26  shows how the invention can be operated in a multi-resolution mode designed to increase the capture range without sacrificing accuracy. Two pattern training modules (not shown) are run on a single training image, with different settings of low-pass filter module  310  and image sub-sample module  320 . In a first setting designed to attenuate fine detail, a low resolution pattern  2600  is generated. In a second setting designed to pass fine detail, a high resolution pattern  2610  is generated. 
   A low resolution run-time module  2620  uses the low resolution pattern  2600 , and a start pose and client map, to analyze run-time image  130  to produce a low resolution pose that is much more accurate than the start pose but not as accurate as can be achieved at higher resolution. A high resolution run-time module  2630  uses the high resolution pattern  2610 , the low resolution pose as a start pose, and the same client map, to analyze run-time image  130  to produce the final pose, rms error, coverage, and evaluated dipole lists. 
   The multi-resolution mode is supervised by overall control module  2640 , as illustrated by the flow chart in FIG.  28 . As part of its operation  2800 , the low resolution rms error, coverage, and clutter values are examined, and if  2802  they do not indicate a reasonable match between image and stored pattern, the operation is aborted  2803  without attempting to run the high resolution module. If the low resolution module produces a good match, high resolution module is run  2804 . If the high resolution module does not produce good results  2806 , it usually means that the image is out of focus, and the user is so-warned. In some embodiments, when this happens, the low resolution results are used instead of the high resolution results  2808 . If the results of the high resolution module are acceptable, the results of the high resolution module are provided to the user  2810  for interpretation, or further processing, according to the particular application. 
   In a preferred embodiment, an overall match score is computed for each resolution step that is equal to the coverage value minus half the clutter value. The low resolution results are used instead of the high resolution results if the high resolution match score is less than some fraction of the low resolution match score. In a preferred embodiment, the fraction used is 0.9. 
   In a preferred embodiment, the methods of U.S. Pat. No. 6,457,032, issued Sep. 24, 2002, entitled “Efficient, Flexible Digital Filtering”, and U.S. Pat. No. 6,408,109, issued Jun. 18, 2002, entitled “Apparatus and method for detecting and sub-pixel location of edges in a digital image” are used for feature extraction, Cognex Corporation&#39;s PatQuick™ tool is used to determine the starting pose, and the multi-resolution style of  FIG. 26  is used. The following parameter settings are used for feature extraction by default. Many other strategies can be devised to suit specific applications. 
   For training the low resolution pattern  2600 , and corresponding run-time module  2620 , the image is sub-sampled by sub-sampler  320  by an equal amount in x and y given by the formula 
       floor   ⁡     (         wh     8       )         
         where w and h are the width and height, respectively, of the pattern  100  in pixels and the floor function gives the largest integer that is less than or equal to its argument. Note that sub-sampling by n means taking every n th  pixel. The low-pass filter  310  uses a filter size parameter (“s” in U.S. Pat. No. 6,457,032, issued Sep. 24, 2002, entitled “Efficient Flexible Digital Filtering”) equal to one less than the computed sub-sample amount. The Cartesian to polar conversion module  340  multiplies the gradient magnitude values by 2.0 to improve precision at the low end, where most gradient values lie.       

   For training the high resolution pattern  2610 , and corresponding run-time module  2630 , the low-pass filter  310  and the sub-sampler  320  are set to pass the source image  300  unmodified. 
   As part of its operation, the PatQuick™ tool reports a “contrast” value in gray levels that is the median gradient magnitude of the pixels in the image on which it is run that correspond to the trained pattern. In a preferred embodiment, this contrast value is used to set the default noise threshold for the peak detector  350 . Many other schemes for setting noise thresholds are known in the art that can be used to achieve equivalent results. 
   In said preferred embodiment, PatQuick™ is run on the training image  100  and the contrast value reported by the tool is saved as part of the pattern parameters  220 . For training the low resolution pattern  2600 , the peak detection module  350  uses a noise threshold equal to 10 gray levels. For training the high resolution pattern  2610 , the peak detection module  350  uses a noise threshold equal to one-quarter of said saved contrast. 
   For the run-time image  130 , when the PatQuick™ tool is used to determine the starting pose the contrast value it reports is examined. For both the low resolution run-time module  2620 , and the high resolution run-time module  2630 , the peak detection module  350  uses a noise threshold equal to that used for the corresponding pattern  2600  or  2610 , but in each case multiplied by the ratio of run-time contrast to the saved train-time contrast. 
   The preferred embodiments described herein use a six-degree-of-freedom coordinate transform to represent the mapping between points in the image and points in the pattern (i.e. the pose), and a least-squares fitting to determine how to use the information provided by the field to modify a given pose so as to produce a new pose that represents a better correspondence between image and pattern features. Many other arrangements can be devised by those of ordinary skill in the art for achieving similar results within the scope of the invention. These other arrangements may have advantages in certain specific applications. 
   For example, the six degree of freedom coordinate transform can be replaced with other analytic models of the mapping between points in the image and points in the pattern. One useful such model is the well-known perspective transform. Another useful model is one that corrects for lens distortions, such as that produced by so-called “fisheye” lenses. In these cases a different least squares solution would be used, and appropriate changes would be made to the pose element  1500 , the compose module  1505 , the normal tensor module  1510 , the map module  1520 , the rotate module  1530 , the sums module  1535 , and the solve module  1540 . The image dipole  1515 , field module  1525 , evaluate module  1545 , and field dipole evaluation module  1550  need not change. 
   In other arrangements, the least squares method can be replaced with other well-known methods for fitting data. In such arrangements, appropriate changes might be made to the rotate module  1530 , the sums module  1535 , and the solve module  1540 . Alternatively, one or more of these modules might be replaced by different modules that are required for the fitting method to be used. 
   In still other arrangements, a non-analytic mapping between points in the image and points in the pattern, such as a 2-dimensional lookup table with interpolation, may be used. In such an arrangement, the pose  1500  is a lookup table mapping image points to pattern points, and the map module  1520  does the lookup and interpolation. The field module  1525  and evaluate module  1545  can be used without modification. The compose module  1505 , normal tensor module  1510 , rotate module  1530 , sums module  1535 , and solve module  1540  are not used. Instead, an intermediate lookup table is produced as follows. For every image dipole  1515 , an entry is made in the intermediate lookup table by adding the force vector obtained from the field module  1525  to the mapped position from map module  1520 . Along with this field-corrected position, the weight obtained from the evaluate module  1545  is also stored in the intermediate table entry. 
   The intermediate table thus produced may be sparse, in that many points will not have been filled in, and it may have errors caused by the occasional unreliable image dipole. It can be used, however, to produce a new pose  1500  by applying a smoothness constraint. For example, each element of the new pose can be determined by a weighted mean or median of some neighborhood of corresponding elements of the intermediate table. Other methods for using smoothness as a constraint are well-known in the machine vision literature. 
   Other modifications and implementations will occur to those skilled in the art without departing from the spirit and the scope of the invention as claimed. Accordingly, the above description is not intended to limit the invention, except as indicated in the following claims.