The Caliper tool uses the projection region to produce a one-dimensional representation of the portion of the image that contains the edges of interest. This one-dimensional projection image will contain not only the edges of interest, but also other edges caused by noise and unwanted information in the original image. Applying a filter to the one-dimensional projection image increases the strength of the edges of interest while at the same time decreasing image noise.

The Caliper tool performs filtering by convolving the one-dimensional projection image with a filter operator. A filter operator is a block of arithmetic operators that is applied sequentially to blocks of pixels within the one-dimensional projection image to produce the pixel values in the filtered image.

Figure 2 illustrates an example of a simple 3-element filter operator. In this case, the filter operator subtracts the value of each pixel’s left neighbor and adds the value of each pixel’s right neighbor to produce the pixel value in the filtered image.

Figure 2. Filtering the projection image

The filtered image no longer visually resembles the input image. However it does have an important new characteristic: a graph of the filtered pixel values reveals that the peaks in the values, both positive and negative, correspond to the location of the edges within the initial image. The Caliper tool uses these peaks in the filtered image to determine the location of edges in the original image.

In addition to producing an image with peaks that correspond to edges in the input image, image filtering also removes noise and spurious edges from the input image. Figure 3 illustrates an example where the image contains two true edges along with spurious edges caused by variations in pixel values around the edge.

When a filter with a size of 1 is applied to the image, both the edges of interest and spurious edges appear as peaks in the filtered projection image.

Figure 3. Filtering with filter size of 1

Figure 4 shows the effect of applying a filter with a width of 2. The peaks that correspond to the edges of interest are broader, and most of the spurious peaks from Figure 3 are no longer present in the filtered projection image.

Figure 4. Filtering with filter size of 2

Once it has located the edges in the original image, the Caliper tool computes a score for each potential edge pattern in the image. The score for each edge pattern in the image is computed based on the similarity between the edge pattern in the image and an ideal edge pattern that you define. The edge patterns in the image are called edge pattern candidates while the ideal edge pattern is called the edge model. You can define edge models that contain a single edge or a pair of edges.

You control the method that the Caliper tool uses to score edge pattern candidates by performing the following steps:

• Define the edge model that describes the edge or edge pair for which you are searching.
• Define one or more scoring methods that define how edge pattern candidates are to be evaluated for similarity to the edge model.

An edge model is a description of the edge pattern that you expect to see in the image. An edge model includes edge spacing, position, and polarity (dark to light or light to dark).

For each edge pattern candidate in the image, the Caliper tool computes a score. You can specify exactly how to evaluate each edge attribute, such as position, strength, or polarity, when computing the overall score for that edge. The tool computes a separate value for each attribute for each edge pattern candidate based on the scoring method that you supply, then computes an overall score for each edge pattern candidate by combining the individual scores.

You can specify the projection region as an affine rectangle and affine rectangle sampling parameters. The projection direction is defined as being parallel to the Po-Py side of the affine rectangle, and the number of Po-Px divisions specified by the affine sampling parameters defines the number of samples in the projection region.

Figure 5 shows how an affine rectangle and affine rectangle sampling parameters determine the projection region.

Figure 5. Project region defined by affine rectangle and sampling parameters

You specify an affine rectangle by supplying all of the parameters required to define the rectangle to a function call or object constructor. Figure 6 shows how the constructor or function call uses the parameters you specify to construct an affine rectangle.

Figure 6. Creating an affine rectangle.

Figure 7 shows how different skew, rotation, and Po values determine the orientation and handedness of an affine rectangle.

Figure 7. Handedness and orientation in an affine rectangle.

Note that each of the rectangles shown in Figure 7 occupies the same position within the coordinate system.

Figure 8 shows the specific steps required to change the handedness of an affine rectangle without moving the rectangle or changing the location of its origin (Po point).

Figure 8. Changing handedness requires changing skew and rotation

You specify the location of an affine rectangle to determine what portion of the input image is included within the rectangle. Affine rectangles that occupy the same part of an image can have their vertex points in different locations.

The orientation and handedness of an affine rectangle determine how the affine rectangle is used. The Caliper tool will use the Po-Py axis of the affine rectangle as the projection direction and the Po-Px axis as the search direction.

Bilinear interpolation considers the values of the four pixels closest to the center of each affine pixel in the affine sampling rectangle. The distance-weighted average of the values of the four pixels is computed and used for the value of the affine pixel.

Figure 10 shows how bilinear interpolation is computed.

Figure 10. Bilinear interpolation

The value of the interpolated affine pixel is computed using the following formula:

Figure 11. Value of interpolated affine pixel

where

Figure 12 shows how bilinear interpolation would produce the affine sampled image shown in Figure 9. The centers of the affine pixels are shown as grey crosses, and the four pixels used to compute each affine pixel are surrounded by a heavy border.

Figure 12. Computing a interpolated sample image

The computed interpolated pixel values are rounded to the nearest whole integer value.

The filter size and type that you specify for the Caliper tool controls how the tool removes noise from the input image and how it accentuates the peaks of interest in the image.

You should specify a filter size that closely matches the size of the edges in your input images. The size of an edge is the number of pixels wide that the edge is. Edges can be sharp, in which case the edge only spans one or two pixels, or edges can be dull, in which case they might span many pixels. Figure 15 illustrates examples of sharp and dull edges; the figure illustrates both types of edges at different magnifications.

Figure 15. Sharp and dull edges

When you specify a filter size that is close to the edge size, the Caliper tool produces stronger edge peaks in the filter image. If you specify a filter size that is too large or too small, the filter image will contain broad, low peaks. Figure 16 illustrates the effect of applying filters of different sizes to sharp and dull edges.

Figure 16. The effect of filter size and edge size on peak size

Figure 16 illustrates that the more closely you match the filter size to the size of the edges of interest within the image, the sharper the desired peaks in the filtered image and the fewer spurious peaks there will be.

The Caliper tool supports a filter kernel that approximates the negative of the first derivative of a Gaussian distribution. This kernel provides better smoothing than traditional boxcar filters.

You specify the half-width of the filter, where the half-width is equal to four times the sigma of the Gaussian distribution. Figure 17 illustrates how you specify the width of the filter.

Figure 17. Filter size for a Gaussian filter

The filter itself is constructed by taking the negative of the first derivative of the Gaussian curve, as shown in Figure 18.

Figure 18. Constructing the filter

The Caliper tool lets you define a minimum peak height or contrast threshold. Any peaks that are lower than your minimum peak height are excluded from the results. This allows you to limit your analysis of edges in the image to just those edges of a certain magnitude.

You specify the contrast threshold as the difference in normalized pixel values between the two sides of the edge.

To evaluate whether an edge pattern candidate in the image is a good match for the edge pattern that you are seeking, you must define a model that describes the edge pattern of interest. You define an edge model by specifying the polarity (light-to-dark or dark-to-light) and position of each edge in the model. You specify the positions of edges relative to a model origin. The Caliper tool lets you define edge models with a single edge or a pair of edges.

Table 1 describes an edge model with a symmetric pair of edges. The origin of the model is centered between the two edges.

Figure 19 shows an idealized graphical representation of the edge model described in Table 1. Note that the position of the model origin is implied by the values of the edge positions.

Figure 19. Idealized representation of an edge model

The Caliper tool bases the score of an edge pattern candidate on how different the edge pattern candidate is from the edge model. You can specify the particular measure that the tool uses to assess this difference. The types of measures that you can specify are called scoring method types. The available scoring method types are

• Position
• Size
• Contrast
• Straddle

When the tool applies a scoring method type to an edge pattern candidate the result is a raw score. The raw score is different for different scoring method types. The scoring method types and their raw scores are described in the following sections.

You can score an edge pattern candidate based on the position of the edge pattern candidate relative to the center of the projection region you specified for this Caliper tool. The position is defined as the distance between the center and the model origin point within the edge pattern candidate.

If you expect the edge of interest to be a specific distance from the center of the projection region, then you can define an absolute positional scoring method and the raw score will be expressed as an absolute distance in pixels.

If you are using an edge pair model and you would like to consider the variation in position between the edge pattern candidate and the center of the projection region relative to the size of the model, you can define a relative positional scoring method. In this case the raw score will be normalized so that a value of 1.0 means that the distance was equal to the size of the model.

If you are using an edge pair model, you can score edge pattern candidates based on how much the width between the edges of the edge pattern candidate varies from the width between the edges of the edge model. You can define a size scoring method to be absolute, in which case the raw score will be returned as an absolute size difference in pixels.

If you would like to consider the size difference relative to the size of the model, you can define a relative size scoring method. In this case the raw score will be normalized so that a value of 1.0 means that the size difference was equal to the size of the model.

You can score edge pattern candidates based on the contrast of the edge pattern candidates. The contrast of an edge is expressed in terms of the change in pixel values divided by the size of the edge in pixels. The raw score for a contrast scoring method is normalized so that a value of 1.0 is equal to a contrast of 256 (the maximum possible value for contrast). If you specify an edge pair model, the raw score is the average of the contrast for the two edges.

If you are using an edge pair model, you can score edge pair candidates based on whether or not the two edges lie on either side of the center of the projection region. This type of scoring method can be used to find objects defined by a pair of edges that are expected to lie under the center of the projection region. The raw score is returned as 1.0 if the center of the projection region is straddled by the edges, 0.0 if the center of the projection region is not straddled by the edges.

For each scoring method, the selected scoring method type produces a raw score. You control the effect that this raw score has on the overall score for an edge pattern candidate by defining a scoring function. A scoring function maps a raw score to a mapped score. The mapped scores for each scoring method for an edge pattern candidate are combined to form the overall score for that edge pattern candidate.

You define a scoring function by specifying low and high input and output values. Figure 20 shows a scoring function.

Figure 20. Scoring function

You define the scoring function by defining values for xc, x1, x0, y1, and y0. The Caliper tool uses the scoring function you define to map raw scores to mapped scores as follows. Raw scores above x0 are mapped to a score of y0. Raw scores below xc are mapped to a score of 0. Raw scores between xc and x1 are mapped to a score of y1. Raw scores between x1 and x0 are mapped linearly to the range of scores between y1 and y0.

The values you define for y0 and y1 must be between 0.0 and 1.0. You can specify any values for xc, x1, and x0, including negative values. You would specify negative values for one or more of the xc, x1, and x0 points if you expected raw scores less than zero.

The scoring function illustrated in Figure 20 would be appropriate for a case where higher raw scores should produce higher mapped scores, such as a contrast scoring method where greater the edge contrast should produce a higher score.

In other cases, such as a positional scoring method where the edge pattern candidate is expected to be at or near the center of the projection region, a higher raw score should result in a lower mapped score. In a case such as this, you specify a value for x1 that is greater than x0, and a value for xc that is greater than x0. Figure 21 shows a scoring function that is appropriate for a case where higher raw scores should result in lower mapped scores.

Figure 21. Scoring function with x1 greater than x0

You must specify one and only one scoring function per scoring method, although you can specify any number of geometric scoring methods per Caliper tool.

If you are using an edge pair model and you specify a two-sided size scoring method type, you can define a scoring function that scores edge pattern candidates that are smaller than the edge model differently from edge pattern candidates that are larger than the edge model. This type of scoring function is called a two-sided scoring function.

Figure 22 shows an example of a two-sided scoring function that is more tolerant of edge pattern candidates that are larger than the edge model than of edge pattern candidates that are smaller. If the edge pattern candidate is exactly the same size as the edge model, the raw score will be 0.0; if the edge pattern candidate is smaller than the edge model, the raw score will be less than 0; if the edge pattern candidate is larger than the edge model, the raw score will be greater than 0.

If you specify a two-sided scoring function, the raw score is normalized so that an edge pattern candidate that is smaller than the edge model by an amount equal to the size of the edge model receives a raw score of -1.0 and an edge pattern candidate that is larger than the edge model by an amount equal to the size of the edge model receives a raw score of 1.0.

Figure 22. Two-sided scoring function

Figure 22 shows a two-sided scoring function with distinct values for y1 and y1h but the same values for y0 and y0h. While it is possible to specify different values for y0 and y0h, the resulting scoring function contains a discontinuity at x0b. Figure 23 shows a scoring function where y0h is different than y0.

Figure 23. Another two-sided scoring function

You can also specify a two-sided scoring function for cases where raw score values outside of a range of values receive higher scores. Figure 24 shows an example of this type of two-sided scoring function.

Figure 24. Two-sided scoring function with low output values in the center

For each edge pattern candidate, once the Caliper tool has computed the raw score for each scoring method and applied the scoring function for that scoring method it computes an overall score for the edge pattern candidate.

The Caliper tool computes the overall score for an edge pattern candidate by taking the Nth root of the product of the mapped scores. For example, if you defined four scoring methods, the Caliper tool would multiply the four scores and take the 4th root of the result.

Note that if any single scoring function produces a value of 0, then the overall score for the edge pattern candidate will also be 0.

Figure 25 shows the overall process of computing the score for an edge pattern candidate.

Figure 25. Scoring an edge pattern candidate