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### lecture16

Course: CS 791, Fall 2009
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791E CS Computer Vision Instructor: Mircea Nicolescu Lecture 16 Thresholding Otsu's method A measure of region homogeneity is variance (i.e., regions with high homogeneity will have low variance). Otsu's method selects the threshold by minimizing the within-class variance of the two groups of pixels separated by the thresholding operator. It does not depend on modeling the probability density functions,...

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791E CS Computer Vision Instructor: Mircea Nicolescu Lecture 16 Thresholding Otsu's method A measure of region homogeneity is variance (i.e., regions with high homogeneity will have low variance). Otsu's method selects the threshold by minimizing the within-class variance of the two groups of pixels separated by the thresholding operator. It does not depend on modeling the probability density functions, however, it assumes a bimodal distribution of gray-level values (i.e., if the image approximately fits this constraint, it will do a good job). 2 Thresholding Means and variances - Assume light objects against dark background - Consider that we have an image with L gray levels and its normalized histogram (for each gray-level value i, P(i) is the normalized frequency of i). - Assuming that we have set the threshold at T, the normalized fraction of pixels that will be classified as background and object will be: 3 Thresholding Means and variances - The mean gray-level value of the background and the object pixels will be: - The mean gray-level value over the whole image (grand mean) is: 4 Thresholding Means and variances - The variance of the background and the object pixels will be: - The variance of the whole image is: 5 Thresholding Within-class and between-class variance - It can be shown that the variance can be written as follows: where: W2(T) is defined to be the within-class variance B2(T) is defined to be the between-class variance. 6 Thresholding Determining the threshold - Since the total variance does not depend on T, the T minimizing W2 will be the T maximizing B2 - Let's consider maximizing B2 we can rewrite B2 as follows: -Start from the beginning of the histogram and test each gray-level value for the possibility of being the threshold T that maximizes B2 7 Thresholding Determining the threshold 8 Thresholding Drawbacks of Otsu's method - The method assumes that the histogram of the image is bimodal (two classes). - The method breaks down when the two classes are very unequal (the classes have very different sizes). - In this case, B2 may have two maxima. - The correct maximum is not necessary the global one. - The selected threshold should correspond to a valley of the histogram. - The method does not work well with variable illumination. 9 Thresholding Effect of illumination on segmentation - How does illumination affect the histogram? 10 Thresholding Handling non-uniform illumination laboratory solution - Obtain an image of just the illumination field. 1. Suppose that f(x, y) = i(x, y) r(x, y), where i(x, y) is non-uniform 2. Project the illumination pattern on a surface with uniform reflectance (e.g., a white surface) g(x, y) = k i(x, y) 1. Normalize f (x, y): h(x, y) = f(x, y) / g(x, y) = r(x, y) / k 1. If r(x, y) can be segmented using T, then h(x, y) can be segmented using T/k 11 Thresholding Handling non-uniform illumination local thresholding - A single threshold will not work well when we have uneven illumination due to shadows or due to the direction of illumination. - The idea is to partition the image into m x m subimages and then choose a threshold Tij for each subimage. - This approach might lead to sub-images having simpler histograms (e.g., bimodal) 12 Thresholding Handling non-uniform illumination local thresholding 13 Thresholding Handling non-uniform illumination variable thresholding - In case of uneven illumination, another useful technique is to approximate the values of the image by a simple function (i.e., plane). - Thresholding can be performed relative to the plane (e.g., points above the plane will be part of the object, and anything below will be part of the background). 14 Thresholding Handling non-uniform illumination variable thresholding 15 Thresholding Drawbacks of thresholding - Pixels assigned to a single class need not form coherent regions as the spatial locations of pixels are completely ignored (Note: Only hysteresis thresholding considers some form of spatial proximity). - Threshold selection is not always straightforward. 16 Region Growing The fundamental drawback of histogram-based region detection is that histograms provide no spatial information (only the distribution of gray levels). Region-growing approaches exploit the important fact that pixels which are close together have similar gray values. Region growing - Start with a single pixel (seed) and add new pixels 17 Region Growing 18 Region Growing More than one seed can be used 19 Region Growing How do we choose the seed(s) in practice ? - It depends on the nature of the problem - If targets need to be detected using infrared images for example, choose the brightest pixel(s). - Without prior knowledge, compute the histogram and choose the gray-level values corresponding to the strongest peaks (e.g., 1 and 7 in the previous example) 20 Region Growing How do we choose the similarity criteria (predicates)? - The homogeneity predicate can be based on any characteristic of the regions in the image, such as: - - - - - - - average intensity variance color texture motion shape size 21 Region Growing Region growing based on simple surface fitting - It has been suggested to use a predicate that checks whether the error in approximating pixel the data by some function is less than some threshold. - A digital image is a set of quantized samples of a continuously varying function. - The idea is fitting an appropriate low-order surface (e.g., planar or biquadratic) over the image data of a region. - If the errors are small, then we can conclude that the pixel values belong to the same region. 22 Region Growing 23 Region Growing Problems - The main assumption of this approach is that regions are nearly constant or have smooth variations in image intensity. - This approach will not work for non-smoothly varying regions (e.g., textured regions). - More sophisticated approaches are required to allow for more realistic intensity variations due to shading. 24 Region Merging Region merging operations eliminate false boundaries and spurious regions by merging adjacent regions that belong to the same object. Merging schemes begin with a partition satisfying condition (4) (e.g., regions produced using thresholding). Then, they proceed to fulfill condition (5) by gradually merging adjacent image regions. 25 Region Merging 26 Region Merging How to determine region similarity? - Based on the gray values of the regions. - Compare their mean intensities. - Use surface fitting to determine whether the regions may be approximated by one surface. - Use hypothesis testing to judge the similarity of adjacent regions (assumes that the intensity values are drawn from a probability distribution). - Based on the weakness of boundaries between the regions. 27 Camera Calibration Camera Calibration - Produces an estimate of the extrinsic and intrinsic camera parameters 28 2D Geometric Transformations Translation - Moves points to new locations by adding translation amounts to the coordinates of the points 29 2D Geometric Transformations Translation cont. - To translate an object, translate every point of the object by the same amount 30 2D Geometric Transformations Scaling - Changes the size of the object by multiplying the coordinates of the points by scaling factors 31 2D Geometric Transformations Scaling cont. - Scale factors affect size as following: - If sx = sy uniform scaling - If sx sy non-uniform scaling - If sx , sy < 1, size is reduced, object moves closer to origin - If sx , sy > 1, size is increased, object moves further from origin - If sx = sy = 1, size does not change 32 2D Geometric Transformations Scaling cont. - Control the location of a scaled object by choosing the location of a point (fixed point) with respect to which the scaling is performed 33 2D Geometric Transformations Rotation - Rotates points by an angle - From triangle OPPx: - From triangle OP'Px': - From the above equations we have: 34 2D Geometric Transformations Rotation cont. - To rotate an object, rotate every point of the object by the same amount (rotate only the endpoints of line segments - redrawing is required) 35 2D Geometric Transformations Rotation cont. - Performing rotation about an arbitrary point 36 2D Geometric Transformations Summary of transformations: - Translation: P' = P + T - problem: cannot represent it as matrix multiplicati...

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