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).
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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:
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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:
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Thresholding
Means and variances
- The variance of the background and the object pixels will be:
- The variance of the whole image is:
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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.
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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
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Thresholding
Determining the threshold
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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.
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Thresholding
Effect of illumination on segmentation
- How does illumination affect the histogram?
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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
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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)
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Thresholding
Handling non-uniform illumination local thresholding
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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).
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Thresholding
Handling non-uniform illumination variable thresholding
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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.
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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
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Region Growing
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Region Growing
More than one seed can be used
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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)
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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
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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.
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Region Growing
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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.
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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.
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Region Merging
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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.
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Camera Calibration
Camera Calibration
- Produces an estimate of the extrinsic and intrinsic camera parameters
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2D Geometric Transformations
Translation
- Moves points to new locations by adding translation amounts to the coordinates of the points
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2D Geometric Transformations
Translation cont.
- To translate an object, translate every point of the object by the same amount
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2D Geometric Transformations
Scaling
- Changes the size of the object by multiplying the coordinates of the points by scaling factors
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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
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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
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2D Geometric Transformations
Rotation
- Rotates points by an angle - From triangle OPPx:
- From triangle OP'Px':
- From the above equations we have:
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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)
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2D Geometric Transformations
Rotation cont.
- Performing rotation about an arbitrary point
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2D Geometric Transformations
Summary of transformations:
- Translation: P' = P + T
- problem: cannot represent it as matrix multiplicati...