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Radiometric Normalization

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  1. Radiometric Normalization Spring 2009 Ben-Gurion University of the Negev

  2. Instructor • Dr. H. B Mitchell email: Sensor Fusion Spring 2009

  3. Radiometric Normalization • Radiometric Normalization ensures that all input measurements use the same measurement scale. • We shall concentrate on statistical relative radiometric normalization. • These methods do not require spatial alignment although they assume the images are more-or-less aligned. • Other methods will be discussed throughout the course Sensor Fusion Spring 2009

  4. Histogram Matching • Input: Reference image A and test image B. • Normalization: Transform B such that (pdf of B) is same as (pdf of A), i.e. find a function such that • The solution is where Sensor Fusion Spring 2009

  5. Histogram Matching • Easy if B has distinct gray-levels • Let be histogram of B • Suppose A has pixels with a gray-level • Then all pixels in A with rank are assigned gray-level rank are assigned gray-level etc Sensor Fusion Spring 2009

  6. Histogram Matching • If gray-levels are not distinct may break ties randomly. Better to use “exact histogram specification”. Sensor Fusion Spring 2009

  7. Exact Histogram Specification • Convolve input image with 6 masks e.g. • Resolve ties using . If no ties exist, stop • Resolve ties using . If no ties exist, stop • etc Sensor Fusion Spring 2009

  8. Midway Histogram Equalization • Warp both input histograms to a common histogram • The common histogram is defined to be as similar as possible to • A solution: Define by its cumulative histogram : • Implementation is difficult. Fast algorithm (dhw) is available using dynamic programming. Sensor Fusion Spring 2009

  9. Midway Histogram Equalization Optical flow with and without histogram equalization Sensor Fusion Spring 2009

  10. Midway Histogram Equalization If input images have unique gray-levels (use exact histogram specification) then midway histogram is trivial: where is kth largest gray levels in A and B Sensor Fusion Spring 2009

  11. Ranking • Ranking may also be used as a robust method of radiometric normalization. • Very effective on small images, less so on large images with many ties. • Solutions? exact histogram specification. fuzzy ranking Sensor Fusion Spring 2009

  12. Ranking. Classical • Classical ranking works as follows: • M crisp numbers • Compare each with . • Result is • The crisp ranks are • where • Note: We may make the eqns symmetrical by redefining : Sensor Fusion Spring 2009

  13. Ranking. Classical • Example. Sensor Fusion Spring 2009

  14. Ranking. Fuzzy • Fuzzy ranking is a generalization of classical ranking. • In place of M crisp numbers we have M membership functions • Compare each with “extended min” and “extended max” . • Result is • The fuzzy ranks are • where Sensor Fusion Spring 2009

  15. Thresholding • Thresholding is mainly used to segment an image into background and foreground • Also used as a normalization method. • A few unsupervised thresholding algorithms are: Otsu Kittler-Illingworth Kapur,Sahoo and Wong etc • Example. KSW thresholding. Consider image as two sources foreground (A) and background (B) according to threshold t. Optimum threshold=maximum sum of the entropies of the two sources Sensor Fusion Spring 2009

  16. Thresholding • Advantage: Unsupervised thresholding methods automatically adjust to input image. • Disadvantage: Quantization is very coarse • May overcome? by using fuzzy thresholding Classical Fuzzy t Sensor Fusion Spring 2009

  17. Aside: Fuzzy Logic • From this viewpoint may regard fuzzy logic as a method of normalizing an input x in M different ways: • We have M membership functions which represent different physical qualities eg “hot”, “cold”, “tepid”. • Then represent x as three values which represent the degree to which x is hot, x is cold and x is tepid. Degree to which x is regarded as hot x Sensor Fusion Spring 2009

  18. Likelihood • Powerful normalization is to convert the measurements to a likelihood • Widely used for normalizing feature maps. • Requires a ground truth which may be difficult. Sensor Fusion Spring 2009

  19. Likelihood. Edge Operators • Example. Consider multiple edge operators Canny edge operator. Sobel edge operator. Zero-crossing edge operator • The resulting feature maps all measure the same phenomena (i.e. presence of edges). • But the feature maps have different scales. Require radiometric normalization. • Can use methods such as histogram matching etc. But better to use likelihood. Why? Sensor Fusion Spring 2009

  20. Likelihood. Edge and Blob Operators • Example. Consider edge and blob operators • Feature maps measure very different phenomena. Radiometric normalization is therefore of no use. • However theory of ATR suggests edge and blob are casually linked to presence of a target. • Edge and Blob may therefore be normalized by semantically aligning them, i.e. interpreting them as giving the likelihood of the presence of a target. Sensor Fusion Spring 2009

  21. Likelihood. Edge and Blob Operators • Edge map E(m,n) measures strength of edge at (m,n) • Blob map B(m,n) measures strength of blob at (m,n) • Edge likelihood measures likelihood of target existing at (m,n) given E(m,n) • Blob likelihood measures likelihood of target existing at (m,n) given B(m,n). • Calculation of the likelihoods requires ground truth data. • Three different approaches to calculating the likelihoods. Sensor Fusion Spring 2009

  22. Likelihood. Platt Calibration • Given training data (ground truth): • K examples of edge values: and K indicator flags (which describe presence or absence of true target): • Suppose the function which describes likelihood of a target given an edge value x is sigmoid in shape: • Find optimum values of and by maximum likelihood Sensor Fusion Spring 2009

  23. Likelihood. Platt Calibration • Maximum likelihood solution is If too few training samples have or then liable to overfit. Correct for this by using modified Sensor Fusion Spring 2009

  24. Likelihood. Histogram • Platt calibration assumes a likelihood function of known shape • If we do not know the shape of the function we have may simply define it as a discrete curve or histogram. • In this case we quantize the edge values and place them in histogram bins. • In a given bin we count the number of edge values which fall in the bin and the number of times a target is detected there. • Then the likelihood function is Sensor Fusion Spring 2009

  25. Likelihood. Isotonic Regression • Isotonic regression assumes likelihood curve is monotonically increasing (or decreasing). • It therefore represents a intermediate case between Platt calibration and Histogram calibration. • A simple algorithm for isotonic curve fitting is PAV (Pair-Adjacent Violation Algorithm). Monotonically increasing likelihood curve of unknown shape Sensor Fusion Spring 2009

  26. Likelihood. Isotonic Regression • Find montonically increasing function f(x) which minimizes • Use PAV algorithm. This works iteratively as follows: • Arrange the such that • If f is isotonic then f*=f and stop • If f is not isotonic then there must exist a label l such that • Eliminate this pair by creating a single entry with which is now isotonic. Sensor Fusion Spring 2009

  27. Likelihood. Isotonic Regression # score init iterations In first iteration entries 12 and 13 are removed by pooling the two entries together and giving them a value of 0.5. This introduces a new violation between entry 11 and the group 12-13, which are pooled together formin a pool of 3 entries with value 0.33 Sensor Fusion Spring 2009

  28. Likelihood. Isotonic Regression • So far have considered pairwise likelihood estimation. • How can we generalize to multiple classes with more than two classes? • Project. Sensor Fusion Spring 2009