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## Radiometric Normalization

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**Radiometric Normalization**Spring 2009 Ben-Gurion University of the Negev**Instructor**• Dr. H. B Mitchell email: harveymitchell@walla.co.il Sensor Fusion Spring 2009**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**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**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**Histogram Matching**• If gray-levels are not distinct may break ties randomly. Better to use “exact histogram specification”. Sensor Fusion Spring 2009**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**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**Midway Histogram Equalization**Optical flow with and without histogram equalization Sensor Fusion Spring 2009**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**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**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**Ranking. Classical**• Example. Sensor Fusion Spring 2009**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**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**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**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**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**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**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**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**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**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**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**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**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**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**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