Radiometric Normalization Spring 2009 Ben-Gurion University of the Negev.
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Transcript of Radiometric Normalization Spring 2009 Ben-Gurion University of the Negev.
Radiometric Normalization
Spring 2009
Ben-Gurion University of the Negev
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
t
Classical Fuzzy
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.
x
Degree to which x is regarded as hot
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.