radiometric normalization spring 2009 ben-gurion university of the negev

Radiometric Normalization

Spring 2009

Ben-Gurion University of the Negev

Sensor Fusion Spring 2009

Instructor

• Dr. H. B Mitchell

email: [email protected]


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


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


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


Histogram Matching

If gray-levels are not distinct may break ties randomly. Better to use “exact histogram specification”.


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


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.



Optical flow with and without 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


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


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 :


Ranking. Classical

Example.


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


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


Thresholding

Advantage: Unsupervised thresholding methods automatically adjust to input image.

Disadvantage: Quantization is very coarse May overcome? by using fuzzy thresholding

t

Classical Fuzzy


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


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.


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?


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.


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.


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


Likelihood. Platt Calibration

Maximum likelihood solution is

If too few training samples have or

then liable to overfit. Correct for this by using modified


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


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



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.


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



So far have considered pairwise likelihood estimation. How can we generalize to multiple classes with more than two

classes? Project.

radiometric normalization spring 2009 ben-gurion university of the negev

Documents

histogram of b

course sensor fusion

wheresensor fusion spring

stopetcsensor fusion

common histogram

histogram matchinginput

histogram matchingeasy

cumulative histogram