Chapter 21

R(x) Algorithm

a) Anomaly Detectionb) Matched Filter

Definition of terms: A pixel can be represented by

n1

n2n

nJ

DC

DC=

DC

x

1 2 3= [ , , ] =

NX x x x x

11 21 N1

12 22 N2

1J 2J NJ

DC DC DC

DC DC DC

DC DC DC

sub image containing N pixels designated by

(1)

(2)

1 2, ...TNs s ss

= 1Ts s

(3)

(4)

The target can be specified as an additive signal with a spatial shape characterized by:

where we have normalized the pixel intensities in the spatial descriptor such that

The spectral signature of the target (in an additive sense) is characterized by:

T1 2[ , ... ]Jb b bb (5)

The Statistical Model suggests that the optimum filter is based on the joint probability distribution of the signal x under the hypothesis

H0 no target andH1 target present

the demeaned data in a local window with spectral covariance M would have a joint probability distribution described by Gaussian statistics as

When a target is present the mean will be shifted by the additive signal and the joint distribution can be expressed as:

1

1

1J -NN 22 2

0 1 2P , ... 2п e

NTn n

N

x M x

x x x M n| |

1

1

1N- 22 2

1 1 2, ... 2п | |

NT

n n n nn

s sJN

NP e

x b M b

x x x Mx

(6)

(7)

A constant false alarm rate (CFAR) detector that will optimize detection (given our assumptions are valid) is the log likelihood ratio test (i.e. the log of the probabilities) expressed as

1

1

1

0

P

P

H

n

H

XX

X

>

< 0Y (8)

simplifying and moving non-data driven terms into a modified

threshold we obtain the detector

1

1

1

0

NT

n n on

H

s

H

—>

<Yb M x

1Ty X b M Xs

or more compactly the output of this linear matched filter can be expressed as

(9)

(10)

o

—

Y

First recall that the covariance matrix can be represented in terms of the Eigen vectors and eigen values as:

where P is the matrix of ordered Eigen vectors

Λ is the diagonal matrix of ordered Eigen values .

Finally, recall that

Equation 10 expressed in terms of the output for each pixel (n) produced by the spectral operator yields

1 2, ,... JP e e e

1 1 T M PΛ P

1 1T T Tn n

ny ·b M x b PΛ P x

Graphical Interpretation

1 2, ,... J

and

(11)

(12)

(13)

(14)

PΛM PΛP 1 TP

Returning to Matrix algebra recall that Λ-1 must satisfy the equation

1 ITΛ Λ (15)

J

2

1

1

λ

10

λ

10

00λ

1

Λ

(16)

this means that Λ-1 must be made up of the reciprocal of the Eigen values i.e.

Stocker figure 1

Stocker figure 3

Stocker figure 4

Post multiplication of the target projection term bTP by Λ-1 in Equation 14 thus yield a J element row vector whose elements are the projection of the additive target spectrum onto the eigen values and optimally weighed by the inverse of the eigen values (i.e. the reciprocal). Finally, the dot product of the row and column vectors yields a matched filter gain weighed by the eigen values (inversely weighed) to accentuate the difference of target from clutter.

a constant false alarm rate CFAR detector can be obtained by normalizing the optimized matched filter using a detector of the form

21

01 1( ) =

X

1T

T T

2

H

>r r

1 <1N H

b M Xs

Xs M Xs b M b(17)

Implementation

where the numerator is the squared matched filter from Equation 10 and the denominator normalizes for variation in the local clutter covariance and the correlation of the target with the local clutter.

The locally adaptive CFAR detector can then be implemented using the following steps

1. Compute the local mean vector on a window of N pixels about a pixel center of interest and demean the data locally according to

2. Compute the local covariance according to

N΄

n-1

1

Nn n

n nx x x x x

1 T

NM XX

M

(18)

(19)

N

1n

1n1JnJ

N

1n

1n11n1

)DC)(DCDC(DCN

1

)DC)(DCDC(DCN

1

M̂

N

1n

2n21n1 )DC)(DCDC(DCN

1

N

1n

JnJJnJ

N

1n

JnJ1n1

)DC)(DCDC(DCN

1

)DC)(DCDC(DCN

1

(20)

recall that X is the J X N matrix made up of the N demeaned J band pixel vectors comprising the local window

Stocker figure 7

The locally adaptive CFAR detector can then be implemented using the following steps illustrated in Figure 7 from Stocker et al. 1990. Con’t

3. Compute the local additive target vector

4. Compute the spectral filter and apply it to the image .

5. Using the normalized target shape vector s, apply the spatial matched filter to the spectrally filtered product and square the result

6. Compute the local normalization value of

b b x

1Tb M 1Tb M X

1Tb M Xs

21Tb M Xs

111

T

N

Xs M Xs

and multiply the output of Step 5 to obtain the r(x) value.

7. Apply the desired CFAR threshold to locate detections.

1 bMb 1T

Unknown target spectrum

In many cases, the target spectrum may not be well known or known at all. Stocker et al.1990 suggest one way to deal with this situation is to generate a filter bank of possable filters designed to span the spectral space (either all space for the unknown target or a possable target range for the poorly known target). However, they point out that there will clearly be a performance loss due to the difference between the postulated target spectrum used in the filter production and the true target spectrum. Further, they show that to maintain this loss at reasonable levels, the size of the filter bank grows extremely large as the number of bands increase (c.f. Figure 10 from Stocker et al. 1990).

Stocker Figure 10

anomaly detector designed to find "target" spectra differing from the clutter but without a specific target spectral signature. In this case, the best estimate of the target signature can be expressed in terms of the normalized target shape as:

Substituting into 10 and including scene-derived estimates for all parameters yields

the overall signal to clutter ratio expressed as:

1

N

n nn

s

b x Xs

11 -1T T TNbM Xs Xs M Xs Xs XX Xs

11 2

TSCR b M b

(22)

(23)

(24)

Stocker figure 8

Stocker figure 11

Stocker figures 12,13

Figure A

From page 9

.. .

.

.

Stocker figure 14