TH E MAXIMUM LIKELIHOOD DETECTION PROCESSOR

The mathematical operations required to perform ML detection can be determined by solving the problem in either the frequency domain or in the spatial domain. The

alternative spatial function (Cook and Bemfield, 1967). Having obtained the frequency domain function it can then be converted back to a spatial function to determine the operations which must be performed on the images to implement the processor.

The "frequency response" for a two-dimensional processor (that is, the mathematical description of its mode of operation in spatial frequency terms) is given by the so-called matched filter equation:

H (w„wz) = S*(w„W2) exp[-j(w,m,wzn)]

Nt(w,,o)2) I 2

I N , ( w „ W 2 ) I 2

is the frequency response of the processor; is the complex conjugate of the signal; is the (power) spectral density of the noise;

is the exponential factor indicating the Fourier domain; are the spatial frequency coordinates corresponding to the horizontal and vertical spatial coordinates (i.e. x and y).

5.4.1 The operation of the M L processor

The operations performed by the ML processor depend on whether or not the noise contaminating the signal, is "white" (Figure 5.3). "White" or "uncorrelated" noise is defined as a random disturbance with a uniform spatial frequency spectrum. This form of noise image can be seen as the snow-like appearance on a television screen which is tuned to a channel that is not broadcasting.

Conversely, noise which does not have a uniform spatial frequency spectrum is referred to as "coloured" or "correlated" noise. The spatial frequency spectrum of coloured noise is typically characterised by a decline in power with increasing spatial frequency. In this case, the corresponding effect in the spatial domain is a statistically correlated image, that is, an image which has spatial structure.

Figure 5.3 Binary images depicting random noise: (a) uncorrelated ("white") noise; (b) correlated (low frequency) noise; (c) correlated (high frequency) noise.

5.4.1.1 The white noise case

Where the noise is white the denominator in equation 5.3 is equated to unity and under these circumstances the operation performed by the ML processor is the mathematical process of cross-correlation of the signal to be detected with each subimage of the image being searched for a match (Cattermole, 1986). The subimage with the greatest probability of being the correct match, that is, representing the same portion of bone, is indicated by the spatial position at which the cross-correlation reaches its maximum value. A three dimensional plot of the output of a detection processor is shown is Figure 5.4 where the conditional probability is plotted on the vertical axis against the spatial position on the image where the match is being sought (indicated by the two horizontal axes, x and y).

5.4.1.2 The correlated noise case

Where the noise accompanying the signals is correlated the operation performed by the ML processor is rather more complicated than in the case of white noise. Under these circumstances the ML processor operates by performing preliminary processing to transform the noise into white noise and then proceeds in the same manner as for the white noise case indicated above (Kailath, 1970). The operation of the ML processor for correlated noise can be thought of as two linear processors acting in series (Figure 5.5).

The first processor transforms the correlated noise to uncorrelated (white) noise by equalising the power at all spatial frequencies. This process is known as whitening or

- 3 0 0 0 0 0 1 - 1 2 9 0 0 0 - - 2 5 5 0 0 0 !! ! !I m l y ^2 0 ^ 0

F ig u re 5 .4 Three-dim ensional mesh plot show ing the output o f a detection processor. The cross-correlation value is plotted on the vertical axis against the spatial position on the image where the match is being sought (indicated by the two horizontal axes, X and y).

d e c o r r e la tio n . T h e se c o n d p r o c e sso r then d e te c ts th e sig n a l (w h ic h is n o w su b m e rg ed in w h ite n o is e ) u sin g c r o ss-c o r r e la tio n as d e sc r ib e d a b o v e . H o w e v e r , b e c a u s e th e sig n a l and n o is e a re in tim a te ly b ou n d to g e th e r , th e fir st p r o c e sso r acts o n , and a lte r s, both th e n o is e and th e s ig n a l. A c c o r d in g ly , th e o u tp u t fro m th e first p r o c e s s o r c o n s is ts o f w h ite n o is e p lu s a s p e c tr a lly d istorted sig n a l. C o n se q u e n tly , the se c o n d p r o c e s s o r m u st b e d e sig n e d to d e te c t th e sp e c tr a lly d isto rted or "w hitened" sig n a l rather than th e o r ig in a l s ig n a l. T h is is a c c o m p lis h e d b y p a ssin g both sets o f im a g e data (fro m e a r lie r and la ter rad iograp h s) th ro u g h th e fir st p r o c e sso r . A s b e fo r e , fo r any g iv e n s u b im a g e in th e fir st r a d iograp h , th e o v e r a ll p r o c e s s o r se le c ts th e s u b im a g e in th e se c o n d rad iograp h w ith the g r ea te st p r o b a b ility o f rep resen tin g the sa m e p o rtio n o f b o n e . A g a in , th is is in d ic a ted b y the sp a tia l p o s itio n at w h ic h th e c r o ss-c o r r e la tio n , p e rfo rm e d b y th e se c o n d p r o c e sso r , r e a c h e s its m a x im u m v a lu e .

INPUT ► OUTPUT cross-correlation processor decorrelation processor

t o t a l l i n e a r p r o c e s s o r

Figure 5.5 Diagrammatic representation of a two-stage detection processor for signals in correlated noise.