Chapter 4 SEGMENTATION

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10 CHAPTER 4. <strong>SEGMENTATION</strong>a greater overall brightness than others. Alternatively, trend can sometimes be removedbefore thresholding an image, for example by using a top-hat transform (§5.5).4.1.2 Multivariate classifiersThere are many ways to extend the concept of a threshold in order to segment a multivariateimage, such as:• Allocate pixel (i, j) to category 1 iff ij,m ≤ t mfor m =1,...,M,where subscript m denotes the variate number, and there are M variates.• Or, more generally, use a box classifier where the conditions for category 1 are:t 1,m
4.1. THRESHOLDING 11where summation is over all the N r pixels in region B r . The simplest case to consider is wherethe variance-covariance matrices in the different categories are assumed to be the same. In thiscase, the common variance-covariance matrix, V , can be estimated as⎛⎞R∑V = ⎝ ∑∑/∑ RTf ij f ij − N r µ r µ T ⎠r N r .r=1(i,j)∈B rDistances in multidimensional space are measured taking into account the variance-covariancematrix V . The squared Mahalanobis distance between pixel value f ij and the rth categorymean is(f ij − µ r ) T V −1 (f ij − µ r ),where the superscript ‘−1’ denotes a matrix inverse.The segmentation rule is to allocate each pixel in the image to the nearest category mean, asmeasured by the above distance. The rule simplifies to a set of linear thresholds. For example,in the two category case, a pixel is allocated to category 1 ifr=1µ T 1 V −1 µ 1 − 2µ T 1 V −1 f ij ≤ µ T 2 V −1 µ 2 − 2µ T 2 V −1 f ij .This is linear discrimination. If the variances are not assumed to be equal, then this resultsin quadratic discrimination. For more details, see for example Krzanowski (1988, §12.2).Fig 4.5 shows this method applied to the bivariate image obtained by combining the MRIproton density and inversion recovery images of a woman’s chest (Figs 1.7(a) and (b)). Fourtraining areas have been selected manually as representatives of1. lung tissue,2. blood in the heart,3. muscle, and other lean tissues,4. adipose tissues, such as subcutaneous fat.These areas are shown superimposed on the inversion recovery and proton density images inFigs 4.5(a) and (b).Fig 4.5(c) is a scatter plot of f ij,2 against f ij,1 for the training pixels, together with the linearseparators of all pixel values into the four classes. For example, the muscle pixels are placed inthe top-left of Fig 4.5(c) because they have high values of proton density but low values fromthe inversion recovery signal, as can be seen in Figs 4.5(a) and (b). The points have been codedin progressively darker shades of grey, to denote lung, muscle, blood and fat pixels respectively.There is substantial overlap between blood and fat pixels in Fig 4.5(c), so that even the trainingset has not been classified precisely.Fig 4.5(d) shows the pixel allocation of the whole image. Air pixels, for which f ij = 0, havebeen excluded from all classes. The lungs, shown as the lightest shade of grey, have been
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