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N − m); j = 1, . . 1. 2: Linear smoothing filters applied to X-ray image: (a) 3 × 3 moving average, (b) Gaussian, σ 2 = 23 , (c) 5 × 5 moving average, (d) Gaussian, σ 2 = 2, (e) 9 × 9 moving average, (f ) Gaussian, σ 2 = 6 23 . CHAPTER 3. FILTERS 6 c c using column weights w−m , . . , wm , followed by a row operation: gij = m l=−m wlr hi,j+l for i, j = (m + 1), . . , (n − m), r r , . . , wm . In order for this to be possible, the array of weights, wkl , must using row weights w−m be expressible as the product of the column and row weights, as follows: wkl = wkc wlr for k, l = −m, .

Sometimes the images will not exactly coincide, in which case they must be distorted in order to match. This process is known as image registration. In order to distort images to match each other, we must have an accurate knowledge of the differences between them. This may be expressed in terms of which parts of the digital images refer to which particular parts of the scene, and should therefore be aligned in the two images. Occasionally this may be known a priori from the nature of the imaging instruments.

Here, and subsequently, ‘tan−1 ’ must produce output over a range of 2π radians, to ensure that cos θkl = akl /rkl and sin θkl = bkl /rkl . The Fortran77 and C functions ‘atan2’ will achieve this). Therefore, as we can see, f is a sum of sine (or cosine) terms. 2. LINEAR FILTERS IN THE FREQUENCY DOMAIN 15 Let us take a simple example, composed of only a few sine terms, to see what this transformation looks like. If f is the 4 × 4 array:   f =   2 1 2 3 5 6 7 6 2 3 2 1  7 6 5 6   ,  then by substituting numbers into the above equations we obtain the result that the real and imaginary parts of f ∗ are:   a=   0 0 0 0 0 0 0 32 0 0 0 0 0 0 0 64       and b =     −8 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0    .

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