Here we extend all things Fourier to two dimensions. Shortly we will be able to model many effects on an image, such as motion or focus blur, by the 2D version of convolution, which is handled especially simply by the Fourier Transform. This enables us to restore an image from many kinds of noise and other corruption. We begin Section 15.1 by showing how the Fourier Transform, and others, may be converted from a 1- to a 2-dimensional transform of a type called separable, reducing computation and adding simplicity. In the Fourier case we may apply the FFT in each dimension individually, and hence speed calculation still further.

In Section 15.1.3 we prove that certain changes in an image result in predictable changes in its transform. We include the effect of both rotation and projection, which are germane to computerised tomography in Chapter 18. In Section 15.1.4 we present consequences of the 2D Convolution Theorem for the Fourier Transform, and offer a polynomial-based proof that purports to show ‘why’ the result holds. Section 15.1.5 establishes connections between correlation and the Fourier Transform, for later use.

We begin Section 15.2 by considering the low-level operation of changing pixels solely on the basis of their individual values, then move on to the possibilites of ‘filtering’ by changing Fourier coefficients. Next we see how the same effect may be accomplished by convolving the original with a matrix of coefficients. We introduce filters that achieve edge-detection in an image.