Transform Methods

In this second lecture we discuss the Discrete Fourier Transform (DFT), and its relation to filtering and convolution. Both signals and images will be considered. Applications to deblur-ring and digital image correlation will be discussed. Most of the discussion can be found in greater detail in the reference [1] A good starting point for this discussion is the well-known exponential Fourier series expansion of a function, x(t), on [0, 1]. x(t) = ∞ k=−∞ c k e k (t) = ∞ k=−∞ c k e 2πikt. (1) The Fourier coefficients can be obtained by scalar products: c k = x(t), e k (t) = 1 0 x(t)e 2πikt dt. (2) Now, in a typical sampling situation, we sample x(t) at the N points 0, 1 N , 2 N ,. .. , N − 1 N. to create a vector X =      x(0) x(