Chapter 3: The Image, its Mathematical and Physical Background. 3.1. Overview 3.2. Linear Integral Transforms 3.3. Images as Stochastic Processes 3.4. Image Formation Physics. 3.1.2. Dirac Delta Function. Heaviside function :. 3.1. Overview. Pulse :. Impulse :. Dirac Delta Function :.

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Chapter 3: The Image, its Mathematical and Physical Background

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○ Analysis: The Fourier sequence F(u), u = 0 , … , N-1 of f(x), x = 0 , … , N-1can be formed from sequences u = 0 , …… , M-1 Recursively divide F(u) and F(u+M); eventually, each contains one element F(w), i.e., w = 0, and F(w) = f(x). 3-27

3.2.5. Sampling theory Continuous function Dense sampling Sparse sampling Objective: looking at the question of how many sampling should be taken so that no information is lost in the sampling process

3.2.7. Wavelet transform Fourier spectrum provides all the frequencies present in a signal but does not tell where they are present. Windowed Fourier transformsuffers from the dilemma: Small range – poor frequency resolution Large range – poor localization Wavelet: wave that is only nonzero in a small region Wave Wavelet

○ Operations on wavelet: (a) Dilation: i) Squashing ii) Expanding (b) Translation: i) Shift to the right ii) Shift to the left (c) Magnitude change: i) Amplification ii) Minification 3-46

Wavelet transform:decomposes a function into a set of wavelets where : wavelets : mother wavelet New variables: scale translation Inverse wavelet transform:synthesize a function from wavelets coefficients