Digital Image Processing Lectures 5 & 6 Mahmood (Mo) Azimi, Professor Colorado State University Fort Collins, Co 80523 email :
[email protected] Each time you have to click on this icon wherever it appears to load the next sound file To advance click enter or page down to go back use page up 1. Parseval’s Theorem and Inner Product Preservation Another important property of FT is that the inner product of two functions is equal to the inner product of their FT’s. ZZ ∞ x(v, u)y∗(u, v)dudv −∞ ZZ ∞ 1 ∗ = 2 X(ω1, ω2)Y (ω1, ω2)dω1dω2 4π −∞ where ∗ stands for complex conjugate operation. When x = y we obtain the well-known Parseval energy conservation formula i.e. ZZ ∞ ZZ ∞ 2 1 2 |x(u, v)| dudv = 2 |X(ω1, ω2)| dω1dω2 −∞ 4π −∞ i.e. the total energy in the function is the same as in its FT. 2. Frequency Response and Eigenfunctions of 2-D LSI Systems An eigen-function of a system is defined as an input function To advance click enter or page down to go back use page up 1 that is reproduced at the output with a possible change in the amplitude. For an LSI system eigen-functions are given by f(u, v) = exp(jω1u + jω2v) Using the 2-D convolution integral ZZ ∞ 0 0 0 0 0 0 g(u, v) = h(u − u , v − v ) exp[jω1u + jω2v ]du dv −∞ change u˜ = u − u0, v˜ = v − v0, then g(u, v) = H(ω1, ω2) exp[jω1u + jω2v] where ZZ ∞ H(ω1, ω2) = h(˜u, v˜) exp[jω1u˜ + jω2v˜]dud˜ v˜ −∞ = F{h(u, v)} To advance click enter or page down to go back use page up 2 is the frequency response of the 2-D system.