Notes 20110125 CIS 6320 Image Processing

From SnOwy - Ed's Wiki Notebook

Contents 1 Assignment: Talks 2 Image Operations and Transformations 3 Inverse Function 4 Function Composition 5 Identity 6 Single Pixel -- Images -- Transformations 7 Neighbourhood Operations

Assignment: Talks

• Applications of ~ in Digital Image Processing (many -- half an hour)
• Send slides by Feb. 11th.

Image Operations and Transformations

Inverse Function

• let f = (A, B, G) be a bijection
• let f^-1 = (B, A, G^-1)
• G^-1 = {(b, a) ∈ B×A | (a, b) ∈ G}
• must be a bijection

Function Composition

• f(g()) is g º f g circle f
• given f = (A, B, F)
• given g = (B, C, G)
• give me g º f = (A, C, H) i.e. H?
• H = {(a, c) ∈ A×C | ∃b∈B,(a,b)∈F∧(b,c)∈G}

Identity

• f = (A, B, G)
• f^-1 = (B, A, G^-1)
• f º f^-1 = (B, B, I)
• where I = {(b, b)}_b∈B
• Let f be a bijection from A to B ...
• fºf^-1=Id_B
• f^-1ºf = Id_A

Single Pixel -- Images -- Transformations

• input image f|Z²
• output image g|Z²
• g(x,y) = T_i(f(x,y))
• where (x, y) is a pixel location
• g = T_iºf
• T_i|R'→R
• L = 256
• T_i -- is an intensity transformation function
• T_i|u |→ u + 10 -- increase brightness
• T_i|u |→ u - 10 -- decrease brightness
• T_i|u |→ 2u -- increase contrast
• T_i|u |→ u/2 -- decrease contrast
• the -- (|→) -- symbol is supposed to be a left-barred right-arrow -- there is no such character :( -- it means associates with.

Neighbourhood Operations

• we have to first discuss distance
• a distance is a function
• d|R²×R² → R
• ... (p, q) |→ d(p, q)
• ∀(p,q) ∈ R²×R², d(p,q)≥0 -- definite, positive
• ∀(p,q) ∈ R²×R², d(p,q) = 0 ←→ p = q -- definite, positive
• ∀(p,q) ∈ R²×R², d(p,q) = d(q,p) -- symmetric
• ∀(p, q, r) ∈ (R²)³), d(p,q) + d(q,r) ≥ d(p,r) -- triangle inequality
• Euclidean Distance
• d_E|R²×R²
• ... ((x,y),(x',y')) |&rarr sqrt((x-x′)²+(y-y′)²)
• Cityblock Distance
• d₄| ...
• ... |→ |x-x′|+|y-y′|
• Chessboard
• d₈| ...
• ... |→ |max(|x-x'|, |y-y'|)
• neighbours -- the pixels whose distance is one from the given pixel
• d4 has four neighbours
• d8 has eight neighbours
• we can now discuss neighbourhood operations ...
• as operations on tuples
• T_i|R⁹→R
• blurring an image
• detect edges
• sharpen an image
• gradient, etc. ...
• aside: convolution in the spatial domain is a neighbourhood operation