Vector Spaces

• Space of vectors, closed under addition and scalar multiplication

Image Averaging as Vector addition

Scaler product, dot product, norm

Norm of Images

Orthogonal Images, Distance,Basis

Roberts Basis: 2x2 Orthogonal

Cauchy Schwartz InequalityU+V≤U+V

Schwartz Inequality

Quotient: Angle Between two images

Fourier AnalysisFourier Analysis

Fourier Transform Pair

• Given image I(x,y), its fourier transform is

Image Enhancement in theFrequency Domain

Image Enhancement in theFrequency Domain

Complex Arithmetic

Fourier Traansform of an Image is a complex matrix

Let F =[F(u,v)]

F = ΦMM I(x,y) ΦNN I(x,y)= Φ*MM F Φ*MM

Where

ΦJJ (k,l)= [ΦJJ (k,l) ] and

ΦJJ (k,l) = (1/J) exp(2Πjkl/J) for k,l= 0,…,J-1

Fourier Transform

Properties

• Convolution Given the FT pair of an image

f(x,y) F(u,v) and mask pair h(x,y) H(u,v)

• f(x,y)* h(x,y) F(u,v). H(u,v) and

• f(x,y) h(x,y) F(u,v)* H(u,v)

Properties of Fourier TransformProperties of Fourier Transform

Properties of Fourier TransformProperties of Fourier Transform

Properties of Fourier TransformProperties of Fourier Transform

Properties of Fourier TransformProperties of Fourier Transform

Image Enhancement in theFrequency Domain

Image Enhancement in theFrequency Domain

Design of H(u,v)

İdeal Low Pass filter

H(u,v) = 1 if |u,v |< r

0 o.w.

Ideal High pass filter

H(u,v) = 1 if |u,v |> r

0 o.w

Ideal Band pass filter

H(u,v) = 1 if r1<|u,v |< r2

0 o.w

İmage EnhancementSpatial SmoothingLow Pass Filtering

Ideal Low pass filterIdeal Low pass filter

Ideal Low Pass FilterIdeal Low Pass Filter

Output of the Ideal Low Pass FilterOutput of the Ideal Low Pass Filter

Gaussian Low Pass FilyerGaussian Low Pass Filyer

Gaussian Low Pass FilterGaussian Low Pass Filter

Gaussian Low Pass FilterGaussian Low Pass Filter

Gaussian Low PassFilterGaussian Low PassFilter

High Pass Filter: Ideal and GaussianHigh Pass Filter: Ideal and Gaussian

Ideal High PassIdeal High Pass

Fourier Transform-High Pas Filtering

Frequency Spectrum of Damaged CircuitFrequency Spectrum of Damaged Circuit

Gaussian Low Pass and High PassGaussian Low Pass and High Pass

Output of Gaussian High Pass Output of Gaussian High Pass

Gaussian Filters: Space and Frequency DomainGaussian Filters: Space and Frequency Domain

Spatial Laplacian Masks and its Fourier Transform

Laplacian FilterLaplacian Filter

Laplacian FilteringLaplacian Filtering