Vector Spaces • Space of vectors, closed under addition and scalar multiplication
Dec 19, 2015
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