Enhancing Images Ch 5:Shapiro, Ch 3:Gonzales

Gray level Mapping Brightness Transform: 1. Position Dependent f(i,j)= g(i,j). e(i,j) g:Clean image e:position dependent noise 2. Position independent

2. Position Independent Gray Level Mapping s=T(r) 2. Position Independent Gray Level Mapping s=T(r)

Negation

2. Gamma Transformation s=T(r) 2. Gamma Transformation s=T(r)

Gamma Correction of CRT

Image Enhancement by Gamma Transform: s=c.r ɣ

Image Enhancement by Gray level mapping: s=c.r ɣ

Image Enhancement by Contrast Stretching

Image Enhancement by Gray level mapping

HİSTOGRAM PROCESSİNG: H(rk)=nk rk: kth gray level, nk: number of pixels with gray value rk HİSTOGRAM PROCESSİNG: H(rk)=nk rk: kth gray level, nk: number of pixels with gray value rk

Histogram Equalization Goal: Find a transformation which yields a histogram with uniform density Histogram Equalization Goal: Find a transformation which yields a histogram with uniform density ?

Algorithm: Histogram Equalization Create an array h with L gray values –Initialize with o value Find the histogram h(r k )= h(r k )+1 Find the cumulative histogram hc(r k )= hc(r k-1 )+ hc(r k ) Set T(r k-1 ) =round [{(L-1)/NM}. hc(r k-1 )] Create the equalized image, s k = T(r k )

Histogram Equalization

Equalized Histogram

Histogram Specification

Histogram Modification

Histogram of a dark image

Histogram Equalization

Specified Histogram

Local Histogram Equalization

Image Subtraction

Convolution or crosscorrelation

Position Dependent Gray Level Mapping Use convolution or correlation: f*h Position Dependent Gray Level Mapping Use convolution or correlation: f*h

Define a mask and correlate it with the image

SMOOTHING

Image Enhancement WITH SMOOTING

Averaging blurrs the image

Image Enhancement WITH AVERAGING AND THRESHOLDING Image Enhancement WITH AVERAGING AND THRESHOLDING

Restricted Averaging Apply averaging to only pixels with brightness value outside a predefined interval. Mask h(i,j) = 1For g(m+i,n+j)€ [min, max] 0 otherwise Q: Study edge strenght smoothing, inverse gradient and rotating mask

Median Filtering Find a median value of a given neighborhood. Removes sand like noise

Median filtering breaks the straight lines Square filter: Cross filter

Image Enhancement with averaging and median filtering

EDGE PROFILES Edges are the pixels where the brightness changes abrubtly. It is a vector variable with magnitude and direction

EDGES, GRADIENT AND LAPLACIAN

SMOOT EDGES, NOISY EDGES

Continuous world Gradient Δg(x,y) = ∂g/ ∂x + ∂g/ ∂y Magnitude: |Δg(x,y) | = √ (∂g/ ∂x) 2 + (∂g/ ∂y) 2 Phase : Ψ = arg (∂g/ ∂x, ∂g/ ∂y) radians

Discrete world Use difference in various directions Δi g(i,j) = g(i,j) - g(i+1,j) or Δj g(i,j) = g(i,j) - g(i,j+1) or Δij g(i,j) = g(i,j)- g(i+1,j+1) or |Δ g(i,j) | = |g(i,j)- g(i+1,j+1) | + |g(i,j+1)- g(i+1,j) |

GRADIENT EDGE MASKS Approximation in discrete grid GRADIENT EDGE MASKS Approximation in discrete grid

GRADIENT EDGE MASKS

GRADİENT MASKS

Edge Detection

GRADIENT OPERATIONS

EDGES, GRADIENT AND LAPLACIAN

Edg Detection with Laplacian

Gaussian Masks

L.O.G LAPLACIAN of GAUSSIAN EDGE MASKS

Laplacian Operator

EDGE DETECTION by L.O.G

Image Enhancement WITH LAPLACIAN AND SOBEL

Image Enhancement (cont.)

Edge Detection with High Boost

Image Enhancement with Laplacian

Marr Hildreth Theory L.L HVS constructs primal sketch based on edges, lines and blobs Therefore L.o.G filters are mathematical representation of HVS at low level

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

Frei-Chen Basis: 3x3 orthogonal

Cauchy Schwartz Inequality  U+V  ≤  U  +  V 

Schwartz Inequality

Quotient: Angle Between two images

Fourier Analysis

Fourier Transform Pair Given image I(x,y), its fourier transform is

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Fourier Transform of an Image is a complex matrix Let F =[F(u,v)] F = Φ MM I(x,y) Φ NN I(x,y)= Φ* MM F Φ* MM Where Φ JJ (k,l)= [Φ JJ (k,l) ] and Φ JJ (k,l) = (1/J) exp(2Πjkl/J) for k,l= 0,…,J-1

Fourier Transform

Properties Convolution Given the FT pair of an image I(x,y) F(u,v) I(x,y)* m(x,y) F(u,v). H(u,v) and I(x,y) m(x,y) F(u,v)* H(u,v)

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Design of H(u,v) Low Pass filter H(u,v) = 1 if |u,v |< r 0 o.w. High pass filter H(u,v) = 1 if |u,v |> r 0 o.w Band pass filter H(u,v) = 1 if r1<|u,v |< r2 0 o.w

Fourier Transform-High Pas Filtering

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Chapter 4 Image Enhancement in the Frequency Domain Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Chapter 4 Image Enhancement in the Frequency Domain Chapter 4 Image Enhancement in the Frequency Domain

Chapter 4 Image Enhancement in the Frequency Domain Chapter 4 Image Enhancement in the Frequency Domain

Chapter 4 Image Enhancement in the Frequency Domain Chapter 4 Image Enhancement in the Frequency Domain

Chapter 4 Image Enhancement in the Frequency Domain Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Spatial Laplacian Masks and its Fourier Transform

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Chapter 4 Image Enhancement in the Frequency Domain Chapter 4 Image Enhancement in the Frequency Domain

Chapter 4 Image Enhancement in the Frequency Domain Chapter 4 Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain

Image Enhancement in the Frequency Domain Image Enhancement in the Frequency Domain