Enhancing Images Ch 5:Shapiro, Ch 3:Gonzales. Gray level Mapping Brightness Transform: 1. Position...

Enhancing Images Ch 5:Shapiro, Ch 3:Gonzales

Gray level Mapping Gray level Mapping

Brightness Transform:

1. Position Dependent

f(i,j)= g(i,j). e(i,j)g:Clean imagee:position dependent noise

2. Position independent

2. Position Independent Gray Level Mappings=T(r)

2. Position Independent Gray Level Mappings=T(r)

NegationNegation

2. Gamma Transformations=T(r)

2. Gamma Transformations=T(r)

Gamma Correction of CRTGamma Correction of CRT

Image Enhancement by Gamma Transform: s=c.rɣImage Enhancement by Gamma Transform: s=c.rɣ

Image Enhancement by Gray level mapping: s=c.rɣImage Enhancement by Gray level mapping: s=c.rɣ

Image Enhancement by Contrast StretchingImage Enhancement by Contrast Stretching

Image Enhancement by Gray level mappingImage Enhancement by Gray level mapping

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

HİSTOGRAM PROCESSİNG:H(rk)=nkrk: 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(rk)= h(rk)+1

• Find the cumulative histogram

hc(rk)= hc(rk-1)+ hc(rk)

• Set T(rk-1) =round [{(L-1)/NM}. hc(rk-1)]

• Create the equalized image, sk= T(rk)

Histogram EqualizationHistogram Equalization

Equalized HistogramEqualized Histogram

Histogram Specification

Histogram Modification

Histogram of a dark imageHistogram of a dark image

Histogram EqualizationHistogram Equalization

Specified HistogramSpecified Histogram

Local Histogram EqualizationLocal Histogram Equalization

Image Subtraction

Convolution or crosscorrelation

Position Dependent Gray Level MappingUse convolution or correlation: f*h

Position Dependent Gray Level MappingUse convolution or correlation: f*h

Define a mask and correlate it with the imageDefine a mask and correlate it with the image

SMOOTHINGSMOOTHING

Image Enhancement WITH SMOOTINGImage Enhancement WITH SMOOTING

Averaging blurrs the imageAveraging 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) = 1 For 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

0 2 1

2 1 2

3 3 2

0 2 1

2 2 2

3 3 2

0 1 1 2 2 2 2 3 3

Median filtering breaks the straight lines

5 5 5 5 5

5 5 5 5 5

0 0 0 0 0

5 5 5 5 5

5 5 5 5 5

Square filter:0 0 0 5 5 5 5 5 5

Cross filter0 0 0 5 5

Image Enhancement with averaging and median filteringImage Enhancement with averaging and median filtering

EDGE PROFILES EDGE PROFILES

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

EDGES, GRADIENT AND LAPLACIAN EDGES, GRADIENT AND LAPLACIAN

SMOOT EDGES, NOISY EDGESSMOOT 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 MASKSApproximation in discrete grid

GRADIENT EDGE MASKSApproximation in discrete grid

GRADIENT EDGE MASKSGRADIENT EDGE MASKS

GRADİENT MASKSGRADİENT MASKS




Edge DetectionEdge Detection

Edge DetectionEdge Detection

GRADIENT OPERATIONSGRADIENT OPERATIONS

EDGES, GRADIENT AND LAPLACIAN EDGES, GRADIENT AND LAPLACIAN

Edg Detection with LaplacianEdg Detection with Laplacian

Gaussian Masks

L.O.G LAPLACIAN of GAUSSIAN EDGE MASKSL.O.G LAPLACIAN of GAUSSIAN EDGE MASKS

Laplacian OperatorLaplacian Operator

EDGE DETECTION by L.O.GEDGE DETECTION by L.O.G

Image Enhancement WITH LAPLACIAN AND SOBELImage Enhancement WITH LAPLACIAN AND SOBEL

Image Enhancement (cont.)Image Enhancement (cont.)

Edge Detection with High BoostEdge Detection with High Boost

Image Enhancement with LaplacianImage 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 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


Fourier Transform 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

• 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)



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















Chapter 4Image Enhancement in the

Frequency Domain


Frequency Domain












Frequency Domain


Frequency Domain


Frequency Domain


Frequency Domain


Frequency Domain


Frequency Domain


Frequency Domain


Frequency Domain



Spatial Laplacian Masks and its Fourier Transform




Frequency Domain


Frequency Domain


Frequency Domain


Frequency Domain









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Enhancing Images Ch 5:Shapiro, Ch 3:Gonzales. Gray level Mapping Brightness Transform: 1. Position...

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