Image Processing in Freq. Domain Restoration / Enhancement Inverse Filtering Match Filtering / Pattern Detection Tomography

Enhancement v.s. Restoration Image Enhancement: –A process which aims to improve bad images so they will “ look ” better. Image Restoration: –A process which aims to invert known degradation operations applied to images.

Enhancement vs. Restoration “Better” visual representation Subjective No quantitative measures Remove effects of sensing environment Objective Mathematical, model dependent quantitative measures

Typical Degradation Sources Low Illumination Atmospheric attenuation (haze, turbulence, … ) Optical distortions (geometric, blurring) Sensor distortion (quantization, sampling, sensor noise, spectral sensitivity, de-mosaicing)

Image Preprocessing Enhancement Restoration Spatial Domain Freq. Domain Point operations Spatial operations Filtering Denoising Inverse filtering Wiener filtering

Examples Hazing

Echo image Motion Blur

Blurred image Blurred image + additive white noise

Reconstruction as an Inverse Problem Distortion H noise measurements Original image Reconstruction Algorithm

Typically: –The distortion H is singular or ill-posed. –The noise n is unknown, only its statistical properties can be learnt. So what is the problem?

Key point: Stat. Prior of Natural Images MAP estimation: likelihoodprior

Image space measurements From amongst all possible solutions, choose the one that maximizes the a-posteriori probability: Bayesian Reconstruction (MAP) P(g|f) P(f) Most probable solution P(f | g)  P(g | f) P(f)

Bayesian Denoising Assume an additive noise model : g=f + n A MAP estimate for the original f: Using Bayes rule and taking the log likelihood :

Bayesian Denoising If noise component is white Gaussian distributed: g=f + n where n is distributed ~N(0,  ) R(f) is a penalty for non probable f data term prior term

Inverse Filtering Degradation model: g(x,y) = h(x,y)*f(x,y) G(u,v)=H(u,v)  F(u,v) F(u,v)=G(u,v)/H(u,v) 

Inverse Filtering (Cont.) Two problems with the above formulation: 1.H(u,v) might be zero for some (u,v). 2.In the presence of noise the noise might be amplified: F(u,v)=G(u,v)/H(u,v) + N(u,v)/H(u,v) Solution: Use prior information  data term prior term

Option 1: Prior Term Use penalty term that restrains high F values: where Solution:

Degraded Image (echo)

F=G/H 

Degraded Image (echo+noise)

The inverse filter is C(H)= H * /(H * H+ ) At some range of (u,v): S(u,v)/N(u,v) < 1 noise amplification. =10 -3

Option 2: Prior Term 1.Natural images tend to have low energy at high frequencies 2.White noise tend to have constant energy along freq. where

Solution: This solution is known as the Wienner Filter Here we assume N(u,v) is constant. If N(u,v) is not constant:

Degraded Image (echo+noise)

Wienner Filtering

Wienner Previous

Degraded Image (blurred+noise)

Inverse Filtering

Using Prior (Option 1)

Wienner Filtering

Matched Filter in Freq. Domain Pattern Matching: –Finding occurrences of a particular pattern in an image. Pattern: –Typically a 2D image fragment. –Much smaller than the image.

Image Similarity Measure: –A function that assigns a nonnegative real value to two given images. –Small measure  high similarity –Preferable to be a metric distance ( non-negative, identity, symmetric, triangular inequality) Can be combined with thresholding: Image Similarity Measures d( - ) ≥ 0

Scan the entire image pixel by pixel. For each pixel, evaluate the similarity between its local neighborhood and the pattern. The Matching Approach

Given: –k×k pattern P –n×n image I –kxk window of image I located at x,y - I x,y For each pixel (x,y), we compute the distance: Complexity O(n 2 k 2 ) The Euclidean Distance as a Similarity Measure

Convolution can be applied rapidly using FFT. Complexity: O(n 2 log n) FFT Implementation Fixed

NaïveFFT Time Complexity Space Integer ArithmeticYesNo Run time for 16× Sec.3.5 Sec. Run time for 32× Sec.3.5 Sec. Run time for 64× Sec.3.5 Sec. Performance table for a 1024×1024 image, on a 1.8 GHz PC: Naïve vs. FFT

NCC: –A similarity measure, based on a normalized cross- correlation function. –Maps two given images to [0,1] (absolute value). –Measures the angle between vectors I x,y and P –Invariant to intensity scale and offset. Normalized Cross Correlation

Note that Thus, The above expression can be implemented efficiently using 3 convolutions. Efficient Implementation

Euclidean distance similarity measure NCC similarity measure

Euclidean distance similarity measure NCC similarity measure

Computer Tomography using FFT

In 1901 W.C. Roentgen won the Nobel Prize (1st in physics) for his discovery of X-rays. CT Scanners Wilhelm Conrad Röntgen

In 1979 G. Hounsfield & A. Cormack, won the Nobel Prize for developing the computer tomography. The invention revolutionized medical imaging. CT Scanners Allan M. Cormack Godfrey N. Hounsfield

f(x,y) 11 22 Tomography: Reconstruction from Projection

Projection: All ray-sums in a direction Sinogram: collects all projections Projection & Sinogram P(  t) f(x,y) t  y x X-rays Sinogram t  

CT Image & Its Sinogram K. Thomenius & B. Roysam

The Slice Theorem spatial domainfrequency domain f(x,y) 11 x y 11 u v Fourier Transform

The Slice Theorem f(x,y) = object g(x) = projection of f(x,y) parallel to the y-axis: g(x) =  f(x,y)dy F(u,v) =   f(x,y) e -2  i(ux+vy) dxdy Fourier Transform of f(x,y): Fourier Transform at v=0 : F(u,0) =   f(x,y) e -2  iux dxdy =  [  f(x,y)dy ] e -2  iux dx =  g(x) e -2  iux dx = G(u) The 1D Fourier Transform of g(x)

Interpolate (linear, quadratic etc) in the frequency space to obtain all values in F(u,v). Perform Inverse Fourier Transform to obtain the image f(x,y). Interpolation Method u v F(u,v)

THE END