EE465: Introduction to Digital Image Processing Image Deblurring Introduction Inverse filtering Suffer from noise amplification Wiener filtering Tradeoff between image recovery and noise suppression Iterative deblurring* Landweber algorithm EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Where does blur come from? Optical blur: camera is out-of-focus Motion blur: camera or object is moving Why do we need deblurring? Visually annoying Wrong target for compression Bad for analysis Numerous applications EE465: Introduction to Digital Image Processing

Application (I): Astronomical Imaging The Story of Hubble Space Telescope (HST) HST Cost at Launch (1990): $1.5 billion Main mirror imperfections due to human errors Got repaired in 1993 EE465: Introduction to Digital Image Processing

Restoration of HST Images EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Another Example EE465: Introduction to Digital Image Processing

The Real (Optical) Solution Before the repair After the repair EE465: Introduction to Digital Image Processing

Application (II): Law Enforcement Motion-blurred license plate image EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Restoration Example EE465: Introduction to Digital Image Processing

Application into Biometrics out-of-focus iris image EE465: Introduction to Digital Image Processing

Modeling Blurring Process • Linear degradation model y(m,n) x(m,n) h(m,n) + blurring filter additive white Gaussian noise EE465: Introduction to Digital Image Processing

Blurring Filter Example FT Gaussian filter can be used to approximate out-of-focus blur EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Blurring Filter Example (Con’t) FT MATLAB code: h=FSPECIAL('motion',9,30); Motion blurring can be approximated by 1D low-pass filter along the moving direction EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing The Curse of Noise z(m,n) y(m,n) x(m,n) h(m,n) + Blurring SNR EE465: Introduction to Digital Image Processing

Image Example BSNR=10dB BSNR=40dB x(m,n) BSNR=10dB BSNR=40dB h(m,n): 1D horizontal motion blurring [1 1 1 1 1 1 1]/7 EE465: Introduction to Digital Image Processing

Blind vs. Nonblind Deblurring Blind deblurring (deconvolution): blurring kernel h(m,n) is unknown Nonblind deconvolution: blurring kernel h(m,n) is known In this course, we only cover the nonblind case (the easier case) EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Image Deblurring Introduction Inverse filtering Suffer from noise amplification Wiener filtering Tradeoff between image recovery and noise suppression Iterative deblurring* Landweber algorithm EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Inverse Filter x(m,n) h(m,n) y(m,n) hI(m,n) x(m,n) ^ blurring filter inverse filter hcombi (m,n) To compensate the blurring, we require EE465: Introduction to Digital Image Processing

Inverse Filtering (Con’t) x(m,n) h(m,n) + y(m,n) hI(m,n) ^ x(m,n) inverse filter Spatial: Frequency: amplified noise EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Image Example motion blurred image at BSNR of 40dB deblurred image after inverse filtering Q: Why does the amplified noise look so bad? A: zeros in H(w1,w2) correspond to poles in HI (w1,w2) EE465: Introduction to Digital Image Processing

Pseudo-inverse Filter Basic idea: To handle zeros in H(w1,w2), we treat them separately when performing the inverse filtering EE465: Introduction to Digital Image Processing

Image Example motion blurred image deblurred image after at BSNR of 40dB deblurred image after Pseudo-inverse filtering (=0.1) EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Image Deblurring Introduction Inverse filtering Suffer from noise amplification Wiener filtering Tradeoff between image recovery and noise suppression Iterative deblurring* Landweber algorithm EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Norbert Wiener (1894-1964) The renowned MIT professor Norbert Wiener was famed for his absent-mindedness. While crossing the MIT campus one day, he was stopped by a student with a mathematical problem. The perplexing question answered, Norbert followed with one of his own: "In which direction was I walking when you stopped me?" he asked, prompting an answer from the curious student. "Ah," Wiener declared, "then I've had my lunch” Anecdote of Norbert Wiener EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Wiener Filtering Also called Minimum Mean Square Error (MMSE) or Least-Square (LS) filtering constant noise energy Example choice of K: signal energy K=0  inverse filtering EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Image Example motion blurred image at BSNR of 40dB deblurred image after wiener filtering (K=0.01) EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Image Example (Con’t) K=0.1 K=0.01 K=0.001 EE465: Introduction to Digital Image Processing

Constrained Least Square Filtering Similar to Wiener but a different way of balancing the tradeoff between Example choice of C: Laplacian operator =0  inverse filtering EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Image Example =0.1  =0.01  =0.001 EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Image Deblurring Introduction Inverse filtering Suffer from noise amplification Wiener filtering Tradeoff between image recovery and noise suppression Iterative deblurring* Landweber algorithm EE465: Introduction to Digital Image Processing

Method of Successive Substitution A powerful technique for finding the roots of any function f(x) Basic idea Rewrite f(x)=0 into an equivalent equation x=g(x) (x is called fixed point of g(x)) Successive substitution: xi+1=g(xi) Under certain condition, the iteration will converge to the desired solution EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Numerical Example Two roots: successive substitution: EE465: Introduction to Digital Image Processing

Numerical Example (Con’t) Note that iteration quickly converges to x=1 EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Landweber Iteration Linear blurring We want to find the root of relaxation parameter – controls convergence property Successive substitution: EE465: Introduction to Digital Image Processing

EE465: Introduction to Digital Image Processing Asymptotic Analysis Assume convergence condition: we have inverse filtering EE465: Introduction to Digital Image Processing

Advantages of Landweber Iteration No inverse operation (e.g., division) is involved We can stop the iteration in the middle way to avoid noise amplification It facilitates the incorporation of a priori knowledge about the signal (X) into solution algorithm More detailed analysis is included in EE565: Advanced Image Processing EE465: Introduction to Digital Image Processing