= α α ( ) ( ) thr. thr. min. Image quantization and false contours. Why 256 quantization levels? min α. min. max. min. min Φ Φ. max. 4. D α α.

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1 L. Yaroslavsky. Course 5.72 Digital Image Processing: Applications Lect. 3. Image Quantization in Image and Transform Domains Optimal element-ise uantization. ε ( ( ˆ ; E I p( D( ε d; ( ( E L { } ( ( p( D( ε d; E R p( D( ε Q d; ( ( Q ( ( D( ˆ Q 2 ( ( Max-Lloyd uantizer: ˆ, arg p D ε ; ˆ d D { ˆ, } ( Compander-expander uantizer. ( ( Q Modified uantization uality criterion: ( ( ˆ ( E I p D d p( D( Δ u / ' d; ( ( ( Q Optimal predistortion function opt ( arg p( D( Δ u / ' d; ( ( are found from Euler-Lagrange euation: { ( ( ( p D Δ u / ' ' } const Examples: ( ( +, < Δ + thr. Threshold criterion: D( uniform uantization +, > Δ thr 2. Threshold criterion: ( ( + (, < Δ δ + thr ( ( ln( / D ; Q ( ln( + / / δ, > Δthr δ ( ( ln( / Image uantization and false contours. Why 256 uantization levels? + ( ( + 2n ( r ( r r r 3. D When n, For p( cexp ( ( ( ( ( 2 2σ 2 ( r ( r r r+ 4. D /, ( ( ( ( ( p( ( p( ( ( ( ( + ( 2n 3 / d 3 / d / ( p / ( p d d σ Φ Φ 3 3σ σ Φ Φ 3 3σ ( ( ( ( 2n / ( p /, here Φ 2n / ( p / d / When p ( const, / / / Practical aspects of element-ise uantization. P-th la uantization: ( sign P d x 2 x exp dx 2π 2. Quantization in the presence of noise. Quantization and correcting image dynamic range. Quantization of holograms. Speckle noise in coherent radiation based imaging systems. Quantization in computer tomography Problems for self-exaation. What role loss function and signal probability density play in uantization optimization 2. Compare Max-Lloyd and compander-expander uantization. What is P-th la uantization? 3. Why 256 uantization levels in logarithmic scale ere selected for image uantization? 4. Describe uantization artefacts for uantization in transform domain. 5. Ho additive noise affects uantization?

2 The principle of element-ise uantization E I ( ( ( D( ε d; p Probability distribution function + ˆ ˆ E L ( ˆ p E p( D( ε d; ( D( ε d; R Q Max-Lloyd uantizer: ( ( { ˆ, } ˆ ( ( ( arg p D ε {, } ( Compander-expander uantizer. ( ( ( ( d ; D( ˆ D( ˆ Compressing nonlinear ˆ Uniform uantization transform Restoration: Uniformly uantized interval representatives Expanding nonlinear ˆ transform Optimal predistortion function ( opt ( Q arg p( D( Δ u / ' d; ( (

3 Element-ise uantization: compander-expander method Quantization: Compressing nonlinear ˆ Uniform uantization transform ( Δ u -.5 (.5 Δ Compressing transformation Uniform uantizer Restoration: Input values Quantization interval indices Uniformly uantized interval representatives Expanding nonlinear ˆ transform (..5 Restored signal values Uniformly uantized values Expanding transformation -.5 Conversion of uantization interval indices into uniformly uantized values

uantization levels Quantization error: StDev.

Stved3 8 uantization levels Quantization

4 Test object Background Stimulus relative

4 False contours and other uantization artefacts 256 uantization levels 32 uantization levels Quantization error: StDev.46 6 uantization levels Quantization error: Stved3 8 uantization levels Quantization error: (St. Dev 6.4 Test object Background Stimulus relative contrast Stimulus Brightness of the background Weber-Fechner la of eye sensitivity to object s contrast

Quantization ith P-th la nonlinearity Initial ECG signal Uniform uantization: 64 levels Uniform uantization: 32 levels Initial ECG signal P-th lo uantization: 64

3 Uniform and non-uniform (P-th la uantization of orthogonal components of image Fourier spectra Initial image Image restored after uniform uantization (3 levels

5 Quantization ith P-th la nonlinearity Initial ECG signal Uniform uantization: 64 levels Uniform uantization: 32 levels Initial ECG signal P-th lo uantization: 64 levels,p.3 P-th lo uantization: 32 levels,p.3 Uniform and non-uniform (P-th la uantization of orthogonal components of image Fourier spectra Initial image Image restored after uniform uantization (3 levels of its spectrum orthogonal components Pth lo uantization of image spectrum 3 Standard deviation of uantization error Nonlinearity index P Image restored after optimal P-th la uantization of its spectrum orthogonal components (P.2, 3 uantization levels

6 Quantization ith noise Image uantized to 6 levels. Observe false contours. Image ith additive noise (std4 uantized to 6 levels Correcting image contrast and uantization ith noise Contrast correcting look-up table and course uantization effects 2 Lo contrast image Contrast corrected image and false contours Destroying false contours by uantization ith additive noise

Speckle noise phenomena in imaging by means of coherent radiation Limitation of the dynamic range of the hologram orthogonal components Quantization of the hologram orthogonal components

2 /8 GrLv 8/8 7/8 6/8 5/8 4/8 3/8 2/8 /8 Standard deviation of speckle versus the threshold parameter of limitation of the ave front orthogonal components dynamic range 2 3 4 5 6

7 Speckle noise phenomena in imaging by means of coherent radiation Limitation of the dynamic range of the hologram orthogonal components Quantization of the hologram orthogonal components Reconstructed images for different degree of limitation of the ave front orthogonal components dynamic range Reconstructed images for different number of uantization levels (64 to 8, from top to bottom /8 GrLv 8/8 7/8 6/8 5/8 4/8 3/8 2/8 /8 Standard deviation of speckle versus the threshold parameter of limitation of the ave front orthogonal components dynamic range Standard deviation of speckle noise as a function of the number of uantization levels Q Orth. component uantization: Speckle contrast for P-th la uantization.2 P Number of uantization levels Speckle contrast in images reconstructed from a hologram of a diffusely reflecting object versus number of uantization levels in P-th la uantizing hologram orthogonal components for different values of nonlinearity index P. One can see that speckle contrast due to the uantization is imized hen P is about.3-,5.

Locality function LOC(.) and loss function LOSS (.) and a priori knowledge on images inp

Locality function LOC(.) and loss function LOSS (.) and a priori knowledge on images inp L. Yaroslavsky. Course 0510.7211 Digital Image Processing: Applications Lecture 12. Nonlinear filters for image restoration and enhancement Local criteria of image quality: AVLOSS( k, l) = AV LOC( m, n;