Transform Coding III
s1 u1 cos φ sin φ − sin φ cos φ
s0 u0 Signal-Independent Unitary Transforms Optimal Transforms & Adaptation
Optimal Unitary Transform Stationary Gaussian sources: KLT General sources: Not straightforward to determine Signal dependent (may change due to signal instationarities)
Backward Adaptive Transform Estimate transform in encoder and decoder based on reconstructed samples Difficult, not often used in practice
Forward Adaptive Transform Determine transform in encoder, include transform specification in bitstream Increased side information Simple variant: Switched transforms
Signal-Independent Transforms Choose transform that provides good performance for variety of signals Not optimal, but often close to optimal for typical signals
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 2 / 38 Signal-Independent Unitary Transforms / Walsh-Hadamard Transform Walsh-Hadamard Transform
For transform sizes N that are positive integer powers of 2 1 AN/2 AN/2 h i AN = √ with A1 = 1 . (1) 2 AN/2 −AN/2
Example: Transform matrix for N = 8
1 1 1 1 1 1 1 1 1 −1 1 −1 1 −1 1 −1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 1 A = √ · (2) 8 8 1 1 1 1 −1 −1 −1 −1 1 −1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1 −1 1 1 −1
Very simple orthogonal transform (only additions & final scaling) Image & video coding: Piece-wise constant basis vectors yield subjectively disturbing artifacts when combined with strong quantization
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 3 / 38 Signal-Independent Unitary Transforms / Discrete Fourier Transform Discrete Fourier Transform (DFT)
Fourier Transform Integral transform representing a signal as integral of frequency components Forward and inverse transforms are given by
∞ ∞ Z Z X (f ) = x(t) · e−2πift dt ⇐⇒ x(t) = X (f ) · e2πift df (3) −∞ −∞
Discrete Fourier Transform Unitary transform (specified by unitary matrix) Discrete version of Fourier transform Input: Discrete signal with finite number N of samples Output: N Fourier coefficients (complex values) Can be derived from Fourier transform using sampling (see illustrations on following slides)
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 4 / 38 Signal-Independent Unitary Transforms / Discrete Fourier Transform Discrete Fourier Transform (DFT)
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 5 / 38 Signal-Independent Unitary Transforms / Discrete Fourier Transform Discrete Fourier Transform (DFT)
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 6 / 38 Signal-Independent Unitary Transforms / Discrete Fourier Transform Discrete Fourier Transform (DFT)
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 7 / 38 Signal-Independent Unitary Transforms / Discrete Fourier Transform Discrete Fourier Transform (DFT)
Discrete Fourier Transform Forward Transform N−1 1 X −i 2πkn u[k] = √ s[n] · e N (4) N n=0 Inverse Transform N−1 1 X i 2πkn s[n] = √ u[k] · e N (5) N k=0
Properties Unitary transform that produces complex transform coefficients For real inputs, it obeys the symmetry u[k] = u∗[N − k] N real samples are mapped onto N real values Fast algorithm: Fast Fourier transform (FFT)
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 8 / 38 Signal-Independent Unitary Transforms / Discrete Fourier Transform Disadvantage of DFT for Transform Coding
Sampling of frequency spectrum causes implicit periodic signal extension Will often observe rather large difference between left and right signal boundary Large difference reduces rate of convergence of Fourier series Strong quantization yields significant high-frequency artifacts
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 9 / 38 Signal-Independent Unitary Transforms / Discrete Trigonometric Transforms Overcome DFT Disadvantage: Discrete Cosine Transform
DFT:
implicit signal replica signal implicit signal replica
DCT-II: DFT
implicit signal replica signal mirrored signal implicit signal replica
Idea of Discrete Cosine Transform (DCT) Introduce mirror symmetry (different possibilities, example: DCT-II) Apply DFT of approximately double size (or four times the size) No discontinuities in periodic signal extension Ensure symmetry around zero: Only cosine terms
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 10 / 38 Signal-Independent Unitary Transforms / Discrete Trigonometric Transforms Discrete Trigonometric Transforms (DTTs)
Discrete Cosine Transforms (DCTs) Introduce mirror symmetry around zero and apply DFT of larger size Imaginary sine terms get eliminated Only cosine terms remain 8 possibilities: DCT-I to DCT-VIII 2 cases for left side: Whole sample or half-sample symmetry 4 cases for right side: Whole sample or half-sample symmetry or whole sample or half-sample anti-symmetry Most relevant case: DCT-II (half-sample symmetry at both sides)
Discrete Sine Transforms (DSTs) Introduce anti-symmetry around zero and apply DFT of larger size Real cosine terms get eliminated Only imaginary sine terms remain Similarly as for DCT: 8 possibilities (DST-I to DST-VIII)
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 11 / 38 Signal-Independent Unitary Transforms / Discrete Trigonometric Transforms The Discrete Cosine Transform (DCT) Family
DCT-I DFT of size 2N − 2 DCT-V DFT of size 2N − 1
DCT-II DFT of size 2N DCT-VI DFT of size 2N − 1
DCT-III DFT of size 4N DCT-VII DFT of size 4N − 2
DCT-IV DFT of size 4N DCT-VIII DFT of size 4N + 2
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 12 / 38 Signal-Independent Unitary Transforms / Discrete Trigonometric Transforms The Discrete Sine Transform (DST) Family
DST-I DFT of size 2N + 2 DST-V DFT of size 2N + 1
DST-II DFT of size 2N DST-VI DFT of size 2N + 1
DST-III DFT of size 4N DST-VII DFT of size 4N + 2
DST-IV DFT of size 4N DST-VIII DFT of size 4N − 2
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 13 / 38 Signal-Independent Unitary Transforms / Discrete Cosine Transform of Type II Derivation of Discrete Cosine Transform Type II (DCT-II)
DFT
implicit signal replica signal mirrored signal implicit signal replica
Specification signal for applying DFT Given: Discrete signal s[n] of size N (i.e., 0 ≤ n < N) Mirror signal with sample repetition at both sides (size 2N) s[n]: 0 ≤ n < N sm[n] = s[2N − n − 1]: N ≤ n < 2N Ensure symmetry around zero by adding half-sample shift s[n − 1/2]: 0 ≤ n < N s+[n] = sm[n − 1/2] = s[2N − n − 3/2]: N ≤ n < 2N
Apply DFT of size 2N to new signal s+[n]
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 14 / 38 Signal-Independent Unitary Transforms / Discrete Cosine Transform of Type II Derivation of Discrete Cosine Transform Type II (DCT-II)
s[n − 1/2]: 0 ≤ n < N s+[n] = s[2N − n − 3/2]: N ≤ n < 2N
DFT of size 2N for extended signal s+[n]
( N)− 2 1 + 1 X −i 2πkn s only known at half-sample u+[k] = s+[n] · e (2N) p(2 ) positions → use m = n − 1/2 N n=0 2N−1 1 X + 1 −i πk m+ 1 = √ s m + · e N ( 2 ) 2 2 N m=0 N−1 2N−1 ! 1 X −i πk n+ 1 X −i πk m+ 1 = √ s[n] · e N ( 2 ) + s[2N − m − 1] · e N ( 2 ) 2 N n=0 m=N yn=2N−m−1
N−1 N−1 ! 1 X −i πk n+ 1 X −i πk N−n− 1 = √ s[n] · e N ( 2 ) + s[n] · e N (2 2 ) 2 N n=0 n=0
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 15 / 38 Signal-Independent Unitary Transforms / Discrete Cosine Transform of Type II Derivation of Discrete Cosine Transform Type II (DCT-II)
Continue derivation
N−1 N−1 ! + 1 X −i πk n+ 1 X −i πk N−n− 1 u [k] = √ s[n] · e N ( 2 ) + s[n] · e N (2 2 ) 2 N n=0 n=0 N−1 N−1 1 X −i πk (n+ 1 ) X −i2πk i πk (n+ 1 ) = √ s[n] · e N 2 + s[n] · e · e N 2 2N | {z } n=0 n=0 1
N−1 1 X −i πk n+ 1 i πk n+ 1 = √ s[n] · e N ( 2 ) + e N ( 2 ) 2N n=0 | {z } πk 1 2 cos( N (n+ 2 ))
DFT of extended signal
r N−1 2 X π 1 u+[k] = · s[n] · cos k n + (6) N N 2 n=0
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 16 / 38 Signal-Independent Unitary Transforms / Discrete Cosine Transform of Type II Derivation of Discrete Cosine Transform Type II (DCT-II)
DFT of extended signal (2N samples) has 2N transform coefficients
r N−1 2 X π 1 k = 0,..., 2N − 1 : u+[k] = · s[n] · cos k n + N N 2 n=0
1 Signal s[n] is completely described by first N transform coefficients
r N−1 2 X π 1 k = 0,..., N − 1 : u+[k] = · s[n] · cos k n + N N 2 n=0
2 Basis functions of derived transform are orthogonal to each other, but they don’t have the same norm
Introduce factors αk so that transform matrix becomes orthogonal
N−1 X π 1 k = 0,..., N − 1 : u[k] = αk · s[n] · cos k n + N 2 n=0
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 17 / 38 Signal-Independent Unitary Transforms / Discrete Cosine Transform of Type II Discrete Cosine Transform of Type II (DCT-II)
Specification of DCT-II Forward transform (DCT-II) and inverse transform (IDCT-II) are given by
N−1 X π 1 u[k] = αk s[n] · cos k n + (7) N 2 n=0
N−1 X π 1 s[n] = αk · u[k] · cos k n + (8) N 2 k=0 with scaling factors ( p 1/N : k = 0 αk = p (9) 2/N : k 6= 0
The orthogonal transform matrix A = {akn} has the elements
π 1 akn = αk · cos k n + N 2
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 18 / 38 Signal-Independent Unitary Transforms / Discrete Cosine Transform of Type II Basis Functions of DCT-II (Example for N = 8)
π 1 b [n] = α · cos k n + k k 8 2
b0 b4
b1 b5
b2 b6
b3 b7
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 19 / 38 Signal-Independent Unitary Transforms / Discrete Cosine Transform of Type II Comparison of DCT-II and KLT
Autocovariance matrix of a first-order Gauss-Markov / AR(1) process
1 %%2 ··· %N−1 % 1 % ··· %N−2 N− 2 %2 % 1 ··· % 2 C SS = σS · (10) ...... . . . . . %N−1 %N−2 %N−3 ··· 1
DCT-II is a good approximation of the eigenvectors of C SS for large %
Unit-norm eigenvectors of C SS approach DCT-II basis vectors for % → 1
Advantages of DCT-II in comparison to KLT DCT-II does not depend on input signal Fast algorithms for computing forward and inverse transform
Justification for wide usage of DCT
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 20 / 38 Signal-Independent Unitary Transforms / Discrete Cosine Transform of Type II KLT Convergence Towards DCT for % → 1
KLT, % = 0.9 DCT-II 0.5 0.5
0 0 Difference between the transform
−0.5 −0.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 matrices of KLT and DCT-II 0.5 0.5
0 0
−0.5 −0.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 2 0.5 0.5 δ(%) = kAKLT(%) − ADCTk2 0 0
−0.5 −0.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0.5 0.5 0 0 0.4 −0.5 −0.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 δ(%) 0.5 0.5 0 0 0.3 −0.5 −0.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0.5 0.5 0 0 0.2 −0.5 −0.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0.5 0.5 0 0 0.1 −0.5 −0.5 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7 0.5 0.5 0 0 0 0.4 0.5 0.6 0.7 0.8 0.9 1 −0.5 −0.5 % 0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 21 / 38 Signal-Independent Unitary Transforms / 2D Transforms Image & Video Coding: 2D Transforms
Separable Transforms Successive 1D transforms of rows and columns of image block Separable orthogonal forward transform
u = A · s · AT (11)
with s — N ×N block of image samples A — N ×N transform matrix u — N ×N block of transform coefficients Inverse transform (separable orthogonal transform)
s0 = AT · u0 · A (12)
Great practical importance: Two matrix multiplications of size N ×N instead of one multiplication of a vector of size 1×N2 with a matrix of size N2 ×N2 Complexity reduction from O(N4) to O(N3)
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 22 / 38 Signal-Independent Unitary Transforms / 2D Transforms Example: Basis Images of Separable 8×8 DCT-II
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 23 / 38 Signal-Independent Unitary Transforms / 2D Transforms Example: Separable DCT-II for 8×8 Image Block
Forward transform for 8 × 8 block of samples: u = A · s · AT
horizontal vertical DCT DCT
original block after 2d DCT
Example calculation of 2d DCT-II: 1 Horizontal DCT of input block: u∗ = s · AT
2 Vertical DCT of intermediate result: u = A · u∗ = A · s · AT
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 24 / 38 Signal-Independent Unitary Transforms / 2D Transforms DCT-II in Practice
Justification for usage of DCT-II Represents signal as weighted sum of frequency components Similar to KLT for highly correlated sources (% → 1) Independent of source characteristics Fast algorithms for computing forward and inverse transform
DCT-II of size 8×8 is used in Image coding standard: JPEG Video coding standards: H.261, H.262/MPEG-2, H.263, MPEG-4 Visual
Integer approximation of DCT-II is used in Video coding standard H.264/AVC (with sizes 4×4 and 8×8) Video coding standard H.265/HEVC (with sizes 4×4, 8×8, 16×16, 32×32) New standardization project H.266/VVC
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 25 / 38 Transform Coding in Practice Transform Coding in Practice
Orthogonal Transform DCT-II or integer approximation thereof Potential extension in H.266/VVC: Switched transform of DCT/DST families (DCT-II, DST-VII, ...) Non-separable transforms
Scalar Quantization Uniform reconstruction quantizers (or very similar designs) Bit allocation by using same quantization step size for all coefficients Usage of advanced quantization algorithms in encoder May use quantization weighting matrices
Entropy Coding of Quantization Indexes Zig-zag scan (or similar scan) Run-level coding, run-level-last coding, or similar approach Alternatively: Adaptive arithmetic coding
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 26 / 38 Transform Coding in Practice / Bit Allocation in Practice Bit Allocation for Uniform Reconstruction Quantizers (URQs)
Remember: Optimal bit allocation: Pareto condition ∂D (R ) k k = const ∂Rk Pareto condition for high rates
2 2 −2Rk Dk = εk · σk · 2 =⇒ Dk (Rk ) = const High rate distortion approximation for URQs 1 D = ∆2 k 12 k Quantization step sizes for optimal bit allocation at high rates 1 D = ∆2 = const =⇒ ∆ = const k 12 k k In practice, (nearly) optimal bit allocation is typically achieved by using the same quantization step size for all transform coefficients
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 27 / 38 Transform Coding in Practice / The JPEG Standard The Image Compression Standard JPEG
Partition color components into 8 × 8 blocks
Transform coding of 8 × 8 blocks of samples
block of 2D scalar entropy samples transform quantization coding sequence of bits
reconstructed 2D inverse decoder entropy block transform mapping decoding
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 28 / 38 Transform Coding in Practice / The JPEG Standard JPEG: Transform of Sample Blocks
Separable DCT-II of size 8×8 (fast implementation possible) Forward transform (in encoder)
horizontal vertical DCT DCT
original block after 2d DCT
Inverse transform (in decoder)
horizontal vertical IDCT IDCT
reconstructed block rec. transform coeffs. Effect of transform: Compaction of signal energy (for typical blocks) Quantization more effective in transform domain
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 29 / 38 Transform Coding in Practice / The JPEG Standard JPEG: Quantization
s'-4 s'-3 s'-2 s'-1 s'0 s'1 s'2 s'3 s'4 u-3 u-2 u-1 u0 u1 u2 u3 u4
-Δ-2·Δ-3·Δ-4·0Δ Δ 1·Δ 3·Δ 4·Δ s
Uniform reconstruction quantizers Equally spaced reconstruction levels (indicated by step size ∆) Simple decoder mapping t0 = ∆ · q Simplest (but not best) encoder: Rounding
q = round(t/∆)
Better encoders use Lagrangian optimization (minimization of D + λR) Quantization step size ∆ determines tradeoff between quality and bit rate
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 30 / 38 Transform Coding in Practice / The JPEG Standard JPEG: Entropy Coding
0.242 0.108 0.053 0.009
0.105 0.053 0.022 0.002
0.046 0.017 0.006 0.001
0.009 0.002 0.001 0.000
probabilities P(qk 6= 0) zig-zag scan (JPEG)
1 Scanning of Quantization indexes Convert matrix of quantization indexes into sequence Traverse quantization indexes from low to high frequency positions JPEG: Zig-zag scan
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 31 / 38 Transform Coding in Practice / The JPEG Standard JPEG: Entropy Coding
2 Entropy Coding of Sequences of Quantization Indexes Lossless coding of sequences of quantization indexes Often long sequences of zeros (in particular at end of sequence) Entropy coding should exploit this property
JPEG: Run-Level Coding (V2V code) Map sequence a symbols (transform coefficients) into (run,level) pairs, including a special end-of-block (eob) symbol level : value of next non-zero symbol run : number of zero symbols that precede next non-zero symbol eob : all following symbols are equal to zero (end-of-block) Assign codewords to (run,level) pairs (including eob symbol)
Example: 64 symbols: 5 3 0 0 0 1 0 1 0 0 1 0 0 0 0 0 0 0 0 0 ... (run,level) pairs: (0,5) (0,3) (3,1) (1,1) (2,1) (eob)
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 32 / 38 Transform Coding in Practice / The JPEG Standard JPEG Compression Example
OriginalLossy Compressed: Image (1024 JPEG×678 image(Quality points, 50)1) 945 KB) 1.940.24 %
100 %
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 33 / 38 Transform Coding in Practice / The JPEG Standard JPEG Compression Example
OriginalLossy Compressed: Image (1024 JPEG×678 image(Quality points, 50)1) 945 KB) 1.940.24 %
100 %
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 33 / 38 Transform Coding in Practice / The JPEG Standard JPEG Compression Example
OriginalLossy Compressed: Image (1024 JPEG×678 image(Quality points, 50)1) 945 KB) 1.940.24 %
100 %
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 33 / 38 Transform Coding in Practice / Hybrid Video Coding Transform Coding in Image and Video Coding
Basic Design of JPEG Transform coding of 8 × 8 image blocks Transform: Separable DCT-II Quantization: Uniform reconstruction quantizers (URQs) with selectable quantization step size (quality parameter) Entropy coding: Zig-zag scan + Run-Level coding
Hybrid Video Coding (see following block diagrams) Partition of each picture into blocks Prediction of blocks: Intra- or intra-picture prediction Transform coding of prediction error blocks Transform coding is a key component Only component that includes quantization Controls trade-off between distortion and rate
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 34 / 38 Transform Coding in Practice / Hybrid Video Coding Hybrid Video Encoder
encoder control transform coefficient levels 푠 [푥, 푦] and quantization parameters bitstream 푘 transform & entropy quantization coding 푢푘[푥, 푦] scaling & input video pictures inv. transform (in coding order and ′ partitioned into blocks) 푢푘[푥, 푦] coding modes, 푠 푘[푥, 푦] intra pred. modes, motion data
∗ 푠푘[푥, 푦] coding mode intra mode decision decision buffer for current picture
intra-picture ∗ 푠푘[푥, 푦] ∗ prediction 푓( 푠푘 푥, 푦 ) in-loop filtering motion-comp. ′ prediction ′ 푠푟[푥+푚푥, 푦+푚푦] 푠푘[푥, 푦]
motion reconstructed video estimation decoded (for monitoring) picture buffer
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 35 / 38 Transform Coding in Practice / Hybrid Video Coding Hybrid Video Decoder
transform coefficient levels bitstream entropy and quantization parameters scaling & decoding inv. transform
′ 푢푘[푥, 푦] 푠 푘[푥, 푦]
∗ 푠푘[푥, 푦]
buffer for intra-picture current prediction intra prediction picture modes ∗ ∗ 푓( 푠푘 푥, 푦 ) 푠푘[푥, 푦]
in-loop filtering
coding modes ′ ′ 푠푟[푥+푚푥, 푦+푚푦] 푠푘[푥, 푦]
motion data motion-comp. decoded video prediction decoded picture buffer
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 36 / 38 Summary Part Summary
Concept of Transform Coding Unitary / orthogonal block transform Scalar quantization of transform coefficients Entropy coding of quantization indexes
Unitary / Orthogonal Transform Goal: Decorrelation, energy compaction KLT is the optimal transform for Gaussian sources KLT achieves complete decorrelation, but is signal dependent Walsh-Hadamard transform: Subjectively unpleasant Discrete Fourier transform (DFT): Problem due to period signal repetition Discrete Cosine transform (DCT) DFT of mirrored signal Reduced blocking artifacts compared to DFT Gauss-Markov: KLT approaches DCT-II for % → ∞
T. Wiegand (TU Berlin) — Image and Video Coding: Transform Coding III 37 / 38 Summary Part Summary
Bit Allocation and Transform Coding Gain
Optimal bit allocation: Pareto condition (same slope for all Dk (R)) For high rates: Optimum bit allocation yields equal component distortion In practice: URQs with same quantization step size for all coefficients For Gauss-Markov: Transform coding gain increases with transform size
Application of Transform Coding Core component of all lossy image and video codecs JPEG, H.262/MPEG-2, H.263, MPEG-4, H.264/AVC, H.265/HEVC Typical transform coding setup in image & video codecs: DCT (or integer approximation thereof) Uniform reconstruction quantizers (with constant quantization step size) Zig-zag scan (or similar scan) Entropy coding of vector of quantization indexes
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