Image Processing Architecture, © Oleh TretiakPage 1Lecture 6 ECE-C453 Image Processing Architecture Lecture 6, 2/3/04 Lossy Video Coding Ideas Technology of DCT and Motion Estimation Oleh Tretiak Drexel University

Image Processing Architecture, © Oleh TretiakPage 2Lecture 6 Decorrelation Ideas Orthogonal Transforms (KLR, DCT)  Main method for intra-frame coding Wavelet  New stuff (JPEG 2000) Predictive coding  Simple  Used for inter-frame coding (video)

Image Processing Architecture, © Oleh TretiakPage 3Lecture 6 Lossy Predictive Coding How to decorrelate?  Predict values  Block coding (DFT)  wavelet Predictive (sample based, feedback) encoder, Differential Pulse Code Modulation (DPCM) Encoder Decoder

Image Processing Architecture, © Oleh TretiakPage 4Lecture 6 Review: Image Decorrelation x = (x 1, x 2,... x n ), a sequence of image gray values Preprocess: convert to y = (y 1, y 2,... y n ), y = Ax, A ~ an orthogonal matrix (A -1 = A T ) Theoretical best (for Gaussian process): A is the Karhunen- Loeve transformation matrix  Images are not Gaussian processes  Karhunen-Loeve matrix is image-dependent, computationally expensive to find  Evaluating y = Ax with K-L transformation is computationally expensive In practice, we use DCT (discrete cosine transform) for decorrelation  Computationally efficient  Almost as good as the K-L transformation

Image Processing Architecture, © Oleh TretiakPage 5Lecture 6 Review: Block-Based Coding Full image DCT - one set of decorrelated coefficients for whole image Block-based coding:  Image divided into ‘small’ blocks  Each block is decorrelated separately Block decorrelation performs almost as well (better?) than full image decorrelation Current standards (JPEG, MPEG) use 8x8 DCT blocks

Image Processing Architecture, © Oleh TretiakPage 6Lecture 6 Rate-Distortion: 1D vs. 2D coding Theory on tradeoff between distortion and least number of bits Interesting tradeoff only if samples are correlated “Water-filling” construction to compute R(d)

Image Processing Architecture, © Oleh TretiakPage 7Lecture 6 Wavelet Transform Filterbank and wavelets 2 D wavelets Wavelet Pyramid

Image Processing Architecture, © Oleh TretiakPage 8Lecture 6 Filterbank Pyramid

Image Processing Architecture, © Oleh TretiakPage 9Lecture 6 Lena: Top Level, next level

Image Processing Architecture, © Oleh TretiakPage 10Lecture 6 This Lecture Idea  Video Coding by Pixel Prediction  Motion Estimation Technology: DCT, and how much it costs Technology: Motion Estimation Algorithms

Image Processing Architecture, © Oleh TretiakPage 11Lecture 6 Video Coding Video: Sequence of images Reason for changes between successive images  Edits  Camera pan, zoom  Intra-frame motion  Intra-frame texture  Noise Model: Successive images are similar Video coding uses intra-frame redundancy to achieve lossy compression

Image Processing Architecture, © Oleh TretiakPage 12Lecture 6 Predicting sequential images f(t-1)f(t)f(t) f(t)–f(t–1)

Image Processing Architecture, © Oleh TretiakPage 13Lecture 6 Motion Compensation Macroblock size  MxN Matching criterion  MAE (mean absolute error) Search window  ±p pixel locations Search algorithm  Full search  Logarithmic search  Parallel Hierarchical One-Dimensional Search  Pixel subsampling and projection  Hierarchical downsampling

Image Processing Architecture, © Oleh TretiakPage 14Lecture 6 Motion Estimation Methods No compensation Full search logarithmic search 3 level hierarchical

Image Processing Architecture, © Oleh TretiakPage 15Lecture 6 DCT Technology DCT Formula How it works  DCT plus quantization DCT implementations and cost  Direct  Separable  Fast  Refinements

Image Processing Architecture, © Oleh TretiakPage 16Lecture 6 What is the DCT? One-dimensional 8 point DCT Input x 0,... x 7, output y 0,... y 7 One-dimensional inverse DCT Input y 0,... y 7, output x 0,... x 7 Matrix form of equations: x, y are one column matrices Note: in these equations, p stands for 

Image Processing Architecture, © Oleh TretiakPage 17Lecture 6 Forward 2DDCT. Input x ij i = 0,... 7, j = 0, Output y kl k = 0,... 7, l = 0,... 7 Matrix form, X, Y ~ 8x8 matrices with coefficients x ij, y kl The 2DDCT is separable! Two-Dimensional DCT Note: in these equations, p stands for 

Image Processing Architecture, © Oleh TretiakPage 18Lecture 6 General DCT One dimension Two dimensions

Image Processing Architecture, © Oleh TretiakPage 19Lecture 6 See 06IPA.xls

Image Processing Architecture, © Oleh TretiakPage 20Lecture 6 Computational Complexity 1D DCT  N input and output samples ~ N 2 = 64 operations (additions + multiplications) 2D DCT - direct implementation  M = N 2 input values, M output values -> M 2 = N 4 2D DCT - separable implementation, Y = TXT T = ZT T, where Z = TX, all matrices are NxN -> 2N 3 operations For N = 8  2D DCT direct — 4096 operations, 64 operations per pixel  2D DCT separable — 1024 operations, 16 ops/pixel Big savings due to separable transform Inverse DFT — same story.

Image Processing Architecture, © Oleh TretiakPage 21Lecture 6 DCT: Encoding in JPEG, MPEG Take 8x8 blocks of pixels Subtract range mean value Compute 8x8 DCT Quantize the DCT coefficients  Typically, many of the samples are equal to zero Lossless entropy coding of the quantized samples Different quantization step is used for different DCT coefficients  y kl — DCT coefficients, q kl — quantizer steps  z kl — quantized values

Image Processing Architecture, © Oleh TretiakPage 22Lecture 6 DCT: Example Data from lena, ‘smooth’ area. RMS error = 3.5 Original DCT DCT, quantizedReconstructed

Image Processing Architecture, © Oleh TretiakPage 23Lecture 6 DCT example Data from lena, ‘busy’ area. RMS error = 7.3 OriginalDCT DCT, quantizedReconstructed

Image Processing Architecture, © Oleh TretiakPage 24Lecture 6 Overview: DCT coding Transformation decorrelates samples Transformed samples are quantized, quantization step depends on the coefficient. Degree of compression and loss can be changed by scaling the quantization steps Many quantized samples are zero —> run length coding At receiver, perform inverse DCT Many calculations! JPEG standard quantization steps

Image Processing Architecture, © Oleh TretiakPage 25Lecture 6 Speeding up the DCT Separable transform - basic speedup Fast DCT transform - like FFT Further speedup through Scaled DCT

Image Processing Architecture, © Oleh TretiakPage 26Lecture 6 Optimized (fast) DCT 1-D Chen DCT diagram. Dashed lines indicate subtraction, — multi- plication by a constant, — multiplication by 0.5 (shift). Characteristics of optimized DCT algorithms

Image Processing Architecture, © Oleh TretiakPage 27Lecture 6 DCT Complexity Direct DCT computation:  64 DCT values, each requires 64 multiplications & additions —> 4096 multiply-accumulate (MA) operations per block  Separable algorithm (operate on rows, then on columns) —> 16 one-dimensional 8 point DCT operations —> 1024 MA operations  Fast implementation ~ Nlog 2 N operations ~ 16x24 = 384 MA ops Special methods ~ many operations involve multiplication by 1 or -1, take advantage of this!

Image Processing Architecture, © Oleh TretiakPage 28Lecture 6 Fast Scaled DCT Picture of a butterfly at last stage of DCT + following quantizer

Image Processing Architecture, © Oleh TretiakPage 29Lecture 6 DCT refinements Multiply-accumulate architectures  Basic operation is a = bc + d, well suited for DCT Super-scalar architectures  Multi-register, multi-ALU processors  Perform several operations in parallel Complexity of scaled DCT algorithms, excluding quantization

Image Processing Architecture, © Oleh TretiakPage 30Lecture 6 Motion Estimation Architecture of Motion Estimation Algorithms and Costs  Full Search  Logarithmic Search  PHODS  Downsample, projection  Hierarchical motion estimation  Other criteria  Multi-image estimation

Image Processing Architecture, © Oleh TretiakPage 31Lecture 6 Baseline Models Previous frame predicts current frame  I(x, y, t) = I(x, y, t-1) + e(x, y, t) Not effective in presence of motion ~ zoom, pan, etc. Prediction to account for motion:  I(x, y, t) = I(x+u, y+v, t-1) + e(x, y, t)  (u, v) — motion (displacement) vector Model works (somewhat) for pan, not for other motion  Compromise: Compute independent motion estimates for rectangular image regions — macroblocks. Macroblocks are, in general, bigger than DCT blocks

Image Processing Architecture, © Oleh TretiakPage 32Lecture 6 Generic Encoder - simplified

Image Processing Architecture, © Oleh TretiakPage 33Lecture 6 Generic Decoder

Image Processing Architecture, © Oleh TretiakPage 34Lecture 6 Motion Compensation Macroblock size  MxN Matching criterion  MAE (mean absolute error) Search window  ±p pixel locations Search algorithm  Full search  Logarithmic search  Parallel Hierarchical One-Dimensional Search  Pixel subsampling and projection  Hierarchical downsampling

Image Processing Architecture, © Oleh TretiakPage 35Lecture 6 Motion Estimation Terminology Issues:  Size of macroblock  Size of search region In video coding standards, M = N = 16

Image Processing Architecture, © Oleh TretiakPage 36Lecture 6 Matching Criterion Matching criterion: what produces the fewest coded bits for the error image Coding for each value of motion vector (u, v) is too time consuming (expensive) In practice, mean absolute error (MAE) is most popular C - current image, R - reference image, (x, y) - macroblock origin

Image Processing Architecture, © Oleh TretiakPage 37Lecture 6 Full-Search Method Compute for (2p+1) 2 values of (i, j). Each location requires 3MN operations Picture dimensions IxJ, F pictures per second  3IJF(2p + 1) 2 operations per second  I = 720, J = 480, F = 30, p = 15 —> 30 GOPS Guaranteed to find best (MAE) displacement How to do it?  Special computers  Smaller p  Faster (suboptimal) algorithm

Image Processing Architecture, © Oleh TretiakPage 38Lecture 6 Logarithmic Search (1D) Goal: find minimum over u in [-p, p] First step: evaluate at -p/2, 0, p/2 (interval ~ p) Next step: choose interval of length p/2 around minimum (2 more evaluations) Continue until interval length is equal to 2. This takes k = ceiling(log 2 p) iterations Example p = 7  Evaluate at -4, 0, 4 —> minimum at -4  Evaluate at -6, (-4), -2 —> minimum at -2  Evaluate at -3, (-2), -1 —> minimum at -3. Done!

Image Processing Architecture, © Oleh TretiakPage 39Lecture 6 Logarithmic Search - 2D First stage requires 3x3 = 9 evaluations Subsequent stages require 8 evaluations k = ceiling(log 2 p) stages (iterations) Rate = 3IJF(8k+1)  p = 15, I = 720, J = 480, F = 30 —> 1 GOPS Can fail to find minimum Bottom line: Faster method, more error than full search

Image Processing Architecture, © Oleh TretiakPage 40Lecture 6 PHODS Parallel Hierarchical One-Dimensional Search 1-st Blue 2-nd Green 3-rd Red Min V Min H ~Twice as fast as logarithmic Less reliable

Image Processing Architecture, © Oleh TretiakPage 41Lecture 6 Other Fast Methods Subsample (do not use all points in macroblock) Projection: Row and column projection of pixels, follow with 1-D search Hierarchical motion estimation  Downsample reference image and current image  Perform low resolution search  Refine

Image Processing Architecture, © Oleh TretiakPage 42Lecture 6 Hierarchical Search Prepare downsampled versions of current and reference images  Full macroblock 16x16  Down 2 macroblock 8x8  Down 4 macroblock 4x4 Full search in Down 4 reference image  16 x speedup, smaller macroblock  16 x speedup, fewer displacement vectors  p = ±16, p’ = ±4 Around point of best match, do local search in Down 2 reference image (3x3 search zone) Repeat for Full reference image (3x3 search zone) Full Down 2 Down 4

Image Processing Architecture, © Oleh TretiakPage 43Lecture 6 Motion Estimation Methods No compensation Full search logarithmic search 3 level hierarchical

Image Processing Architecture, © Oleh TretiakPage 44Lecture 6 Comparison

Image Processing Architecture, © Oleh TretiakPage 45Lecture 6 More Speedup Simpler comparison criteria  Binarize difference, count pixels that do not match  PDC (Pixel Difference Classification)  Binarize current and reference  BPROP (count matching pixels)  DPC (count different pixels)  BMP (operations done on bitplanes) Produce 3-25 fold speedup

Image Processing Architecture, © Oleh TretiakPage 46Lecture 6 Big Picture on Speedup Speedup methods are less accurate  Same Bit Rate, lower SNR  Same SNR, higher bit rate  Binary criteria lose about 0.5 dB Suppose we have adequate computing power? Can we do better?  Sub-pixel motion estimation  First find best match with pixel accuracy in displacement vectors  Interpolate images for half-pixel shifts

Image Processing Architecture, © Oleh TretiakPage 47Lecture 6 Multipicture Motion Estimation Estimate on basis of past and future  Non-sequential image transmission  More chances to find good match  More calculations

Image Processing Architecture, © Oleh TretiakPage 48Lecture 6 Video Compression - Summary Video — sequence of images Can use intraframe compression  Motion JPEG Interframe compression offers great potential for savings No motion compensation — lower compression Motion compensation — greater compression All video standards provide for motion compensation  Compensation done on macroblocks, multiple motion vectors per image  Tradeoff between computing requirement and image quality