1 / 13

Lossy Compression

E[X(i,j)X(i-1,j)] E[X 2 (i,j)]. E[X(i,j)X(i,j-1)] E[X 2 (i,j)]. Lossy Compression. Based on spatial redundancy Measure of spatial redundancy: 2D covariance Cov X (i,j)=  2 e - a (i*i+j*j) Vertical correlation r 1 = Horizontal correlation r 2 = For images we assume equal correlations

Gabriel
Download Presentation

Lossy Compression

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. E[X(i,j)X(i-1,j)] E[X2(i,j)] E[X(i,j)X(i,j-1)] E[X2(i,j)] Lossy Compression • Based on spatial redundancy • Measure of spatial redundancy: 2D covariance • CovX(i,j)= 2e-a(i*i+j*j) • Vertical correlation r1= • Horizontal correlation r2= • For images we assume equal correlations • Typically e-a=r1= r2= 0.95 • Measure of loss (or distortion): • MSE between encoded and decoded image

  2. Rate-Distortion Function • Tradeoff between bit rate (R) of compressed image and distortion (D) • R measured in its per encoder output symbol • Compression ratio = encoder input bits/R • D normalized by the variance of the encoder input • Possible SNR definition = 10 log10 D-1 • For images that can be modeled as uncorrelated Gaussian • R(D)=0.5log2 D-1 • More realistic images • See graph • How do you make these graphs?

  3. Sample vs. Block-based Coding • Sample-based • In spatial or frequency domain • Like the JPEG-LS • Make a predictor function (often weighted sum) • Compute and quantize residual • Encode • Block-based • Spatial: group pixels into blocks, compress blocks • Transform: group into blocks, transform, encode

  4. Which Transformation? • Considerations • Packing the most energy in the least number of elements • Minimizing the total entropy of the sequence • Decorrelating elements in the input blocks maximally • Coding complexity • DFT, KLT, DCT, DHT? See the effects • DCT-based coding • High compaction efficiency for correlated data • Orthogonal and separable • Fast, approximate DCT algorithms available

  5. The JPEG standard • Operating modes • Lossless • Sequential DCT-based • Progressive DCT-based • Hierarchical

  6. JPEG: The Process • Preprocess colors • Divide the image into 8 pixel x 8 pixel non-overlapping blocks – why 8? • Transform each block into 2D DCT • Encode • Stage 1: predictive coder for DC or run-length coder for AC coefficients • Stage 2: Huffman or arithmetic coding

  7. JPEG: Color Processing • Maximum number of color components = 256 • Each sample may be 8 or 12 bits in precision • Conversion of a decorrelated color space • Y-Cb-Cr, YUV, CIELAB • Subsample chrominance components • Interleave components • Maximum MCU components=4 • Maximum number of data units within MCU = 10

  8. JPEG: Quantization Tables • 8 X 8 quantization table for each image component • Q(i,j): quantization step for corresponding DCT element • 1  Q(i,j)  255 • Psycho-visual experiments • Bit-rate control • Total number of block = B • yk[i,j] : (i,j) output of k-th block bits per DCT coeff target av. bit-rate for 8-bit images

  9. JPEG: Entropy Coding • Baseline processing • Total of 4 code tables allowed • Different code tables for luminance and chrominance • DC coefficients • DC differentials are computed and have the range [-2047, 2047] • The range is divided into 12 size categories • Category i needs i bits to represent the value • DC residuals are represented as [size, amplitude] pairs • Size is Huffman encoded

  10. JPEG: Entropy Coding • AC coefficients • Take value in the range [-1023, 1023] • 10 size categories • Only non-zero coefficients need to be encoded • Processed in zig-zag order • More efficient run-length encoding of AC coefficients • Represented as (run/size, amplitude) • If run > 15, possibly several (15/0) symbols are used

  11. JPEG: Progressive Coding • Coding is performed sequentially but in multiple scans • The first scan should produce the full image without all details, and details are provided in successive scans • Spectral selection • Each block divided into frequency bands • Each band transmitted during a different scan • Successive approximation • Given a frequency band, DCT coefficients are divided by 2k • MSBs are transmitted first

  12. Subband Coding • Pass an image through an n-band filter bank • Possibly subsample each filtered output • Encode each subband separately • Compression may be achieved by discarding unimportant bands • Advantages • Fewer artifacts than block-coded compression • More robust under transmission errors • Selective encoding/decoding possible • More expensive

  13. Wavelet Compression • Special case of subband compression • Space and frequency-limited mother function y(t) • Function family: ymn(t) = a0-m/2y(a0-mt - nb0) • If a0=2 and b0=1, the ymn(t) function is an orthonormal basis function • f(t) = • The a0-m term scales the signal • Scaling function fmn(t) = 2-m/2f(2-mt - n) Unscaled high-pass filter downscaled low-pass filter

More Related