JPEG Transform coding

8/13/2019 JPEG Transform coding

1/34

Transform Coding


2/34

Why Transform?

Purpose of transform Remove correlation between input samples

Transform most energy of an input block into a few coefficients

Small coefficients can be discarded by quantization without too much

impact to reconstruction quality

Block transform vs. global transform Block transform: DCT

Global transform: Wavelet


3/34

Block Transform

Divide input data into blocks

Encode each block separately (sometimes with

information from neighboring blocks)

Most DCT-based image/video/audio coding standards Reduce delay and the complexity (e.g. memory

requirement)


4/34

2-D DCT Example

Apply 8x8 DCT to each 8x8 block Histograms of source and DCT coefficients


5/34

Matrix Representation of Transform

Linear transform is an NN matrix:

yN1=TNN* xN1

Inverse Transform: x=T-1*y

Orthogonal Transform: T1= TT T*TT= I

For orthogonal transform:

Rows/Cols are orthogonal to each others


6/34

Optimum Transform

If Ax = x, : Eigenvalues of A x: Eigenvectors of A

If A symmetric A has N orthogonal eigenvectors xi:

Let U = [x1, x2, , xN]

U1AU= =diag(1, 2,, N)

Matrix A can be diagonalized by its eigenvectors.

Usually RxxRxx is not diagonal since data are correlated If we choose T= Rxx Ryy= T RxxT Karhunen-Love Transform (KLT)

Different components of Y are uncorrelated.Desired for compression.

KLT is not practical Need to estimate Rxx

Finding eigenvectors are time consuming Need to send T as side information

Fast implementation of the KLT is generally not available

Find a nice approximation of the KLT for a given Rxx.


7/34

Discrete Cosine Transform (DCT)

AR(1) signal x(n) = x(n 1) + e(n) A good approximate for natural image 0.9


8/34

2-D Separable DCT

Apply T to each row

Then apply T to each column


9/34


10/34

JPEG

Joint Photographic Experts Group ISO/IEC JTC1 SC29 WG1

Formed in 1986 by ISO and CCITT (ITU-T)

Became International Standard (IS) in 1991

ISO/IEC IS 10918-1 (ITU-T T.81): Requirements andguidelines

ISO/IEC IS 10918-2 (ITU-T T.83): Compliance testing

ISO/IEC IS 10918-3 (ITU-T T.84): Extensions

Digital Coding of Continuous-Tone Still Images(grayscale or color)

Compression ratio 10 to 50; 0.5 to 2 bpp.


11/34

Baseline JPEG

8x8

DCT

DCT


12/34

JPEG Quantization

Uniform mid-tread quantizer Larger step sizes for chroma components

Different coefficients have different step sizes

Smaller sizes for low frequency coefficients (more bits)

Larger sizes for low frequency coefficients (less bits)

Human visual system is not sensitive to error in highfrequency.


13/34

Entropy Coding

Zig-Zag Scanning

First DCT Coefficient: DC

Others (63 Coefficients): AC


14/34

Entropy Coding

DCT


15/34

DC Prediction

DC Coefficients: Average of a block DC of neighboring blocks are still similar to each

others: redundancy

The redundancy can be removed by differential coding:

e(n) = DC(n)DC(n-1)

Only encode the prediction error e(n)


16/34

JPEG for Color Image

Color converter: RGB to YUV Level offset: subtract 2^(N-1). N: bits / pixel.

Quantization: Different step size for different coeffs

DC: Predict from DC of previous block

AC: Zigzag scan to get 1-D data

Run-level: joint coding of non-zero coeffs and number ofzeros before it.


17/34

Quality Factor

Actual step size: Scale the basic table by a quality factor.

Actual Q table = scaling x Basic Q table:

Q: quality factor 50: scaling = 50 / quality;

quality factor > 50: scaling = 2 - quality / 50;


18/34

Rate Control

Controlling Quality

Q: Quality Factor (0


19/34

JPEG Example


20/34

JPEG Quantization


21/34

JPEG Example


22/34

JPEG Implementation


23/34

JPEG Implementation


24/34

JPEG 2000

Image Coding System (JTC 1.29.14, ISO 15444)

Goals

Low bit-rate compression e.g., below 0.25 bpp for highly

detailed gray-level images Lossless and lossy compression in a single bitstream

Large images (More than 64K by 64K)

Single decompression architecture

Transmission in noisy environments

Computer generated imagery

Compound documents: bi-level and gray-scale


25/34

JPEG Implementation


26/34

JPEG Implementation


27/34

JPEG Implementation


28/34

JPEG Implementation


29/34


30/34

Progressive JPEG

Baseline JPEG encodes the image block by block: Decoder has to wait till the end to decode and display the entire image.

Progressive: Coding DCT coefficients in multiple scans: The first scan generates a low-quality version of the entire image

Subsequent scans refine the entire image gradually.

Two procedures defined in JPEG:

Spectral selection:

* Divide all DCT coefficients into several bands (low, middle, highfrequency subbands)

* Bands are coded into separate scans

Successive approximation:

* Send MSB of all coefficients first.

* Send lower significant bits in subsequent scans.


31/34

Objective Quality Measure

PSNR (dB): Peak Signal-to-Noise-Ratio For 8-bit Images of size M x N:


32/34


33/34

JPEG 2000

Image Coding System (JTC 1.29.14, ISO 15444)

Goals

Low bit-rate compression e.g., below 0.25 bpp for highly

detailed gray-level images Lossless and lossy compression in a single bitstream

Large images (More than 64K by 64K)

Single decompression architecture

Transmission in noisy environments

Computer generated imagery

Compound documents: bi-level and gray-scale


34/34

JPEG Transform coding

Documents

Transcript of JPEG Transform coding