Compression

1
Compression

• "lossless" fx,y ? gx,y Decompress (
  Compress ( fx,y )
• lossy quality measures
• e2rms 1/MN  ??( gx,y - fx,y )2
• SNRrms 1/MN  ?? gx,y2 / e2rms
• subjective how does it look to the eye
• application how does it influence the final
  results
• for both
• the attained compression ratio
• the time and memory needed for compression
• the time and memory needed for decompression

2
Coding redundancy
Average number of bits for code 2 2.7
bits Compression ratio Cr 3/2.7 1.11
3
Interpixel redundancy
Run Length Encoding (RLE) For whole binary
image Cr 2.63
4
Psycho-visual redundancy
5
General Model
The "mapper" transforms the data to make it
suitable for reducing the inter-pixel
redundancies. This step is generally reversible
and can reduce the amount of data, e.g. RLE, but
not in transformations to the Fourier or Discrete
Cosinus domains. The "quantizer" reduces the
precision of the output of the mapper according
to the determined reliability criteria. This
especially reduces psycho-visual redundancies and
is irreversible. The "symbol encoder" makes a
static or variable length of code to represent
the quantizer's output. It reduces the coding
redundancy and is reversible.
6
Information theory
Questions such as "what is the minimum amount of
data needed to represent an image" will be
answered in "information theory. The generation
of information is modeled as a statistical
process that can be measured in the same manner
as the intuition of information. An event E with
a probability P(E) has I(E) - logr P(E)   
r-ary units of informationP(E) 1/2  then 
I(E) -log2 1/2 1 bit information If a source
generates symbols ai with a probability of P(ai),
then the average information per output is H(z) 
- ? P(ai) logr P(ai)      the insecurity of
entropy of the source This is maximal when
every symbol has an equal probability (1/N). It
indicates the minimal average length (for r2 in
bits per symbol) needed to code the symbols.
7
Huffman coding

• Under the condition that the symbols are coded
  one by one, an optimal code for the set of
  symbols and probabilities is generated. block
  code
• every source symbol is mapped to a static order
  of code symbols
• instantaneous code every code is decoded without
  reference to the previous symbols
• and is uniquely decodable

8
Lempel Ziv Welch coding
This translates variable length arrays of source
symbols (with about the same probability) to a
static (or predictable) code length. The method
is adaptive the table with symbol arrays is
built up in one pass over the data set during
both compression and decompression. Just as
Huffman, this is a symbol encoder which can be
used directly on the input or after a mapper and
quantizer. It is used in GIF, TIFF, PDF and in
Unix compress.
9
Predictive coding
1D pn round (? i1m  ai fn-i ) , first f must
be passed in another way 2D p(x,y) round
(a1fx,y-1 a2 fx-1,y)
10
Lossy predictive coding
11
Delta modulation
Delta Modulation is a simple but well known form
of it pn   ?  pin with  ?  lt 1 (here, pi
stands for the "predictor input")qn   ? sign(
en) and can be represented by a 1-bit value - ?
or ?  
12
Differential Pulse Code modulation
With DPCM, pn ??i1m  ?i pin-i. Under the
assumption that the quantization error (en-qn) is
small, the optimal values of ?i can be found by
minimizing Een2 E fn-pn2 . These
calculations are almost never done for each
single image but rather for a few typical images
or models of them.
Original image
13
4 prediction methods
fig. 8.24 prediction 0.97 f(x,y-1) 0.5(f(x,y-1)f
(x-1,y)) 0.75 (f(x,y-1)f(x-1,y))
-0.5f(x-1,y-1) 0.97 f(x,y-1) or 0.97
f(x-1,y)
14
Lloyd-Max quantizer
Instead of one level more levels can be used.
They might be unequal, e.g. factor 2 beween them.
With a Lloyd-Max quantizer the steps are
optimized to achieve a minimum error. Adjusting
the level  ? ( for each n, e.g. 16 pixels) with a
restricted amount (for example 4) scale factors
yields a substantial improvement of the error in
the decoded image with a small reduction of the
compression ratio.
15
Adaption
Using the 3 point prediction with the best of 4
quantizers per 16 bits error 8 for compression
in bits/pixel 1.0 1.125 2.0 2.125 3.0
3.125
16
Transform coding
JPEG makes use of 88 sub-images, DCT
transformation, quantization of the 64 coeffients
by dividing with a quantization matrix e.g. fig.
8.37b , a zigzag ordering fig. 8.36d of the
matrix followed by a Huffman encoder, separately
for the DC component. It uses a YUV color model,
for the U and V component blocks of 2 by 2 pixels
are combined into 1 pixel. The quantization
matrixes can be scaled to yield several
compression ratios. There are standard coding
tables and quantization matrices, but the user
can also indicate others to obtain better results
for a certain image.
17
examples
25 DCT 8x8 zoomed org 2 x 2 4x4
8x8
DCT norm arr
341 (3.42) 671 (6.33)
18
Wavelet transform
Type wavelet Operaties per pixel
Aantal 0s (lt1,5)
19
4 wavelets
20
Wavelet Compression ratios
341 (2.29) 671 (2.96)
1081 (3.72) 1671 (4.73)
21
JPEG 2000

• Uses wavelets (optionally on parts, tiles, of
  image)
• different for error-free and lossy compression
• gray and color images (upto 16 bit signed values)
• conversion to (about) YCbCr color space
• Cb, Cr components peak around 0
• complicated coding of wavelet values
• organised in layers and finally packets
• allowing more and more refined decoding (and
  storage)
• and access to parts of the image

22
Fractal compression
GIF original Image (161x261 pixels, 8
bits/pixel), JPEG compression 151, JPEG
compression 361, Fractal compression 361.
23
MPEG (1,2,4) video

• I-frame (Intraframe or independent frame), JPEG
  like
• P-frame (predictive frame) difference between
  frame and prediction from I-frame, motion
  compensated
• B-frame (bidirectional) previous I or P and next
  P

24
File formats

• The header contains information about
• type black and white, 8-bit gray level/color,
  3-byte color
• size number of rows, columns and bands, number
  of images
• compression method, possible parameters thereof
• data format for example band or colors per pixel
  or separated
• origin of the image or conditions during
  acquisition
• manipulations previously done on the image
• A few well-known formats are
• GIF for binary, gray level and 8-bits color
  images
• TIFF a multi-type format with many possibilities
• JFIF JPEG coded, for color or gray images of
  natural origin
• MPEG for a series of images
• PBM, PGM, PPM the PBMPLUS formats
• BMP Microsoft's format