08-3 - Image Compression

8/12/2019 08-3 - Image Compression

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methods and standards

Spring 2008 ELEN 4304/5365 DIP 1

by Gleb V. Tcheslavski: [email protected]

http://ee.lamar.edu/gleb/dip/index.htm

JPEGJPEG while being one of the most popular continuous tone imagecompression standards defines three basic coding schemes:1) A lossy baseline coding system based on DCT;

, precision, or progressive reconstruction applications;

3) A lossless independent coding system for reversible compression.

In a baseline format, the image is subdivided into 8x8 pixel blocks,which are processed left to right, top to bottom. For each block, its 64

pixels are level-shifted by subtracting 2 k-1, where 2 k is the maximum


number of intensity levels. Next, a 2D DCT of the block is computed,quantized, and reordered using the zigzag pattern to form a 1Dsequence of quantized coefficients. Next, the nonzero AC coefficientsare coded using a variable-length code. The DC coefficient isdifference coded relative to the DC coefficient of the previous block.


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JPEGJPEG decompression begins from decoding DC and AC coefficients

and recovering an array of quantized coefficients from a 1D zigzag.

De-normalized DCT coefficients


IDCT array Upscaled IDCT reconstructedimage

JPEG

The error between theoriginal and reconstructedimages is due to the lossynature of the JPEGcompression. The rms erroris approximately 5.8 intensitylevels.



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JPEGJPEGapproximation withcom ression 25:1rms error 5.4

JPEGapproximation withcompression 52:1,


rms error .

Predictive codingPredictive coding is based on eliminating the redundancies of closelyspaced pixels in space and/or in time by extracting and coding onlythe new information in each pixel. The new information is defined asthe difference between the actual and predicted value of the pixel.

Lossless predictive coding

Lossless predictivecoding system


Predictor generates theexpected value of eachsample based on aspecified number of

past samples.


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Predictive codingPredictors output is rounded to the nearest integer and is used to

compute prediction errore n n n=

Prediction error is encoded by a variable-length code to generate thenext element of the encoded data stream. The decoder reconstructse(n) from the encoded data and performs the inverse operation

( ) ( ) ( )n e n f n= +


, ,Often, the prediction is a linear combination of m previous samples:

1

( ) ( )m

ii

f n round f n i = =

Predictive codingWhere m is the order of the linear predictor and i, i = 1,m are

prediction coefficients, f (n) are the input pixels. The m samples usedfor prediction can be taken from the current scan line (1D linear

pre c ve co ng , rom e curren an prev ous neLPC), or from the current image and the previous images in an imagesequence (3D LPC). The 1D LPC:

1

( ) ( , )m

ii

f n round f x y i =

=

Spring 2008 ELEN 4304/5365 DIP 10

which is a function of the previous pixels in the current line. Notethat the prediction cannot be formed for the first m pixels. These

pixels are coded by other means (f.e. Huffman code).


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Predictive codingFor the image shown, form a

first-order (m = 1) LPC in form = , ,A predictor is called a previous

pixel predictor, and the coding procedure is differential( previous pixel ) coding.

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up-scaled by 128.

The average prediction error0.26The entropy reduction is due to removal of spatial redundancy.

Predictive codingThe compression achieved in predictive coding is related to theentropy reduction resulting from mapping an input image into a

prediction error sequence. Therefore, the pdf of the prediction error isn genera g y pea e at an as re at ve y sma compare to

the input image) variance. It is often modeled by a zero-meanuncorrelated Laplacian pdf:

21

( )2

e

e

e

e

p e e

=

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where e is the standard deviation of e.


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Predictive codingTwo successive frames of Earth

taken by NASA spacecraft.Usin the first-order m = 1Usin the first-order m = 1LPC:

[ ] ( , , ) ( , , 1) f x y t round f x y t = with = 1, the pixel intensitiesin the second frame can be

predicted from the intensities in

LPC:

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the first frame; the residualimage

Considerable decrease in standard deviation and in the entropyindicates significant compression that can be achieved.

Predictive codingMotion compensated prediction residuals

Since successive frames in a video sequence are often quite similar,coding their differences can reduce temporal redundancy and providesignificant compression. On the other hand, when a frame sequencecontains rapidly moving objects, the similarity between neighboringframes is reduced. The attempt to use LPC on images with littletemporal redundancy may lead to data expansion . Videocompression systems avoid the problem of data expansion by:1. Tracking object movement and compensating for it during the

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pre ct on an erenc ng process;2. Switching to an alternative coding method when there is

insufficient inter-frame correlation (similarity between frames) tomake predictive coding advantageous.


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Predictive codingBasics of motioncompensation:

Each video frame is divided into non-overlapping rectangular regions typically of size 4x4 to 16x16 macroblocks . The movement ofeach macroblock with respect to the previous frame ( reference frame )

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is encoded in a motion vector that describes the motion by definingthe vertical and horizontal displacement from the most likely

position. This displacement is usually specified to the nearest pixel, pixel, or pixel precision. If sub-pixel precision is used, predictionmust be interpolated from a combination of pixels in the frame.

Predictive codingAn encoded frame that is based on the previous frame (forward

prediction) is called a predictive frame (P-frame); the frame that isalso based on the subsequent frame (backward prediction) is called a

- -.stream to be reordered.Finally, some frames are encoded without referencing to any of theneighboring frames (like JPEG) and are encoded independently. Suchframes are calledintraframes or independent

frames I-frames and are

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ideal starting points for thegeneration of predictionresiduals. Also, I-frames can

be easily accessed withoutdecoding the stream.


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Predictive coding Motion estimation is the key concept of motion compensation. During

it, the motion of objects is measured and encoded into motion vectors.The search for the best motion vector requires specification of . ,

the basis of maximum correlation or minimum error betweenmacroblock pixels and the predicted (or interpolated) pixels for thechosen reference frame. One of the most frequently used error measures is the mean absolute distortion (MAD):

1( , ) ( , ) ( , )

m n

AD x y f x i y j p x i dx y j dy= + + + + + +

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1 1i jmn = =

where x and y are the coordinates of the upper-left pixel of the mxnmacroblock being coded, dx and dy are displacements from thereference frame, and p is an array of predicted macroblock pixels.

Predictive codingTypically, dx and dymust fall within alimited search regionaroun eacmacroblock.

Values from 8 to 64 pixels are common, and the horizontal searcharea often is significantly larger than the vertical search area.Another, more computationally efficient measure is the sum of

absolute distortions (SAD) that omits the 1/ mn factor.

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For the specified selection criterion (say, MAD), motion estimationis performed by searching for the dx and dy minimizing MAD( x,y)over the allowed range of motion vector displacements block

matching . An exhaustive search is efficient but expensive; there arefast algorithms that are inexpensive but dont guarantee optimum.


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Predictive codingTwo imagesdiffering by 13frames.

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Motion vectors:highly correlated variable-length code

Difference image:stdiv of error =12.73; entropy = 4.2

Motion-compens.difference image

Predictive codingMotion compensated prediction residual was computed by dividingthe latest figure into 16x16 macroblocks and comparing eachmacroblock to all possible 16x16 macroblock in the earlier framewithin 16 pixels position. The MAD criterion was used. Theresulting standard deviation was 5.62 and the entropy was 3.04

bits/pixel.

We observe that there is no motion in the lower portion of the imagecorresponding to the space shuttle. Therefore, no motion vectors are

shown. The macroblocks in this area are redicted from similarl

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located macroblocks in the reference frame.


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Predictive codingPrediction accuracy can be increased using sub-pixel motioncompensation.

Prediction Residual with 1residual with nomotioncompensation:Stdev = 12.7;Entropy = 4.17

pixel motioncompensation:Stdev = 4.4;Entropy = 3.34

Residual with Residual with

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pixel motioncompensation:Stdev = 4;Entropy = 3.35

pixel motioncompensation:

Stdev = 3.8;Entropy = 3.34

Predictive codingMotion estimation is a computationally intensive process. Fortunately,only the encoder must estimate the macroblock motion. The decoder

of the reference frames that were used in the encoder to form the

prediction residuals.

For this reason, most video compression standards do not includemotion estimation. Instead, compression standards focus on thedecoder: place constraints on macroblock dimensions, motion vector

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prec s on, or zon a an ver ca sp acemen ranges, e c.


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Predictive coding

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Predictive codingMost of the video compression standards use an 8x8 DCT for I-frameencoding but specify a larger area (16x16 macroblocks) for motioncompensation. Additionally, even the P- and B-frame predictionres ua s are trans orm co e ue to e ect veness o coe c entquantization. The H.264 and MPEG-4 AVC support intraframe

predictive coding (in I-frames) to reduce spatial redundancy

Spring 2008 ELEN 4304/5365 DIP 24


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Predictive coding

A typical motion-compensatedvideo encoder

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An encoder exploits redundancies within and between adjacent video

frames, and the psychovisual properties of the human visual system.

Predictive codingEncoders input is a sequence of macroblocks. For color video, eachmacroblock consists of a luminance block and 2 chrominance blocks.The human eye has less spatial acuity for color than for luminance,t ere ore, t e c rom nance oc s are o ten samp e at a t ehorizontal and vertical resolution the luminance block.

luma

chroma

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Grayed elements of the encoder are JPEG encoder that may operateon conventional macroblock (I-frames) or their differences (P- andB-blocks). Inverse mapper performs IDCT.


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Predictivecoding

1 minute HD(1280x720) full-colorvideo containing 150frames fade-in from

black (frames 21, 44),to black (frames 1595,1609, 1652), abruptchanges (frames 1303,

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1304). H.264compression requires

44.56 MB of storage ascompared to about 5GB uncompressed.

MPEG-2

Abbr. NamePictureCoding Types

ChromaFormat

Aspect RatiosScalablemodes

MPEG-2 Profiles

SP Simple profile I, P 4:2:0square pixels,4:3, or 16:9

none

MP Main profile I, P, B 4:2:0square pixels,4:3, or 16:9

none

SNR SNR Scalable

profileI, P, B 4:2:0

square pixels,4:3, or 16:9

SNR (signal-to-noise ratio)scalable

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SpatialSpatiallyScalable

profileI, P, B 4:2:0

square pixels,4:3, or 16:9

SNR- orspatial-scalable

HP High profile I, P, B 4:2:2 or 4:2:0 square pixels,4:3, or 16:9

SNR- orspatial-scalable


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MPEG-2

Abbr. Name Frame rates (Hz) Max h. Max v- Max luminanceMax bitrate

MPEG-2 Levels

. . (Mbit/s)

LL Low Level 23.976, 24, 25,29.97, 30 352 288 3,041,280 4

ML MainLevel

23.976, 24, 25,29.97, 30

720 576

10,368,000;Hp: 14,475,600 for4:2:0 and11,059,200 for 4:2:2

15

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H-14 High 144023.976, 24, 25,29.97, 30, 50,59.94, 60

1440 115247,001,600Hp with 4:2:0:62,668,800

60

HL High Level23.976, 24, 25,29.97, 30, 50,59.94, 60

1920 115262,668,800Hp: with 4:2:0:83,558,400

80

MPEG-2 Allowed Resolutions720 480, 704 480, 352 480, 352 240 pixel (NTSC)720 576, 704 576, 352 576, 352 288 pixel (PAL)

Allowed Aspect ratios (Display AR)4:316:9(1.85:1 and 2.35:1, among others, are often listed as valid DVD aspect ratios,but are actually just a 16:9 image with the top and bottom of the frame maskedin black)

Allowed Frame rates29.97 frame/s (NTSC)25 frame/s (PAL)

Note: By using a pattern of REPEAT_FIRST_FIELD flags on the headers of encoded

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pictures, pictures can be displayed for either two or three fields and almost any picture display rate (minimum of the frame rate) can be achieved. This is mostoften used to display 23.976 (approximately film rate) video on NTSC. Audio+videobitrate

Video peak 9.8 Mbit/sTotal peak 10.08 Mbit/sMinimum 300 kbit/s


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Lossy predictive coding

Lossy predictive codingis achieved by including

the error-free output bythe nearest integer.

The decompressed sequence

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( ) ( ) ( ) f n e n f n= +is also a predictors input. mapped prediction error

Lossy predictive coding Delta modulation

Delta modulation is a simple form of lossy predictive coding, wherethe redictor and uantizer are defined as

( ) ( 1)

( ) 0( )

f n f n

for e ne n

otherwise

= + >=

where is a prediction coefficient (normally < 1) and is a positive

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constant. The quantizer output can be represented by a single bit.


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Lossy predictive coding

Calculations needed tocompress and reconstructan input sequence {14 1514 15 13 15 15 14 20 26 2728 27 27 29 37 47 62 75 77

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= 1, = 6.5.

The process begins with the error-free transfer of the first inputsample 14 to the decoder. The remaining outputs are computedaccording to the equations.

Lossy predictive codingWhen is too small to represent the inputs change, a distortioncalled slope overload occurs. On the other hand, if is too large torepresent the inputs smallest change, granular noise appears. Inimages, these distortions show up as blurred edges and grainy(noisy) surfaces (distorted smooth areas).

These distortions are common to all lossy predictive coding formsand arise from a combination of quantization and prediction methodsused. Predictors are normally designed with the assumption of no

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quant zat on error, an quant zers are es gne to m n m ze t eyrown errors.


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Optimal predictorsUsually, the predictor is chosen to minimize the encoders mean-

square prediction error:2

2 =

with the constrain that

( ) ( ) ( ) ( ) ( ) ( ) f n e n f n e n f n f n= + + =

and ( ) ( 1)m

f n f n =

Spring 2008 ELEN 4304/5365 DIP 35

1i=

Therefore, the optimization criterion is minimal mean-square

prediction error, the quantization error is assumed to be negligible( (n) e(n)), and the prediction is a linear combination of m

previous samples.

Optimal predictorsThe resulting predictive coding approach is called a differential

pulse code modulation (DPCM). Therefore, we need to select the m prediction coefficients that minimize

{ }2

2

1

( ) ( ) ( )m

ii

E e n E f n f n i =

=

The minimization can be done by differentiating the above equation

with res ect to each coefficient, e uatin the derivatives to zero, and

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solving them:1

= R r


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Optimal predictorswhere R -1 is the inverse of the mxm autocorrelation matrix

{ } { } { }( 1) ( 1) ( 1) ( 2) ( 1) ( )2 1 2 2

E f n f n E f n f n E f n f n m

E n n E n n

{ } { } { }( ) ( 1) ( ) ( 2) ( ) ( ) E f n m f n E f n m f n E f n m f n m

R =

And r and are the m-element vectors:

{ }( ) ( 1)2

E f n f n

E n n

1

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{ }( ) ( ) E f n f n m

r =

m

=

The coefficients depend only on the autocorrelations of the samplesin the original sequence and can be determined by matrix operations.

Optimal predictorsThe variance of the prediction error resulting from using theseoptimal coefficients would be

2 2 2m

T = = 1

e ii=

However, computations of autocorrelations is very difficult in practice. Usually, a set of global coefficients is computed assuming asimple input model and substituting the correspondingautocorrelations.. For instance, when a 2D Markov image source

with the se arable autocorrelation function

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{ } 2( , ) ( , ) i jv h E f x y f x i y j =and a generalized 4 th order linear predictor

1 2 3 4 ( , ) ( , 1) ( 1, 1) ( 1, ) ( 1, 1) f x y f x y f x y f x y f x y = + + + +


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Optimal predictorsare assumed, the resulting optimal coefficients are

1 2 3 4 0h v h v = = = = v

the considered image. The sum of prediction coefficients usually is

1

1m

ii

=

which ensures that the predictors output is within the allowed rangeand reduces the im act of transmission noise usuall seen as

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horizontal streaks). Reducing the decoders susceptibility to theinput noise is important since a single error may propagate to all

future outputs! That is, the decoders output may become unstable. Ifthe sum is strictly less than one, an input error will affect only asmall number of outputs.

Optimal predictorsConsidering the prediction error resulting from DPMC coding themonochrome Lena image assuming a zero quantization error andusing each of the 4 predictors:

( , ) 0.97 ( , 1) ( , ) 0.5 ( , 1) 0.5 ( 1, ) ( , ) 0.75 ( , 1) 0.75 ( 1, ) 0.5 ( 1, 1)

0.97 ( , 1) ( , )

0.97 ( 1, )

f x y f x y

f x y f x y f x y

f x y f x y f x y f x y

f x y if h v f x y

f x y otherwise

= = + = +

=

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where( 1, ) ( 1, 1)

( , 1) ( 1, 1)

h f x y f x y

v f x y f x y

= =

are the horizontal and vertical gradients at point ( x,y).


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Optimal predictorsObserve that the 4 th adaptive predictor is designed to improve edge

rendition by computing a local measure of the directional properties.

computed for the 4 predictors.

The visually perceptive errordecreases as the predictorsorder increases.

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The standard deviations are11.1, 9.8, 9.1, and 9.7 intensitylevels.

Optimal quantizationThe staircase quantization functiont = q ( s) is an odd function of s (i.e.q(- s) = -q( s)) and can be describedcompletely by the L/2 values of siand t i shown in the first quadrant ofthe graph. These break points definefunction discontinuities and arecalled the quantizers decision and

reconstruction levels.

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y t e convent on, s s mappe to t i t es n si , s i+ 1 .

The quantizer design problem is to select the best si and t i for a particular optimization criterion and input pdf p( s).


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Optimal quantizationIf the optimization criterion (that can be either statistical or psycho-

visual measure) is the minimization of the mean-square quantizationerror E {( si t i)2} and p( s) is an even function, the conditions form n ma error are

1

1

( ) ( ) 1, 2,... 2

0 0

1,2,... 2 1

i

i

s

i s

i ii

s t p s ds i L

i

t t s i L

+

=

= += =

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2

i i

i i

i L

s st t

=

= =

Optimal quantizationTherefore, the reconstruction levels are the centroids of the areaunder p( s) over the specified decision intervals; the decision levelsare halfway between the reconstruction levels; q is an odd function.

e quan zer sa s y ng e a ove con ons s op ma n e mean-square error sense and is called the L-level Lloyd-Max quantizer.

For a unit variance Laplacian pdf, the 2-, 4-, and 8-level Lloyd-Maxdecision and reconstruction levels (computed numerically) are:

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Optimal quantizationThe 3 quantizers shown provide fixed output rates of 1, 2, and 3

bits/pixel. If a non-unit variance pdf is considered, the reconstructionand decision levels are obtained by multiplying the shown values by

e s an ar ev a on o e cons ere p .The last row in the table shows the step size satisfying

1 1i i i it t s s = = The Lloyd-Max quantizer is not adaptive. However, adjusting thequantization levels based on the local behavior of an image can bever beneficial. In theor slowl chan in re ions can be finel

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quantized, while the rapidly changing areas are quantized morecoarsely. This approach reduces both the granular noise and theslope overload, while requiring a minimal increase in code rate andincreased quantizer complexity.

Wavelet compressionThe idea of wavelet-based compression is similar to the idea of DCTcompression: the transform coefficients can be stored moreefficiently than the pixels themselves since a transform decorrelates

e n orma on s ore n p xe s. e rans orm as s unc ons nthis case wavelets) pack most of the visual information in fewcoefficients, the remaining coefficients can be quantized coarsely orzeroed with little image distortion.

A typical

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waveletcodingsystem


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Wavelet compressionTo encode a 2 Jx2J image, an analyzing wavelet and a minimumdecomposition level J-P are selected and used to compute DWT of theimage. If the wavelet has the complementary scaling function , theas wave e rans orm can e use . e rans orm conver s a arge

portion of the original image to horizontal, vertical, and diagonalcoefficients with zero mean and Laplacian-like distribution. Many ofthese coefficients carry little visual information and can be quantizedand coded to reduce inter-coefficient and coding redundancy.

Since the wavelet transform is both computationally efficient and

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inherently local (since basis functions have a limited duration), imagesubdivision into block is not needed, which eliminate the blocking

artifact and is the major difference compared to the transform codingsystems.

Wavelet compression

The selection of a wavelet affects the computational complexity of

Wavelet selection

images of acceptable error.

When a wavelet has a companion scaling function, transformationscan be implemented as a sequence of filtering operations. The abilityof the wavelet to pack information into a small number of transformcoefficients determines its compression and reconstruction

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per ormance.

The most frequently used are Daubechies wavelets and biorthogonalwavelets.


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Wavelet compression3-scale

Haarwavelet

3-scale

Daubechieswavelet

rans orm

3-scales melet

rans orm

3-scaleCohen-

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transform Daubechies-Feauveau

biorthogonalwavelettransform

Wavelet compressionSymlets are an extension of Daubechies wavelets with increasedsymmetry. As seen in the table, computational intensity increases(from 4 to 28 multiplications and additions per coefficient) whenmov ng o more comp ex wave e s, as we as e n orma on

packing performance.

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The coefficients below 1.5 were set to zero. The potentialcompression ability of biorthogonal wavelet is almost 10% higherthan the one of Haar wavelet.


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Wavelet compressionDecomposition level selection

Since a P -scale fast wavelet transform involves P filter banks, thenumber of iterations in the computation of the forward and inversetransforms increases with the number of decomposition levels. Inmany applications (like searching image database or transmittingimages for progressive reconstruction), the resolution of the storedor transmitted images and the scale of the lowest usefulapproximation normally determine the number of transform levels.

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or a biorthogonal

wavelet anda globalthreshold 25

Wavelet compression

The most important factor affecting wavelet coding compression andreconstruction error is coefficient uantization. The most widel

Quantizer design

used compressors are uniform. However, the effectiveness of thequantization can be improved significantly by

1) Introducing a larger quantization interval around zero (a dead zone );

2) Adapting the size of the quantization interval from scale to scale.

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In either case, the selected quantization intervals must be transmittedto the decoder with the encoded image bit stream. The intervalsthemselves may be determined heuristically or automaticallycomputed based on the image being decoded.


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Wavelet compression : JPEG-2000For instance, the irreversible component transform of JPEG-200 is:

0 0 1 2( , ) 0.299 ( , ) 0.587 ( , ) 0.114 ( , )Y x y I x y I x y I x y= + +1 0 1 2

3 0 1 2

, . , . , . ,

( , ) 0.5 ( , ) 0.41869 ( , ) 0.08131 ( , )Y x y I x y I x y I x y

=

where I 0 , I 1 , and I 2 are the level-shifted input components and Y 0 , Y 1 ,and Y 2 are the corresponding decorrelated components. If the inputcomponents are the red, green, and blue planes, the equation

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r .transformation is to improve compression efficiency since thetransformed components Y 1 and Y 2 are difference images whosehistograms are highly peaked around zero.

Wavelet compression : JPEG-2000After the image has been level shifted and optionally decorrelated,its components can be divided into tiles rectangular arrays of

p xe s t at are processe n epen ent y prov ng a s mp emechanism to access and/or manipulate a limited region of a codedimage.

For instance, an image with a 16:9 aspect ratio can be subdividedinto tiles such that one of its tiles is a subimage with a 4:3 aspect

ratio. That tile could be reconstructed without accessing the other

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tiles in the compressed image.

If the image is not subdivided into tiles, it is called a single tile .


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Wavelet compression : JPEG-2000 Next, the 1D DWT of the rows and columns of each tile component

is computed. For error-free compression, the transform uses a biorthogonal 5-3 coefficient scaling and wavelet vector. A rounding proce ure s use or e rans orm coe c en s av ng non- n egervalues.In lossy applications, a 9-7 coefficient scaling-wavelet vector isused. In either case, the transform is computed using either the fastwavelet transform or a complementary lifting-based approach.

The coefficients

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used to constructthe 9-7 FWT

analysis filter bank

Wavelet compression : JPEG-2000The complementary lifting-based implementation involves 6sequential lifting/scaling operations:

[ ] 0 1(2 1) (2 1) (2 ) (2 2) 3 2 1 3Y n X n X n X n i n i + = + + + + + < +[ ]

[ ]0 1

0 1

(2 ) (2 ) (2 1) (2 1) 2 2 2

(2 1) (2 1) (2 ) (2 2) 1 2 1 1

(2 ) (2 ) (2 1) (2 1

Y n X n Y n Y n i n i

Y n Y n Y n Y n i n i

Y n Y n Y n Y n

= + + + < ++ = + + + + + < += + + +[ ] 0 1

0 1

0 1

) 2

(2 1) (2 1) 2 1

(2 ) (2 ) 2

i n i

Y n K Y n i n i

Y n Y n K i n i

08-3 - Image Compression

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