Topic19 Image Coding-II

ECEECE--G311G311

22--D Signal and Image ProcessingD Signal and Image Processing

Image CodingImage Coding--IIII

ECE G311 – Image Coding-II Slide 2 of 36

OutlineOutlineWaveform Coding

Pixel CodingPCM

Entropy CodingHuffmanGroup CodingLZW Coding

Predictive CodingDelta ModulationLine-by-Line DPCM2D DPCMLossless DPCM


Coding TechniquesCoding TechniquesImage coding schemes can be categorized in the following generalgroups:

Waveform Coding: Coding of image intensities.Pixel Coding: PCM, Lemple-Ziv coding, Huffman coding, Facsimile Coding, Bit-plane Coding.Predictive Coding: Line-by-line DPCM, 2-D DPCM, Tree encoding.

Transform Coding: Coding of transform coefficients.Zonal CodingThreshold Coding

Image Model Coding: Coding of image model coefficients.Other Techniques:

Hybrid CodingAdaptive Coding


Waveform CodingWaveform CodingIn this approach, the coding is performed on spatial-domain image values which are assigned a codeword.There are three techniques:

Pixel-coding:In this techniques. Each pixel is processed independently.Generally speaking, this techniques is also used to achieve loss-less coding.

Predictive coding:In this technique, correlations among a group of pixels is used to remove redundant data.The difference or the new information is then coded.

Binary Image coding:This technique is used in facsimile (or fax) transmission.Presence of a large amount of white or black space is used to compress data.


Pixel CodingPixel CodingIn this approach, each pixel is assumed to be an independent random variable with a prescribed probability.Let the number of gray-levels in an image be equal to L, i.e.,

with probabilities:

Basic Principle:Assign code-words of length li to each level value xi so that the average bit-rate

is minimum.

0 1 1, , , : Gray-levelsLx x x −K

0 0 1 1 1 1Pr( ) , Pr( ) , , Pr( )L Lx p x p x p− −= = =K

1

0

L

i ii

R p−

=∑ l


Pulse Code Modulation (PCM)Pulse Code Modulation (PCM)Let the gray-levels be equally likely, i.e.,

Then the entropy H is given by

Thus we can assign B-bit words to each level and achieve the average bit-rate R equal to H.This results is a fixed-length (or block) coding and is the classic PCM technique.When the gray-level probabilities are not equally likely, then entropy H is not an integer and PCM is not the best approach.However, for practical considerations, images sometimes are stored as B (= 8) bits/pixel.

0 1 1 with 2BLp p p L−= = = =L

1 1 1

2 20 0 0

1 1 1 1 1log log2

L L L

BH B BL L L L

− − −⎛ ⎞⎛ ⎞= − = − = =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠∑ ∑ ∑


Entropy CodingEntropy CodingWhat we need a variable length coding technique that can assign a uniquely decodable (prefix-free) code-words to achieve bit-rate R very close to the entropy H.Such techniques are called Entropy Coding techniques.The best optimal solution of attaining R to H was proposed by David Huffman (1950) as a part of his term project in the Information theory class at MIT. It is now known as Huffman coding technique.Huffman’s trick, in today’s jargon, was to “think outside the box.”He studied binary code tree to establish properties that an optimal prefix-free code should have.These few properties led to a simple recursive procedure for constructing an optimal code.


Entropy Coding (2)Entropy Coding (2)Example-1: To illustrate Huffman coding technique consider our 4×8, 256-level image from an earlier example.

21 21 21 95 169 243 243 24321 21 21 95 169 243 243 24321 21 21 95 169 243 243 24321 21 21 95 169 243 243 243

with gray-level histogram

3/81/81/83/8

1244

12

2195

169243

ProbabilityCountGray-Level


Entropy Coding (3)Entropy Coding (3)Example-1: (Continued)Reduction: Arrange gray-levels in the decreasing order of their probabilities

1/8169

1/41/895

3/83/83/8243

5/83/83/821

Reduction-2Reduction-1ProbabilityGray-Level


Entropy Coding (4)Entropy Coding (4)Example-1: (Continued)Assignment: Arrange gray-levels in the decreasing order of their probabilities

1/8169

1/41/895

3/83/83/8243

5/83/83/821

Reduction-2Reduction-1ProbabilityGray-Level

0

1

11 1

00

01

0000

010

011

The average bit-rate for this assignment is3 3 bits1 1

Huff 8 8 8 8 symbol1 2 3 3 1.875 1.81 R H= × + × + × + × = < =


Entropy Coding (5)Entropy Coding (5)Example-2: (from Gonzalez and Woods)

Huff 0.4 1 0.3 2 0.1 3 0.1 4 0.06 5 0.04 52.2 bits/sym2.14 bits/sym

R

H

= × + × + × + × + × + ×==


Entropy Coding (6)Entropy Coding (6)MATLAB Implementation: IPT does not have any functions for Huffman encoding. However G&W book toolbox provides several functions for Huffman data and image encoding.

CODE = HUFFMAN(P) returns a Huffman code as binary strings in cell array CODE for input symbol probability vector P. Each word in CODE corresponds to a symbol whose probability is at the corresponding index of P. Uses the following two functions.SC = REDUCE(P) create a Huffman source reduction tree in a MATLAB cell structure by performing source symbol reductions until there are only two reduced symbols remaining.MAKECODE(SC,CODEWORD) scan the nodes of a Huffman source reduction tree recursively to generate the indicated variable length code words. >> p = [0.1875,0.5,0.125,0.1875];

>> c = huffman(p)c = ‘011’ ‘1’ ‘010’ ’00’


Entropy Coding (7)Entropy Coding (7)MATLAB Implementation (continued): The following G&W book toolbox function performs Huffman image encoding.

Y = MAT2HUFF(X) Huffman encodes matrix X using symbol probabilities in unit-width histogram bins between X's minimum and maximum values. The encoded data is returned as a structure Y:

Y.code The Huffman-encoded values of X, stored in a uint16 vector. The other fields of Y contain additional decoding info.>> x = imread(‘Tracy.tif’); >> y = mat2huff(x);>> cr1 = imratio(x,y)cr1 = 1.2191>>>> save CTracy y;>> cr2 = imratio(‘tracy.tif’,...

,’CTracy.mat’);cr2 = 1.2365

Difference due to MATLAB data-format overhead


Entropy Coding (8)Entropy Coding (8)MATLAB Implementation (continued): The following G&W book toolbox function performs Huffman image decoding.

X = HUFF2MAT(Y) decodes a Huffman encoded structure Ywith uint16 fields. The output X is of class double.>> load CTracy;>> xhat = huff2mat(y);>> x = imread(‘Tracy.tif’);>> rmse = compare(x,xhat)rmse = 0

Thus Huffman technique is a lossless coding technique which achieves the bit-rate very close to the entropy.In fact, there is no other uniquely decodable assignment that can achieve the bit-rate < RHuff.


Entropy Coding (9)Entropy Coding (9)Disadvantage:

For a large number of levels, this scheme is time consuming.For example, if L = 256 (8-bit image) then we will need 254 source reductions and then 254 code-word assignments.If the probability distribution changes, then we will have to repeat this procedure.

Therefore, in practice, we consider other variable-length coding schemes that are suboptimal yet simple to implement.Truncated Huffman Coding:

In this approach, most probable L1 < L levels are Huffman coded.The remaining (L – L1) levels are coded using a prefix-code followed by a suitable fixed-length code.


Entropy Coding (10)Entropy Coding (10)Modified Huffman Coding:

Here again the most probable L1 < L levels are Huffman coded.Next, the level-index i in {xi, i = 0, 1, …, L – 1} is represented as

For example,

Now the remaining (L – L1) levels can be represented by the indices k and l.These remaining levels are coded by a prefix code for k and a terminator code which is the Huffman code for l.

11

1

0 10 ( / ) 1

Li kL

k L L≤ ≤ −⎧

= + ⎨ ≤ ≤ −⎩

ll

51

1Let 256, 8 32 2

then 0 7, 0 31

LL LL

k

= = ⇒ = =

≤ ≤ ≤ ≤l


Entropy Coding (11)Entropy Coding (11)Group (BN) Codes:

When the gray-level probabilities obey a power law of the form

for some positive β, then group codes are nearly optimal.In this scheme, each code is constructed using two parts:

Continuation bits, denoted by C, and Information bits that are fixed-length PCM codes.

In BN codes, N information bits are used for each continuation bit.The advantage of these group codes is the simplicity in design.Many text images, when converted into run-lengths (to be discussed later) follow the power-law distribution.Hence group codes are used in Fax transmission.

( )( ) 1 ; 0,1, , 1ip x i i Lβα −= + = −K


Entropy Coding (12)Entropy Coding (12)Example-3: Consider an image with 6 levels.

RB2 = 3.33RB1 = 2.6RHuff = 2.2H = 2.15

C00C01C10C11

C00C00C01C00

C0C1

C0C0C0C1C1C0C1C1

10001101010100001001

0.40.30.110.080.060.05

x0x1x2x3x4x5

B2B1HuffmanProbabilityLevel

Code-words using B1:

0 2 5 1 001010 010111 or 100000111101x x x x →


Entropy Coding (14)Entropy Coding (14)Example-4: (from Gonzalez and Woods)


Entropy Coding (16)Entropy Coding (16)Lempel-Ziv-Welch (LZW) Code: This coding scheme has gained a wide acceptance especially for computer files.

It is used in .zip, uuencode, uudecode, gzip, .tiff, etc. algorithms.The scheme uses variable-to-variable-length codes, in which both the number of source symbols encoded and the number of encoded bits per codeword are variable. Moreover, the code is space-varying.It does not require prior knowledge of the source statistics, yet over time it adapts so that the average codeword length R per source letter is minimized. Such algorithms are called universal Algorithms.


Entropy Coding (16)Entropy Coding (16)Lempel-Ziv-Welch (LZW) Code: (continued)

Two schemes were developed in 1977-78. The first, LZ77, uses string-matching on a sliding window; whereas the second, LZ78, uses an adaptive dictionary.Although newer schemes have improved upon them, LZW scheme provides a simple approach to understanding universal data compression algorithms.Basic Idea:

Parse the data as it comes from the source into variable-length blocks called phrases.When a new sequence (old phrase + new symbol) is observed, a new phrase is created.New phrase is encoded as the position of the old phrase concatenated by the new symbol.

This scheme is ill suited for hand calculations (needs software).


Entropy Coding (17)Entropy Coding (17)Example-5: Consider the 4×4, 8-bit image shown on the right. (G&W)

A dictionary of 512 words is created as shown below on the right.The first 256 entries are for basic symbols and are considered as initial conditions.The remaining entries are initially empty.The image is processed left-to-right and top-to-bottom.The coding process is shown in the table on the next slide.

39 39 126 12639 39 126 12639 39 126 12639 39 126 126

01...

255−...−

01...

255256...

511

EntryDictionary Location

Initial 512-word Dictionary


Entropy Coding (18)Entropy Coding (18)Example-5: (continued)

39-3939-126126-126126-39

39-39-126

126-126-39

39-39-126-126

126-39-39

39-126-126

256257258259

260

261

262

263

264

3939126126

256

258

260

259

257126

3939126126393912612639391261263939126126

393912612639

39-39126

126-12639

39-3939-39-126

126126-3939

39-126126

Dictionary Entry

Dictionary Location

(Codeword)Encoded Output

Pixel Being Processed

Currently Recognized Sequence

Thus the original (16×8=) 128-bit image is coded with (10×9=) 90-bit image with the compression ratio of 1.42.


Entropy Coding (19)Entropy Coding (19)

Tracy.tif Tracy_LZW.tif

>> cr = imratio('Tracy.tif','Tracy_LZW.tif')cr =

1.6729

which is a larger CR than the Huffman encoding because LZW uses correlations between pixels which Huffman does not.

Example-6: (G&W)


Predictive CodingPredictive CodingThis technique exploits interpixel redundancy to achieve a significant amount of data compression.Let

and

Then

is the prediction error (also known as the Innovations Process).It is a causal, causally invertible, and uncorrelated random field.

1 2( , ) : zero-mean random fieldx n n

1 2

1 2 1 2 1 2 1 2

( , ) causal optimal predictor

E ( , ) past signal values ( , ), ( , ) ( , )

x n n

x n n x k k k k n n⎡ ⎤= <⎣ ⎦

)

1 2 1 2 1 2( , ) ( , ) ( , )e n n x n n x n n− )


Predictive Coding (2)Predictive Coding (2)Basic Principle:Remove the mutual redundancy between successive pixels by quantizing and coding only the new information contained in the innovations process or in the error difference image.In general, the optimal predictor (conditional mean) is a nonlinear function of the past pixel values.However, in the case of a Gaussian random field, the optimal estimator is also linear which is easier to implement.For ease of implementation, we will consider only linear (and hence suboptimum) predictors.In this case, the prediction error is still causal, causally invertible, but is not uncorrelated (or white).


Predictive Coding (3)Predictive Coding (3)Example-7:Consider the following four predictors:

The error images are:

1 1 2 1 2

2 1 2 1 2 1 2

3 1 2 1 2 1 2 1 2

1 24 1 2

1 2

1 2 1 2

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

0.97 ( , 1) h v( , )

0.97 ( 1, ) otherwise

h ( 1, ) ( 1, 1

x n n x n nx n n x n n x n nx n n x n n x n n x n n

x n nx n n

x n n

x n n x n n

= −= − + −= − + − − − −

− Δ ≤ Δ⎧= ⎨ −⎩

Δ = − − − −

)

)

)

)

1 2 1 2

)

v ( , 1) ( 1, 1)x n n x n nΔ = − − − −

1 2 1 2ˆ( , ) ( , ); 1, 2,3,4k ke x n n x n n k= − =


Predictive Coding (4)Predictive Coding (4)Example-7: (continued)

The higher-order predictor gives the best result.

1 1 2( , )e n n 2 1 2( , )e n n

3 1 2( , )e n n 3 1 2( , )e n n


Predictive Coding (5)Predictive Coding (5)Optimum 2-D Linear Predictor: Using the image formation model, the one-step linear predictor of x(n1, n2) is given by

1 21 2

1 2 , 1 1 2 2( , )

( , ) ( , )a

k kk k

x n n a x n k n k∈

− −∑R

)

where Ra is the model support for the NSHP {ak1,k2} coefficients.

These coefficients are determined by minimizing the MSE

This minimization is done using the orthogonality principle which results in the normal set of equations:

2 21 2 1 2 1 2E ( , ) E ( , ) ( , )e n n x n n x n n⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

)

or in matrix notation1 2, 1 1 2 2 1 2( , ) ( , )x xa R k k R k k− − =∑ l l l l

1−= ⇒ =Ra r a R r


Predictive Coding (6)Predictive Coding (6)Optimum 2-D Linear Predictor: (continued)

1−= ⇒ =Ra r a R rwhere

R is the autocorrelation matrix, r is the autocorrelation vector, and a is the vector of unknown coefficients.

The prediction error is zero mean and its mean-squared value or variance is given by

1 21 2

2 2, 1 2

( , )( , )e x k k x

k ka R k kσ σ= − − −∑

21 2 where is the variance of the RF ( , ).x x n nσ


Predictive Coding (7)Predictive Coding (7)Predictor Geometry:

1 2( , )n naR

Encoder Structure:

TRANSFORMATION

QUANTIZER CODEWORDASSIGNMENT

1 2( , )x n n BitStream

1 2( , )e n n

PREDICTOR1 2( , )x n n)

Uncorrelated,Small-variance signal

High-correlated,Large-variance signal

+−


Predictive Coding (8)Predictive Coding (8)Linear Predictive Coding (LPC) Techniques:Based on the complexity in the predictor and the quantizer, we have three approaches:

Delta Modulation (DM):Uses a 1st-order one-step predictor and one bit in the quantizer.Differential Pulse Code Modulation (DPCM):Uses higher-order predictors and more bits in the quantizer. There are two possibilities:

Line-by-Line DPCM:This approach predicts one entire scan-line using a 1-D vector predictor.2-D DPCM:This approach predicts one pixel at a time using 2-D scalar predictor.


Predictive Coding (9)Predictive Coding (9)1. Delta Modulation:

This is the simplest predictive coder which employsOne-step (first-order) predictor:

1-bit Quantizer:

Encoder:

1 2

1 2 h 1 2 1 2( , )

( , ) ( , 1) ( , )x n n

x n n x n n e n nρ= − +)

1442443

1 21 2

1 2

( , ) 0( , ) ; : Step-size

( , ) 0e n n

e n ne n n

+Δ >⎧= Δ⎨−Δ <⎩

1 2( , )x n n 1 2( , )e n n

1 2( , )x n n)

+−

+Δ

−Δ

[ ]1 2 1 2( , ) sgn ( , )e n n e n n= Δ%

1 2 1 2 1 2

1 2 h 1 2 1 2 1 2

If ( , ) and are small, then ( , ) ( , ).Hence, ( , ) ( , 1) ( , ) ( , ).

e n n e n n e n nx n n x n n e n n x n nρ

Δ ≈− + ≈

%

% %


Predictive Coding (10)Predictive Coding (10)1. Delta Modulation: (continued)

How should the predicted value be computed? Consider

1 2( , )x n n 1 2( , )e n n

1 2( , )x n n)

+−

+Δ

−Δ

[ ]1 2 1 2( , ) sgn ( , )e n n e n n= Δ%

?PREDICTOR FILTER

■ Decoder: Must generate pixel value from the quantized error.

1 2 1 2 1 2 1 2( , ) ( , ) ( , ) ( , )e n n x n n x n n x n n+ = ≈)

% % %

1 Pixel Delay

hρ

1 2( , )e n n% 1 2( , )x n n%

1 2( , )x n n)%

++

1 2( , )x n n≈

1 2( , )x n n)%

1 2( , 1)x n n −%



In the receiver, the decoder has to generate predictor estimates based on Hence, it is necessary that the encoder also make use of in order to maintain synchronization between the transmitter and the receiver.Therefore, we have to use in order to generateshown below:

1 2 1 2( , ), or equivalently, ( , ).x n n e n n% %

e%

1 2( , )e n n% , asx)%

1 Pixel Delayhρ

1 2( , )e n n%

1 2( , )x n n%

1 2( , )x n n)%

+

+

1 2( , 1)x n n −%

1 2( , )x n n)%

Filter



Encoder:

1 2( , )x n n 1 2( , )e n n+−

+Δ

−Δ

Decoder:

1 Pixel Delay

hρ

1 2( , )e n n% 1 2( , )x n n%++

1 Pixel Delayhρ

1 2( , )e n n%

+

+


Predictive Coding (13)Predictive Coding (13)Delta Modulation: (continued)

Disadvantages:Uses on the horizontal correlation but not higher-order or 2-D correlations.Slope overload.Granularity.Limited compression ratios (< 3).

Granularity



Example-8:

Original Image Delta Modulation: ρh = 0.97


Predictive Coding (15)Predictive Coding (15)2. Line-by-Line DPCM

This technique treats each scan line as a 1-D signal and performs an LPC on it.It employs a higher-order 1-D predictor and more bits in the quantizer. Therefore, slope-overload and granularity problems are avoided.However true 2-D correlations are not used.Encoder and Decoder Equations: Pth-order predictor

[ ]1 2 1 21

1 2 1 2 1 2 1 2 1 2

1 2 1 2 1 21

( , ) ( , )

( , ) ( , ) ( , ); ( , ) Q ( , )

( , ) ( , ) ( , )

Ppp

Ppp

x n n a x n n p

e n n x n n x n n e n n e n n

x n n a x n n p e n n

=

=

= −

= − =

= − +

∑

∑

)% %

)% %

% % %


Predictive Coding (16)Predictive Coding (16)2. Line-by-Line DPCM: (continued)

Analysis:

If P = 1 then

Hence for this case it can be shown that

For example, if

or if

1 2 1 2 1 2 1 2If ( , ) ( , ), then ( , ) ( , )e n n e n n x n n x n n≈ ≈% %

1 h : Horizontal correlation coefficient.a ρ=

PCM 1D-DPCM 2 2h

1 1log2 1

R Rρ

⎛ ⎞− = ⎜ ⎟

−⎝ ⎠0.97, thenhρ =

PCM 1D-DPCM 2 bpp CR 8 6 1.33R R− = ⇒ = =

0.999, thenhρ =

PCM 1D-DPCM 4.5 bpp CR 2.28R R− = ⇒ =


Predictive Coding (17)Predictive Coding (17)2. Line-by-Line DPCM: (continued)

Example-9

Original Image Line-by-Line DPCM: ρh = 0.97, 4-bit Quantizer


Predictive Coding (18)Predictive Coding (18)3. 2-D DPCM

This is the most general predictive technique and employs a causal NSHP model in the predictor and hence 2-D correlations are used.Encoder and Decoder Equations: Similar to 1-D LPC

[ ]

1 21 2

1 21 2

1 2 , 1 1 2 2( , )

1 2 1 2 1 2

1 2 1 2

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

( , ) ( , )

( , ) ( , ) ( , );( , ) Q ( , )

( , ) ( , ) ( , )

a

a

k kk k

k kk k

x n n a x n k n k

e n n x n n x n ne n n e n n

x n n a x n k n k e n n

∈

∈

= − −

= −

=

= − − +

∑

∑

R

R

)% %

)%

%

% % %


Predictive Coding (19)Predictive Coding (19)3. 2-D DPCM: (continued)

The predictor coefficients are given by

where R is the autocorrelation matrix, r is the autocorrelation vector, and a is the vector of unknown coefficients.

The prediction error is zero mean and its mean-squared value or variance is given by

1−= ⇒ =Ra r a R r

1 21 2

2 2, 1 2

( , )( , )e x k k x

k ka R k kσ σ= − − −∑

21 2 where is the variance of the RF ( , ).x x n nσ


( ) ( )2 2v h2

v h 2(1,0) (0,1)

(1,1)(0,0)

x x x xx x

xx

R RR

R

σ ρ σ ρσ ρ ρ

σ= = =


Example-10: Consider a zero-mean separable RF with

Note that

Then

This gives a very simple prediction mask:

2121 2 v h( , )x xR σ ρ ρ= lll l

yields=Ra r

1,0 v 0,1 h 1,0 v h(1,0) (0,1) (1,1)

; ;(1,0) (1,0) (1,0)

x x x

x x x

R R Ra a a

R R Rρ ρ ρ ρ= = = = = = −

v h v

h 1 2( , )n nρ ρ ρρ

−


Predictive Coding (21)Predictive Coding (21)3. 2-D DPCM:

Example-10: (continued)Note that

Thus this predictor requires only the horizontal and vertical correlations which are easy to compute.If the image is highly correlated thenHence

Therefore, we are successful in converting a highly correlated, high variance image into an uncorrelated, low variance “difference” image.

[ ][ ]

1 2

1 2 1 2 1 2 1 22 2 2 2

v h

E ( , ) 0

E ( , ) ( , ) 0,( , ) ( , ) Uncorrelated

(1 )(1 )e x

e n n

e n n e k k k k n n

σ σ ρ ρ

=

= ≠ ⇒

= − −

h h 1 (but 1).ρ ρ≈ ≈ <

2 2 2 2h v(1 )(1 ) 1 e xρ ρ σ σ− − ⇒


Predictive Coding (22)Predictive Coding (22)3. 2-D DPCM:

Example-10: (continued)The rate reduction is given by

For example, if ρh = ρv = 0.97, then

If ρh = ρv = 0.997, then

2

PCM 2D-DPCM 2 2

2 2 2v h

1 log2

1 1log2 (1 )(1 )

x

eR R

σσ

ρ ρ

⎛ ⎞− = ⎜ ⎟

⎝ ⎠

⎛ ⎞= ⎜ ⎟

− −⎝ ⎠

PCM 2D-DPCM 4 bpp CR 8/ 4 2R R− = ⇒ = =

PCM 2D-DPCM 7.4 bpp CR 13R R− = ⇒ =



Encoder:

1 2( , )x n n 1 2( , )e n n+−

NSHP Predictor

1 2( , )e n n%

+

+

Decoder:

1 2( , )e n n% 1 2( , )x n n%++

QUANTIZER

NSHP Predictor



Example-11

Original Image 2-D DPCM: 4 bpp


Predictive Coding (25)Predictive Coding (25)

DPCM Result Images DPCM Error Images

1-bit 1.125-bits

2.125-bits

3.125-bits

2-bits

3-bits

1-bit 1.125-bits

2.125-bits

3.125-bits

2-bits

3-bits

3. 2-D DPCM (continued): Example-12


Predictive Coding (26)Predictive Coding (26)4. Lossless DPCM

To preserve the quality of images, one could perform DPCM without a quantizer. The required encoder and decoder are:

Encoder:

Decoder:

BitStream

CODEWORDASSIGNMENT

1 2( , )x n n 1 2( , )e n n


+−

NEARESTINTEGER

1 2( , )e n n1 2( , )x n nSymbol

decoder


BitStream



MATLAB Implementation (G&W Book Toolbox)

Y = MAT2LPC(X,a);

Encodes matrix X using 1-D lossless predictive coding. A linear prediction of X is made based on the coefficients in a. If a is omitted, a = 1 (for previous pixel coding) is assumed. The prediction error is then computed and output as encoded matrix Y.X = LPC2MAT(Y,a);

Decodes input matrix Y based on linear prediction coefficients in a and the assumption of 1-D lossless predictive coding. If a is omitted, filter a = 1 (for previous pixel coding) is assumed.



Example-13: Consider the aligned matchstick image encoded using the following predictor

The resulting integer-valued error image is computed and coded using the mat2lpc function.

[ ]( , ) round ( , 1)x m n x m n= −)

>> x = imread(‘aligned_matches.tif’);

>> e = mat2lpc(x);

>> entropy(e)

ans = 5.9727 % Less than image entropy

% equal to 7.3505



Example-13: (continued)>> imshow(mat2gray(e));

>> y = mat2huff(e);

>> cr = imratio(x,y);

cr = 1.3311 % more than 1.0821 for

% the original image

>> [h,f] = hist(e(:)*512,512);

>> bar(h,f,’k’);

>> xhat = lpc2mat(huff2mat(y));

>> compare(x,xhat)

ans = 0

Topic19 Image Coding-II

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Transcript of Topic19 Image Coding-II