Wavelet Transform Based Image Compression CODECS

A Dissertation On

“Wavelet Transform Based Image Compression CODECS” By

Sandip D. Lulekar (2006MEC007)

Supervisors Dr. S. V. Bonde

Dr. T. R. Sontakke

Sant Gajanan Invention & Advanced Research Center (SGIARC)

Shri Sant Gajanan Maharaj College of Engineering, SHEGAON (M.S.) 2

Outline 1. 2. 3. 4. 5. 6. 7. 8.

Introduction Problem Definition Block Diagram of CODEC Wavelet Transform Algorithms (EZW, SPIHT & SPECK) Design of 2D-DWT core Results Conclusions & Future Work 3

1. Introduction 1.1 Image Compression • Digital Image – matrix of pixels • Uncompressed multimedia data – (graphics, audio & video) requires considerable storage capacity & huge transmission bandwidth • Data compression – represents information in a compact form • Image compression – creates image file sizes of manageable & transmittable dimensions 4

1.2 Need for Image Compression

5

1.3 Principles behind Image Compression • • 1. 2. 3.

Redundancy reduction – removing duplication from the signal source (image/video) Irrelevancy reduction – omits parts of the signal that will not be noticed by the signal receiver (HVS) Spatial Redundancy – correlation between neighboring pixels Spectral redundancy – correlation between different color planes or spectral bands Temporal redundancy – correlation between adjacent frames in a sequence of images 6

2. Problem Definition • Heterogeneous network • Providing efficient compression is not sufficient • Main challenge – how a single flexible bit-stream can be provided to optimally service each user • Solution – some sort of scalable & embedded functionalities need to be provided by the CODEC • Wavelet Transform – algorithms: • EZW, SPIHT & SPECK

based

image

compression

7

• Scalability – a potential in the coded bit-stream, which allows the decoder to usefully decode from only parts of the bit-stream • In order to meet certain requirements such as quality, spatial resolution & temporal resolution • Embeddedness – a single coded file can be used to decode the image 1. MATLAB Implementation – EZW, SPIHT & SPECK 2. CADENCE Implementation – 2D DWT core 8

3. Block Diagram of CODEC 1. COder

2. DECoder Entropy Decoder

Decoder

Inverse Quantizer

Inverse WT 9

4. Wavelet Transform on Image 4.1 Background • Signals – in their raw form are time-amplitude representation • Transformation – These time-domain signals are often needed to be transformed into other domains like frequency domain • Transform – of a signal is just another form of representing the signal • It does not change the information content present in the signal 10

• Transformation of signals – helps in identifying distinct information which might otherwise be hidden in the original signal • Depending on the application – the transformation technique is chosen, and each technique has its own advantages & disadvantages • Transform Types 1. FT (Fourier Transform) – DFT, STFT & FFT 2. DCT (Discrete Cosine Transform) 3. WT (Wavelet Transform) – CWT, DWT 11

4.2 Why Wavelet Transform? 1. Fourier Transform (FT) • Frequency content – of the signal is very important • Most popular transform – used to obtain the frequency spectrum of a signal • Suitable for stationary signal – signals whose frequency content does not change with time •

Drawback – tells how much of each frequency exists in the signal, but it does not tell at which time these frequency components occur 12

2. Short Time Fourier Transform (STFT) • Signals – image & speech have different characteristics at different time or space • i.e. they are non-stationary • To analyze – these signals, both frequency & time information are needed simultaneously • Hence STFT was introduced • Input signal – chopped into sections, & each is analyzed for its frequency content separately •

Drawbacks – fixed window width – gives constant resolution at all frequencies 13

3. Wavelet Transform (WT) • A Wave – is an oscillating function of time or space • Wavelets – are localized waves, & have their energy concentrated in time or space and are suited to analysis of non-stationary signals • Wavelet Transform – provides a better time-frequency representation of the signal • •

Multi-resolution Technique – by which different frequencies are analyzed with different resolutions Image compression – based on the ideas of sub-band decomposition or Discrete Wavelet Transform 14

4.3 Pyramidal Decomposition LL

HL

LH

HH

15

4.4 2D-DWT of Image LL

HL

LH

HH

Image

16

5. Algorithms 5.1 EZW (Embedded Zero-tree Wavelet) •

Introduced by J. M. Shapiro in 1993 1. Larger Wavelet coefficient – contains more information & therefore the EZW algorithm encodes the larger wavelet coefficients first 2. Maximum & average absolute coefficient – values tend to get smaller as one move from the lower frequency subbands to the higher frequency subbands 17

5.1.2 Zero Tree Structure • Zero-tree structure – is a tree in which the parent-object has four child objects • Parent – coefficients at coarse scale • Children – coefficients corresponding to same spatial location at the next finer scale of similar orientation • zero-tree structure decreases from parent to child 18

• Significant Test – if a coefficient is found to be insignificant with threshold ‘tO’ then all the children will be insignificant too & therefore ‘branch’ will not contain any important information • A whole tree – could be encoded as a single symbol, resulting in data reduction • Output – produced by the EZW algorithm is progressive in nature • Resolution – as more data is added to the compression process, the more detailed image will be reconstructed • E in EZW – Progressive coding is also known as Embedded coding 19

5.1.3 EZW Algorithm

20

• (DL) – coordinates of the coefficients to be coded • (SL) – coefficients already coded as significant 1. Positive (P) – If the coefficient c is significant according to current threshold ‘to’ and positive 2. Negative Significant (N) – If the coefficient c is significant according to current threshold ‘to’ and negative 3. Isolated Zero (IZ) – If the coefficient c is insignificant according to current threshold ‘to’ and one or more of its descendants significant 4. Zero-tree root (ZT) – If current coefficient c and all of its descendants are insignificant (zero) according to current threshold ‘to’ 21

5.1.4 Dominant Pass

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5.1.5 Subordinate Pass

23

5.2

SPIHT (Set Partitioning In Hierarchical Trees)

5.2.1 Introduction • Introduced – by Amir Said & William Pearlman in 1996 • Produces – an embedded bit stream from which the best reconstructed images can be extracted • Idea – is based on partitioning of sets, which consists of coefficients or representatives of whole sub-trees • Root of the tree – is excluded from the computation of the significance attribute • Classify – the coefficients of a wavelet transformed image into the three different sets: i.e. – LIP, LIS, LSP 24

1. List of Insignificant Pixels (LIP) – which contains the coordinates of those coefficients which are insignificant with respect to the current threshold 2. List of Significant Pixels (LSP) – which contains the coordinates of those coefficients which are significant with respect to threshold 3. List of Insignificant Sets (LIS) – which contains the coordinates of the roots of insignificant subtrees • Sets of coefficients – in LIS are refined and if coefficients become significant they are moved from LIP to LSP, During the compression procedure 25

• Difference – to EZW, is the definition of the significance • A significance function Sn(t) – which decides the significance of the set of coordinates with respect to the threshold 2n Sn(T) = 1 , if max (i,j)ЄT{|Ci,j|}>2n 0 , else Notations • H – Roots of the all spatial orientation trees • O(i, j) – Set of offspring of the coefficient (i, j) • D(i, j) – Set of all descendants of the coefficient (i, j) • L(i, j) – D(i, j) - O(i, j) 26

5.2.2 Parent-Child Relationship in SPIHT

• Significance is computed for the sets D(i, j) and L(i, j) 27

• SPIHT Algorithm 1.Initialization Compute initial threshold LIP: all root nodes (in low pass subband) LIS: all trees (type D) LSP: empty 2.Sorting Pass a) Check significance of all coefficients in LIP • If significant, output 1 followed by a sign bit & move it to LSP • If insignificant, output 0 28

b) Check significance of all trees in LIS For type-D tree • If significant, output 1 & proceed to code its children • If a child is significant, output 1, sign bit, & add it to LSP • If a child is insignificant, output 0 and add it to the end of LIP • If the child has descendants, move the tree to the end of LIS as type L, otherwise remove it from LIS • If insignificant, output 0 For type-L tree • If significant, output 1, add each of the children to the end of LIS as type D and remove the parent tree from LIS • If insignificant, output 0 3. Refinement pass, like EZW • Decrease the threshold by a factor of 2 • Go to Step 2. 29

5.3

SPECK (Set Partitioning Embedded BloCK)

• Different – from others in that it does not use trees, which span and exploit the similarity across different subbands • it makes use of sets or groups of pixels-called blocks • Main idea – is to exploit the clustering of energy in frequency & space in hierarchical structures of the transformed images • Significance testing – on sets determines whether the maximum magnitude in it is above a certain threshold • Results – of these tests determines the path taken by the coder to code the source samples 30

5.3.1 SPECK coding • Image – X is represented by an indexed set of transformed coefficients C{i, j} located at pixel position (i, j) • Pixels – are grouped together in sets, which comprise of regions in the transformed image • Followed by the ideas of SPIHT completes the SPECK algorithm • Condition – a set T of pixels is significant with respect to threshold n, if max (i,j)ЄT{|Ci,j|}>2n Otherwise it is insignificant 31

5.3.2 Partitioning of Image X

• SPECK – algorithm makes use of rectangular regions of image • Regions – or sets therefore referred to as sets of type S, can be of varying dimensions • Dimension – of a set S depends on the dimension of the original image & the subband level of the pyramidal structure at which the set lies 32

• • • •

We maintain two lists: LIS – List of Insignificant Sets LSP – List of Significant Pixels The LIS contains sets of type S of varying sizes which have not yet been found significant against threshold n

• SPECK Algorithm • The actual algorithm consists; 1. Initialization step 2. Sorting Pass 3. Refinement Pass 4. Quantization step 33

5.3.3 SPECK Encoder

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5.3.4 SPECK Decoder

35

6. Design of 2D-DWT core • 2D-DWT/IDWT – is a heart • Hence its core (HDL-Code) was designed & verified • Consist of computation, data storage, and control blocks • EDA Tools – CADENCE Design Systems Inc., platform • NC-VHDL – all the blocks and the interface were described • RTL compiler – after the RTL core description was verified, the gate level circuit was synthesized • NCSim – for the verification of the whole functions we simulated the core with the behavioral description 36

6.1 Original DWT

37

6.2 Modified DWT

38

6.3 Architecture

39

6.4 CADENCE EDA Tools Flow

40

7. Results 7.1 MATLAB Platform Tree Coding EZW

Block Coding SPECK

SPIHT

Wavelet type: biorthogonal 4.4, Level: 7 Algorithm

Image

Size

MSE

PSNR(dB)

EZW

Lena

SPIHT

Lena

256x256 512x512 256x256 512x512

155.66 79.04 14.84 6.72

26.21 29.15 36.41 39.85 41

Wavelet type: biorthogonal, Level: 10 Algorithm

Image

Size

MSE

PSNR(dB)

Lena

128x128

135.30

26.19

Penny

512x512

4.83

41.29

SPECK

42

1. EZW Algorithm

43

2. SPIHT Algorithm

44

3. SPECK Algorithm

45

7.2 CADENCE EDA Tools Platform • Benchmark example – such as a Lena image used • Simulation results – were compared with the results obtained from the MATLAB programs • Synthesized – system composed of 16,187 equivalent 2-input NAND gates, this gate count is considerably small • 8-bit, B/W – Lena image with 256 X 256 pixels required 850,000 clock cycles

46

1. DWT Core Input

47

DWT Core Output

48

2. DWT Control Input

49

DWT Control Output

50

3. IDWT Control Input

51

IDWT Control Output

52

4. RWTU Input

53

RWTU Output

54

5. Module: DWT Top

55

Module: MUX

56

8. Conclusions & Future Work 8.1

Conclusions

• Results – show that these algorithms are successfully implemented • Implementation – creates a single file of coded bitstream • As more no. bits – are added to the right side of the bitstream more fine details/resolution we get • Hence the problem – of embeddedness & scalability in a image compression CODEC as stated in the problem definition are completely eliminated 57

• It can be used in future image compression systems • 2D-DWT core – shows a very good performance • From the satisfactory – simulation and synthesized results we can conclude that 2D-DWT core works properly • Goal – of project work is satisfied

58

8.2 Future Work • M-EZW, M-SPIHT & L-SPECK algorithms can also be implemented • Currently working on SOC Encounter EDA Tool, so that complete VLSI chip of our proposed architecture of the 2D-DWT core can be obtained • 2D-DWT core can be used in various image processing ASIC chips

59

Thank You …!!!

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References [1] R. Sudhakar, Ms R. Karthinga, S. Jayaraman, “Image Compression Using Coding of Wavelet Coefficients – A Survey”, International Congress for Global Science & Technology (ICGST)- International Journal on Graphics, Vision & Image Processing (GVIP), GVIP Special Issue On Image Compression, pp 1 – 13, 2007. [2] Robi Polikar, “The Engineer’s Ultimate Guide to Wavelet Analysis”. http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html [3] K. Sayood, “Introduction to Data Compression”, 2nd Edition, Academic Press, Morgan Kaufmann Publishers, 2000. [4] Jerome M Shapiro, “Embedded Image Coding Using Zerotrees of Wavelet Coefficients”, IEEE Transaction on Signal Processing, vol. 41, no. 12, pp 3445 – 3462, December 1993. 61

[5] Amir Said & Pearlman W.A., “A New, Fast and Efficient Image Coder based on Set Partitioning In Hierarchical Trees”, IEEE Transaction on Circuit and Systems for Video Technology, vol. 6, no. 3, pp 243 – 250, June 1996. [6] Pearlman W. A., Islam A., Nagraj N. & Said A., “Efficient LowComplexity Image Coding with a Set-Partitioning Embedded Block Coder”, IEEE Transaction on Circuit and Systems for Video Technology, vol. 14, no. 11, pp 1219 – 1235, November 2004. [7] Seonyoung Lee & Kyeongsoon Cho, “Design of a Twodimensional Discrete Wavelet Core for Image Compression”, Journal of the Korean Physical Society, vol. 38, no. 3, pp 224 – 231, March 2001. [8] Incisve Simulation, CADENCE Lab Manual, Version 5-8.3, http://www.cadence.com 62