wavelet selection for image compression using spiht...

INTERNATIONAL JOURNAL OF COMPUTATIONAL INTELLIGENCE AND HEALTHCARE INFORMATICS

Vol. 4, No. 2, July-December 2011, pp. 81-87, Published by Serials Publications, ISSN: 0973-7413

Wavelet Selection for Image Compression using SPIHT Algorithm

P. Srinivasan1 and P. Prakasam2

1,2Centre for Advanced Research, Muthayammal Engineering College, Rasipuram-637408, Tamilnadu, India.1E-mail: [email protected], 2E-mail: [email protected]

ABSTRACT: In this paper, SPIHT (Set Partitioning In Hierarchical Trees) algorithm has been studied and tested for 512 × 512Lena, Baboon and Barbara images with various wavelet family. The performance of algorithm has been investigated usingMean Square Error and Peak Signal to Noise Ratio. The achieved results of Lena image show that bior 6.8 reconstructs theimage with PSNR of 42.16 dB, bior 4.4 with 42.07 dB and sym 6 with 41.84 dB for .9 bpp. Similarly the results of Baboonimage show that bior 6.8 reconstruct the image with PSNR of 27.974 dB, bior 4.4 with 27.888 dB, sym 6 with 27.823 dB, db 4with 27.717 dB, sym 4 with 27.715 dB, db 2 with 27.327 dB and sym 2 with 27.327 dB for .9 bpp. The results of Barbara imageshow that bior 6.8 reconstruct the image with PSNR of 35.142 dB, sym 6 with 34.845 dB, bior 4.4 with 34.717 dB, sym 4 with34.374 dB and db 4 with 34.371 dB for .9 bpp. Finally bioorthogonal wavelet family has recommended for Wavelet Imagecoding using SPIHT algorithm.

Keywords: Wavelet Transform, SPIHT Method, Bit error rate, SNR

1. INTRODUCTION

Image compression addresses the problem of reducing theamount of data required to represent a digital image. Theunderlying basis of the reduction process permits (a)individual images to be transmitted faster; (b) more parallelchannels to be transmitted faster; (c) a reduction oftransmitted power requirements; (d) more compact imagestorage; (e) reducing the volume of data to be transmitted(Rafal c. Gonzalez et.al., Raghuveer M. Rao et. al. and JulienReichel et. al., 2001). The compression is applied prior tostorage or transmission of the image. At later, some timethe compressed image is decompressed to acquire theoriginal image or an estimation of it. Psychophysicalexperiments indicate that a human only comprehends visualinformation at a rate of about 50 bits per second. Anadvantage of wavelet transforms is that the windows vary(Raghuveer M. Rao et. al. and Smith M. J. T et. al. 1998).In order to isolate signal discontinuities, one likes to havesome very short basis functions. At the same time, in orderto obtain detailed frequency analysis, one likes to have somevery long basis functions. A way to achieve this is to haveshort high-frequency basis functions and long low-frequencyones. This happy medium is exactly what you get withwavelet transforms. One thing to remember is that wavelettransforms do not have a single set of basis functions likethe Fourier transform, which utilizes just the sine and cosinefunctions. Instead, wavelet transforms have an infinite setof possible basis functions. Thus wavelet analysis(Villasenor. D, et.al. 1995) provides immediate access toinformation that can be obscured by other time-frequencymethods such as Fourier analysis. In this paper the acquired

image is compressed using wavelet based SPIHT (Set Parti-tioning In Hierarchical Trees) algorithm and various imageshas taken as example and analyzed using different waveletsand different bits per pixels.

2. MATERIALS AND METHOD

2.1 SPIHT (Set Partitioning In Hierarchical Trees)Algorithm Adheres to the Following Steps:(Antonini, M. et. al., 1992)

1. Obtain the DWT of the input images and represent thewavelet transform coefficients using a finite numberof bits, which results in a very fine quantization of theDWT image.

2. Quantize the coefficients to 0, 1, or –1 with the thresholdmagnitude set to 2 14. If a wavelet coefficient has avalue less than –2 14, it is set to –1; if it has a valuegreater than 2 14, it is set to 1; and it is set to 0 otherwise.The resulting image is called a significance map.

3. The significance map is tree encoded using the spatialorientation trees.

4. The threshold is lowered to 2 13 and the codingprocedure is repeated on the pixels consideredinsignificant in the previous step. The resulting binarycode is appended to the code of the previous step.

5. The previous steps are repeated successively withdecreasing thresholds 2 12, 2 11 and so on.

2.2 Algorithm

Define the following variables for SPIHT (Set partitioningin hierarchical tree) algorithm(Antonini, M. et. al., 1992).

mailto:[email protected]

mailto:[email protected]

82 International Journal of Computational Intelligence and Healthcare Informatics

• O(i, j): Set of coordinates of all offspring of node (i, j):Children only

• D(i, j): Set of coordinates of all descendants of node(i, j): children, grandchildren, great-grand, etc

• H(i, j): Set of all tree roots (nodes in the highest pyramidlevel): parents

• L(i, j): D(i, j) – O(i, j) (all descendents except theoffspring): grandchildren, great-grand, etc.

LSP- List of Significant Pixel

LIP - List of Insignificant Pixel

LIS - List of Sorting Pass

3.1 Wavelet Selection

The wavelets chosen as the basis of the forward and inversetransforms affect all aspects of wavelet coding system designand performance. They impact directly on the computationalcomplexity of the transforms and less directly the system’sability to compress and reconstruct image of acceptableerror. In this Paper, Haar, Bioorthogonal (Bior 4.4, Bior 6.8),Daubechies (db1, db2, db4) and Symmetric (sym2, sym4,sym6) wavelets has been chosen as the wavelet basisfunction. The above mentioned wavelets are used tocompress the image and Mean Square Error and PSNR (PeakSignal to Noise Ratio) were analyzed. The wavelet imagecoding has been done using SPIHT algorithm and it isimplemented in the platform of MATLAB.

3.2 Implementation Procedure

1. Load an image to be compressed.

2. Compute the wavelet based image compression usingSPIHT algorithm with different wavelets as motherwavelet for different bits per pixel.

3. Reconstruct the compressed image using inverse SPIHTprocedure.

4. Also plot the error image.

5. Compute Mean Square Error and Peak Signal to NoiseRatio for reconstructed image.

6. Based on the analysis, conclude the suitable waveletfor different data rates.

3.3 Lena Image

The 512 × 512 Lena image to be compressed, reconstructedimage and error image using SPIHT algorithm is shown inFig. 1.

Initialization LSP and output sign of coeff: 0/1 = –/+n = |_log2 (max |coeff|)_| if Sn(k, 1)=0, then add (k, 1) to the

LIP = All elements in H end of the LIP

LSP = Empty else (type L)

LIS = D’s of Roots Send Sn(L(i, j))

Significance Map Encoding If Sn(L(i, j))=1(“Sorting Pass”)Process LIP add each (k, 1) ε O(i, j)

for each coeff (i, j) in LIP to the end of the LIS as an

Output Sn(i, j) entry of type D

If Sn(i, j) = 1 remove (i, j) from the LIS

Output sign of coeff(i,j): 0/1 = -/+ end if on type

Move (i, j) to the LSP End loop over LIS

Endif Refinement Pass

End loop over LIP Process LSP

Process LIS for each element (i, j) in LSP – except

for each set (i, j) in LIS those just added above

if type D Output the nth most

Send Sn(D(i, j)) significant bit of coeff

If Sn(D(i,j))=1 End loop over LSP

for each (k,l) ε O(i, j) Update

output Sn(k, l) Decrement n by 1

if Sn(k, l)=1, then add (k,l) to the Go to Significance Map Encoding Step

3. IMPLEMENTATION AND RESULTS

Like other transform coding techniques, wavelet transformcoding is based on the idea that the co-efficient of atransform that de correlates the pixels of an image has beencoded more efficiently than the original pixels themselves.If the transforms basis function packs most of the importantvisual information into a small number of co-efficient, theremaining co-efficient can be quantized coarsely or truncatedto zero with a little image distortion. The principle differencebetween the wavelet-based system and other transformcoding system is the omission of the transform coder’s subimage processing stages. Wavelet transforms are bothcomputationally efficient and inherently local (i.e., theirbasis functions are limited in duration), subdivision of theoriginal image is unnecessary.

The mean square error and PSNR of Lena image istabulated in table 1 and 2 respectively.

Figure 1: Lena – Original Image, Reconstructed Image and Error Image

Wavelet Selection for Image Compression using SPIHT Algorithm 83

Table 1Mean Square Error for Lena Image

Wavelet Haar Bior4.4 Bior6.8 Db1 Db2 Db4 Sym2 Sym4 Sym6Type/ bpp

0.1 126.326 70.916 71.411 126.326 96.334 80.739 96.334 81.48 78.043

0.2 71.298 33.639 33.984 71.298 49.461 39.563 49.461 38.765 37.22

0.3 44.924 21.338 21.128 44.924 29.989 23.96 29.989 23.8 22.466

0.4 32.608 14.132 14.123 32.608 21.523 16.825 21.523 16.826 15.623

0.5 23.843 10.49 10.207 23.843 15.458 11.789 15.458 11.729 10.876

0.6 18.049 8.018 7.898 18.049 11.838 9.181 11.838 9.221 8.548

0.7 14.118 6.388 6.281 14.118 9.48 7.278 9.48 7.331 6.843

0.8 11.581 4.997 4.909 11.581 7.703 5.768 7.703 5.828 5.324

0.9 9.344 4.035 3.955 9.344 6.138 4.562 6.138 4.638 4.255

Table 2Peak Signal to Noise Ratio (PSNR) – Lena Image

Wavelet Haar Bior 4.4 Bior 6.8 Db1 Db2 Db4 Sym2 Sym4 Sym6Type/ bpp

0.1 27.116 29.623 29.593 27.116 28.293 29.06 28.293 29.02 29.207

0.2 29.6 32.862 32.818 29.6 31.188 32.158 31.188 32.246 32.423

0.3 31.606 34.839 34.882 31.606 33.361 34.336 33.361 34.365 34.616

0.4 32.998 36.629 36.632 32.998 34.802 35.871 34.802 35.871 36.193

0.5 34.357 37.923 38.042 34.357 36.239 37.416 36.239 37.438 37.766

0.6 35.566 39.09 39.156 35.566 37.398 38.502 37.398 38.483 38.812

0.7 36.633 40.077 40.151 36.633 38.363 39.511 38.363 39.479 39.778

0.8 37.494 41.144 41.221 37.494 39.264 40.521 39.264 40.475 40.869

0.9 38.426 42.072 42.16 38.426 40.251 41.539 40.251 41.467 41.842

The tabulated results show that bior 6.8 reconstructsthe image with PSNR of 42.16 dB, bior 4.4 with 42.07 dB

and sym6 with 41.84 dB for .9 bpp. The error analysisand SNR analysis of Lena image is shown in Figs. 2 and3 respectively.

Figure 2: Reconstruction Error Analysis – Lena Image Figure 3: Signal to Noise Ratio Analysis – Lena Image

Reconstruction error and SNR analysis show thatBioorthogonal wavelet family gives better estimation ofLena image with low mean square error.

3.4 Baboon Image

The 512 × 512 baboon image to be compressed, reconstruc-ted image and error image using SPIHT algorithm is shownin Fig. 4.


The mean square error and PSNR of Baboon image is tabulated in table 3 and 4 respectively.

Figure 4: Baboon – Original Image, Reconstructed Image and Error Image

Table 3Mean Square Error - Baboon Image


0.1 560.026 509.634 501.905 560.026 526.237 508.658 526.237 509.657 506.172

0.2 448.548 388.803 384.833 448.548 413.74 394.96 413.74 392.015 387.068

0.3 360.934 304.041 297.456 360.934 327.614 307.477 327.614 308.142 303.52

0.4 300.782 248.475 244.421 300.782 269.532 250.656 269.532 251.552 247.334

0.5 258.142 206.379 202.605 258.142 227.496 209.903 227.496 210.974 206.607

0.6 221.487 171.964 167.74 221.487 192.793 177.48 192.793 177.432 173.829

0.7 190.362 145.791 141.359 190.362 163.023 150.356 163.023 150.435 147.83

0.8 163.996 122.401 119.905 163.996 139.33 127.078 139.33 126.975 124.273

0.9 139.962 105.743 103.672 139.962 120.334 122.077 120.334 110.037 107.334

Table 4Peak Signal to Noise Ratio (PSNR) - Baboon Image


0.1 20.649 21.058 21.125 20.649 20.919 21.067 20.919 21.058 21.088

0.2 21.613 22.234 22.278 21.613 21.964 22.165 21.964 22.198 22.253

0.3 22.557 23.302 23.397 22.557 22.977 23.253 22.977 23.243 23.309

Table Cont’d


The tabulated results show that bior 6.8 reconstructsthe image with PSNR of 27.974 dB, bior 4.4 with 27.888dB, sym 6 with 27.823 dB, db 4 with 27.717 dB, sym 4with 27.715 dB, db 2 with 27.327 dB and sym 2 with 27.327dB for .9 bpp. The error analysis and SNR analysis ofBaboon image is shown in Figs. 5 and 6 respectively.

Table 4 Cont’d

0.4 23.348 24.178 24.249 23.348 23.825 24.14 23.825 24.125 24.198

0.5 24.012 24.984 25.064 24.012 24.561 24.911 24.561 24.889 24.979

0.6 24.677 25.776 25.884 24.677 25.28 25.639 25.28 25.641 25.73

0.7 25.335 26.494 26.628 25.335 26.008 26.36 26.008 26.357 26.433

0.8 25.983 27.253 27.342 25.983 26.69 27.09 26.69 27.094 27.187

0.9 26.671 27.888 27.974 26.671 27.327 27.717 27.327 27.715 27.823

The 512 × 512 barbara image to be compressed, reconstructed image and error image using SPIHT algorithm isshown in Fig. 7.

Figure 6: Signal to Noise Ratio Analysis – Baboon ImageFigure 5: Reconstruction Error Analysis - Baboon Image

Reconstruction error and SNR analysis show thatbioorthogonal wavelet, Symmetric wavelet and Daubechieswavelet families give better estimation of Baboon imagewith low mean square error. However Haar wavelet is lesscomplexity wavelet, it reconstructs the image with highmean square error.

Figure 7: Barbara – Original Image, Reconstructed Image and Error Image


The mean square error and PSNR of Barbara image istabulated in table 5 and 6 respectively. The tabulated resultsshow that bior 6.8 reconstructs the image with PSNR of

35.142 dB, sym 6 with 34.845 dB, bior 4.4 with 34.717 dB,sym 4 with 34.374 dB and db 4 with 34.371 dB for .9 bpp.

Table 5Mean Square Error for Barbara Image


0.1 352.939 273.2 267.455 352.939 302.543 276.853 302.543 279.514 273.387

0.2 240.729 164.875 154.462 240.729 191.194 165.923 191.194 166.875 159.123

0.3 181.282 112.154 106.524 181.282 139.868 116.641 139.868 117.032 109.309

0.4 145.999 77.123 71.064 145.999 105.357 84.353 105.357 83.88 77.032

0.5 113.242 56.902 52.507 113.242 77.331 61.167 77.331 61.098 55.916

0.6 87.736 44.359 40.503 87.736 61.239 47.368 61.239 47.528 43.003

0.7 71.952 34.347 31.284 71.952 49.493 37.34 49.493 37.407 33.842

0.8 59.414 26.928 24.353 59.414 39.715 29.977 39.715 29.88 26.687

0.9 48.416 21.947 19.902 48.416 32.211 23.77 32.211 23.753 21.31

The error analysis and SNR analysis of Barbara imageis shown in Figs. 8 and 9 respectively.

Table 6Peak Signal to Noise Ratio(PSNR) - Barbara Image


0.1 22.654 23.766 23.858 22.654 23.323 23.708 23.323 23.667 23.763

0.2 24.316 25.959 26.243 24.316 25.316 25.932 25.316 25.907 26.114

0.3 25.547 27.633 27.856 25.547 26.674 27.462 26.674 27.448 27.744

0.4 26.487 29.259 29.614 26.487 27.904 28.87 27.904 28.894 29.264

0.5 27.591 30.58 30.929 27.591 29.247 30.266 29.247 30.271 30.656

0.6 28.699 31.661 32.056 28.699 30.261 31.376 30.261 31.361 31.796

0.7 29.56 32.772 33.178 29.56 31.185 32.409 31.185 32.401 32.836

0.8 30.392 33.829 34.265 30.392 32.141 33.363 32.141 33.377 33.868

0.9 31.281 34.717 35.142 31.281 33.051 34.371 33.051 34.374 34.845

Reconstruction error and SNR analysis show thatbioorthogonal wavelet, Symmetric wavelet family givebetter estimation result of Barbara image with low meansquare error. However, other wavelet families reconstructthe image with low PSNR.Figure 8: Reconstruction Error Analysis - Barbara Image

Figure 9: Signal to Noise Ratio Analysis - Barbara Image


4. CONCLUSION

JPEG images are considered for image encoding usingSPIHT algorithm. The algorithm has been studied anddeveloped for implementation of 512 × 512 Lena, Baboonand Barbara images. The performance of algorithm has beeninvestigated using Mean Square Error and Peak Signal toNoise Ratio. Implemented and analyzed results of Lenaimage show that bior 6.8 reconstructs the image with PSNRof 42.16 dB, bior 4.4 with 42.07 dB and sym 6 with 41.84dB for .9 bpp. Implemented and analyzed results of Baboonimage show that bior 6.8 reconstructs the image with PSNRof 27.974 dB, bior 4.4 with 27.888 dB, sym 6 with 27.823dB, db 4 with 27.717 dB, sym 4 with 27.715 dB, db 2 with27.327 dB and sym2 with 27.327 dB for .9 bpp. Imple-mented and analyzed results of Barbara image show thatbior 6.8 reconstructs the image with PSNR of 35.142 dB,sym6 with 34.845 dB, bior4.4 with 34.717 dB, sym4 with34.374 dB and db4 with 34.371 dB for .9 bpp. From theanalysis, the conclusion has been made, that the biorthogonalwavelet family is well suitable for Lena Image compression,the biorthogonal, Symmetric and Daubechies waveletfamilies give the better compression for Baboon image andbioorthogonal, symmetric wavelet families are more suitablefor Barbara image. Finally bioorthogonal wavelet family is

recommended for Wavelet Image coding using SPIHTalgorithm. In further, this algorithm is extended to motionimage compression like video image, 3-D image etc,. Videopicture may be partitioning into frame by frame and thisalgorithm may be applied to frame encoding.

References

[1] Antonini M., Barlaud M., Mathieu P., Daubechies I., “ImageCoding Using Wavelet Transform”, IEEE Transaction on ImageProcessing, 1(2), April 1992 pp. 205-220.

[2] Julien Reichel and Murat Kunt, “Integer Wavelet Transform forEmbedded Lossy to Loss Less Image Compression”, IEEETransactions on Image Processing, 10(3), March 2001, pp.383-392.

[3] Rafal C. Gonzalez and Richard E. Woods, Chapter Number 7,“Digital Image Processing”, Second Edition, Pearson Education,pp. 349-408

[4] Raghuveer M. Rao, and Ajit S. Bopardikar, Chapter Number, 1and 2, “Wavelet Transforms, Introduction to Theory andApplications”, Second Indian Reprint, Addison Wesley, pp.1-50

[5] Smith M.J.T., and A. Docef, “A Study Guide for Digital ImageProcessing, 1st Ed. Singapore: World Scientific, 1998.

[6] Villasenor D.B. Belzer, and J. Liao, “Wavelet Filter Evaluationfor Image Compression”, IEEE Trans. Image Process., 4(8), pp.1053-1060, Aug. 1995.

wavelet selection for image compression using spiht...

Documents