2004 International Conference on Image Processing ( U P )

MULTIDIMENSIONAL SIGNAL COMPRESSION USING MULTI-SCALE RECURRENT PATTERNS WITH SMOOTH SIDE-MATCH CRITERION

Eddie B L Filhot Murilo B de Curvalhot Eduurdo A B du Silvai

Genius Institute of Technology AV Acai 875 BL E Manaus - AM 69075-904 BRAZIL

t PEWCOPPWDEUPOLI Universidade Federal do Rio de Janeiro Cx P 68504 Rio de Janeiro - RJ 21945-970 BRAZIL

1 TETICTC Universidade Federal Fluminense R Passos da Pbtria 156 Niteroi RJ 24210-240 BRAZIL

efilhogeniusorgbr murilotelecomuffbr eduardolpsufrjbr

ABSTRACT The recently proposed method for image compression hased on multi-scale recurrent pattcms the M M P (Multidimensional Multi- scale Parser) has been shown to perform well for large class of images specially for those containing lest or graphics However i ts petiormance for coding smooth gray scale images was st i l l dis- tant from the state-of-the art In this paper we propose an extcn- sion for it the SM-MMP (Side-match MMP) In it as in MMP a multidimensional signal is recursively segmented into variable- length hlocks and each segment is encoded using expansions and contractions of vectors in a dictionary The dictionary is updated whilc the data i s being encoded using concatenations o f expanded and contracted versions of previously encoded blocks However unlike MMP in SM-MMP the dictionaries xe built considering smoothness constraints around block boundaries similarly to what happcns in side-match vector quantization methods This allows it to pctiorm better than MMP when the images are smooth without sacrificing i ts performance for images containing text or graph- ics Indded our simulation results show that the proposcd method is effective yielding improvements of the order of 15 dB over the original M M P for grayscale images while preserving the high performance of the original M M P for graphics text and mixed im- ages Keywords Rrcrrrrent Patrem Murchirrg Multi-scale Drcomposi- rion Miilridimensioitd Sigiiul Compression Vecror Qiianiizuriim Sidr Murdz

1 INTRODUCTION

In a recent work [I] we have introduced the Multidimensional Multi-scale Parser algorithm (MMP) a lossy data compression method based on approximate pattern matching with scahs Simu- lation results have shown that M M P is very eFfective to compress a variety of image sources When applied to images containing text or graphics as well as compound images with text graphics and grayscale imagcs the MMP algorithm outperformed other coders representing the spate-or-the-art in image compression However for pure grayscale images although its performance was better than the one of some transform hascd coders such as the DCT- hased JPEC it did not perform as well as other coders based on the DWT as the SPIHT [5] and JPEC2000 [4] This can be ex- plained by the fact that the success of transform-based schemes

~ ~~ ~~

This work was accomplished through the pmnrnhip between UFAM (Universidade Fedcnl do Amazonas) and UFRJICOPPE with the finan- cial suppen pruvidcd by SUFRAMA (Superintendencia da Zona Fnnca de Mnnaus)

relies on the supposition that the image data i s essentially o f a lowpass nature so that most of their important information will he clustered in the low frequency transform coefficicnts leading tu efficient encoding schemes for the quantized cocfficients Haw- ever this assumption i s not true for a large class of image data as lor example text and graphics A l l the tests were done with grayscale images hut it i s easy to extend the algorithm to work with n-dimensional data Unlike most methods uscd in image and video compression the MMP encodes a source without us- ing the transformation-quantizstionentmpy coding paradigm In addition i t makes no strong a priori assumption about the signal i t attempts to compress Thus although i t performs well for a large class of image data thc MMP is not as successful lo compress grayscale images as it i s to compress mixed images when com- pared to DWT-based ctxlers Based on the above one possible way to improve the performance of M M P with grayscale images is lo assume some modcl for grayscale image sources and fine tune the algorithm accordingly Ideally this should be done in such a way that the good performance of MMP with lest graphics an mixed image sources is not impaired

In this paper we propose an extension lo M M P that we refer to as SM-MMP (Side Match-Multidimensional Multi-scale Parser) In it as in the original MMP a signal is encoded by first segmcnt- ing i t into variable-length blocks Each block is encoded using expansions and contractions of vectors from a dictionary The dic- tionary is updated as the data is being encoded with concatena- tions of expansions and contractions o f previously encoded vec- tors However unlike the original MMP the SM-MMP adopts an underlying model for the source When attempting to encode a given block SM-MMP first measures the behavior of the im- age in a causal neighborhood of the block Then i t builds a sub- dictionary referred to as slate dicrionary by selecting from the dictionary only those vectors which meet some smoothness crite- ria relative to the causal neighborhood This way the state dic- tionary can he of a size ranging from a minimum predefined size up to the size of the complete dictionary This is equivalent to set the probability of occurrence of those vectors that do not meet the smwthness criteria to zero (they are out o f the slate dicrionov) which i s a reasonable assumption if the image is smooth By re- ducing the dictionary to a state dictionary which contains only the most prohable vectors the SM-MMP has the potential to lower the mean rate while keeping the mean distortion unchanged Note that this technique is similar to that used in the side-match vector quantization scheme [ 2 ]

The remaining o f this paper is organized as follows In sec- tion 2 the original version of the MMP algorithm i s presentcd In section 3 we propose the side match extension lhe SM-MMP Ex-

0-7803-8554-304$2000 02004 IEEE 3201

perimental results with image data are presented in section 4 and section 5 presents the conclusions

2 THE MMP ALGORITHM

The MMP algorithm is a method to lossy compress data using up- pmrimnrr mnrchin~ wilh scoles This is an extension o f ordinary approximate pattern matching where we allow vectors of dityerent lengths to be matched In order to do this we use a scale transfor- matinn Ti lip c) RN to adjust the sizes of the vectors prior to the miltching attempt [I] In the case of image sources we can use a two-dimensional scale transformation to allow the matching of two matrices of ditkrent sizes

In a two-dimensional MMP an input matrix X i s parsed in L non-overlapping blocks and each hlock is represcnted by a scale- transformed version s o f a matrix Si in a dictionary V This process is illustrated in figure I(a)

(b)

Fig I Segmentation in MMP

The segmentation o f the input matrix X can be represented by a segmentation tree S as illustrated in figure I(h) Each node I Q of S is assminted to a block X of size (-LJ N x 2-151 N ) where p i s the depth o f the node rij in the segmentation tree and 1x1 i s the largest intcgcr that is not greater than a A node nj of the segmentation tree has either two children nodes nZj+ and i i21+2 or no child at all A ride that has no child is a leaf node In M M P only the lealt nodes of the segmentation tree are associated to matrices in the dictionary that are used to approximate the input matrix

The dictionary in M M P is updated as follows Whenever the approximations X2jf and X2+ associated to the children nodes n2j+l and n2+2 are available MMP forms an estimate Xj o f the block associated lo the parent node nj as the concatenation o f the approximations to the blocks associated to the two children nodes In the example o f figure I when and XG are available we can concatenate them to get a new approximation X2 This new ap- proximation can then be included in the dictionary This procedure can he executed at the encoder side and repeated at the decoder side We just need to output to the decoder the information regard- ing the segmentation tree and the dictionary indexes corresponding

Fig 2 Up and left neighbors

As in the original MMP wc use the dictionary containing ma- trices of size n x m Lo encode the block X3 However we USE a smoorlraess criruion to modify the probability of occurence as- sociated to each element of the dictionary In our model those vectors in the dictionary that do not meet the smoothness critenon have their probabilities set to zero This i s equivalent to reduce the cadinality o f the dictionary We call the reduced dictionary the stare rlicrioizary To build the state dictionxy we make decisions bascd on the nrgosir metric rj = R (U3 L7 Si) defined as

R(UjLS) =

3202

The nrgosirv can he seen as a measure of how different are border pixels from neighboring hlixks As we assume that the image being prnccssed is smooth the first elements in the sfure dicrionuy are the ones that present the smallest nrgosirier

To encode a given block Xi we choose a state dictionary D that i s composed of the N least rugose elements S of the dictio- nary D according to equation ( I ) that is the N elements with smaller values of rij The size N of the state dictionary depends on the level of acriviry or thc blocks Uj and LJ We evaluate the activity of a block X of size 71 x m hy the function 4(X) defined as

The acrivit i s a measurc of the uncertainty about the other pixels that do not helong to the border of the block A high level of uncertainty about them means that we need a bigger srnre d i e riunay for a more precise encoding

The size N of the state dictionary used t o cncodr a block X3 isdetermined by the ratio N ( (A (Ui) + A (L)) 2) A where A i s the maximum value of A(X) for the hole image and N i s the maximum sizc allowed for the state dictionary

updating procedure by forcing the algorithm to learn not only the concatenations of hlocks hut also shifted versions of those Th is allows the dictionary 2 to grow faster reducing the mean distor- tion Note that the mean rate i s nut significantly affecled since the size N of the state dictionary V3 i s independent o f thc size of D (see equation (2))

To further improve the performance we modilied the dictionary-

4 EXPERIMENTAL RESULTS

We have implemented the original MMP and the SM-MMP We used the programs to lossy compress gray-scale still image data The input images were initially divided in I6 x 16 blocks that were sequentially processcd by the algorithms The initial dictionaly contained only vectors at scale 1 x 1 and was set to Vo = (0 2 253) The vectors at all other scales were oh- tained by the use of a sepuroble two-dimensional scale transfor- mation that was implementcd using classical sampling-rate change operations as described in [IT Since the block size i s 16 x 16 9 different scales have been used (see figure I(a))

Figures 3 4 5 and 6 show the R-D performance of the al- gorithms with the images Lena F-16 PPI205 and PPIZOY all o f size 512 x 512 The results lor the Side-Match vector quantizer described in 131 Gust for Lena and Flh) and for the SPIHT algo- rithm 151 are also shown for comparison Since very similar results are obtained for JPEG2000 [41 and SPlHT algorithms we have not included the results for the first (in fact results Tor JPEC2000 tend to be slightly worse than the results for SPIHT since due to JPEG2000s much larger flexibility i ts headers are much longer what decreases its codingefficiency) The test images Lcna and F- 16 were downloaded from the location httplsipiuscedu The im- ages PPI205 and PPI209 were scanned respectively from pages 1205 and 12W of the IEEE Transucrions on lmupe prucessing volume 9 number 7 July 2000 PPI205 contains only text and formulas while PPI029 i s a compound o f gray-scale (two com- pressed versions oflena) tcxt formulas and graph show us that

07 02 03 04 05 06 07 08 09 1 11

R PPp)

Fig 3 R-D pertormancc with Lena 512 x 512

01 02 03 04 05 06 07 08 09 1 11

R (I

Fig 4 R-D performance with F-I6 512 x 512

i) The SM-MMP algorithm outperformed MMP for almost all images of the test group and presented equivalent results for PP1205

i i) Both SM-MMP and M M P algorithms outperformed SPIWT z 2dB for PPI209 (compound grayscale text and graphs) and hy FZ 4d8 for PPI205 (text and equations)

iii) The SPlHT algorithm outperformed SM-MMP for the image Lena The pertormancc with F-I6 below 05 bpp was equiva- lent for the two algorithms

iv) SM-MMP outperformed the Side-Match VQ (31 for the F16 test image at a l l rates For the image Lena SM-MMP per- formed heller above 04 bpp Note although the rate range provided in reference [3] i s very narrow we can see a clear tendency for the improvement provided by SM-MMP over Side-Malch V Q 10 increase with the rate (we did not have ac- cess for results of Sidc-Match VQ for either other images or other rates) Figure 7 shows a delail ofthe image Lena compressed by SM-

MMP and MMP It i s quire clear the reduction of the blocking effect by the use o f the smoothness criterion in SM-MMP

5 CONCLUSIONS

In this paper we have proposed SM-MMP a method for multi- dimensional signal compression based on MMP We were able to achieve significant gains over the original MMP for smooth images by introducing an underlying statistical model to the image source This was obtained without rxr i f ic ing its performance for images with texts and graphics The adopted model is quite simple based on the side-match vector quantisation approach We expect even better results as we improve the image model using For example Gibbs distribution models

3203

perimental results with image data are presented in section 4 and section 5 presents the conclusions

2 THE MMP ALGORITHM

The MMP algorithm is a method to lossy compress data using up- pmrimnrr mnrchin~ wilh scoles This is an extension o f ordinary approximate pattern matching where we allow vectors of dityerent lengths to be matched In order to do this we use a scale transfor- matinn Ti lip c) RN to adjust the sizes of the vectors prior to the miltching attempt [I] In the case of image sources we can use a two-dimensional scale transformation to allow the matching of two matrices of ditkrent sizes

In a two-dimensional MMP an input matrix X i s parsed in L non-overlapping blocks and each hlock is represcnted by a scale- transformed version s o f a matrix Si in a dictionary V This process is illustrated in figure I(a)

(b)

Fig I Segmentation in MMP

The segmentation o f the input matrix X can be represented by a segmentation tree S as illustrated in figure I(h) Each node I Q of S is assminted to a block X of size (-LJ N x 2-151 N ) where p i s the depth o f the node rij in the segmentation tree and 1x1 i s the largest intcgcr that is not greater than a A node nj of the segmentation tree has either two children nodes nZj+ and i i21+2 or no child at all A ride that has no child is a leaf node In M M P only the lealt nodes of the segmentation tree are associated to matrices in the dictionary that are used to approximate the input matrix

The dictionary in M M P is updated as follows Whenever the approximations X2jf and X2+ associated to the children nodes n2j+l and n2+2 are available MMP forms an estimate Xj o f the block associated lo the parent node nj as the concatenation o f the approximations to the blocks associated to the two children nodes In the example o f figure I when and XG are available we can concatenate them to get a new approximation X2 This new ap- proximation can then be included in the dictionary This procedure can he executed at the encoder side and repeated at the decoder side We just need to output to the decoder the information regard- ing the segmentation tree and the dictionary indexes corresponding

Fig 2 Up and left neighbors

As in the original MMP wc use the dictionary containing ma- trices of size n x m Lo encode the block X3 However we USE a smoorlraess criruion to modify the probability of occurence as- sociated to each element of the dictionary In our model those vectors in the dictionary that do not meet the smoothness critenon have their probabilities set to zero This i s equivalent to reduce the cadinality o f the dictionary We call the reduced dictionary the stare rlicrioizary To build the state dictionxy we make decisions bascd on the nrgosir metric rj = R (U3 L7 Si) defined as

R(UjLS) =

3202

The nrgosirv can he seen as a measure of how different are border pixels from neighboring hlixks As we assume that the image being prnccssed is smooth the first elements in the sfure dicrionuy are the ones that present the smallest nrgosirier

To encode a given block Xi we choose a state dictionary D that i s composed of the N least rugose elements S of the dictio- nary D according to equation ( I ) that is the N elements with smaller values of rij The size N of the state dictionary depends on the level of acriviry or thc blocks Uj and LJ We evaluate the activity of a block X of size 71 x m hy the function 4(X) defined as

The acrivit i s a measurc of the uncertainty about the other pixels that do not helong to the border of the block A high level of uncertainty about them means that we need a bigger srnre d i e riunay for a more precise encoding

The size N of the state dictionary used t o cncodr a block X3 isdetermined by the ratio N ( (A (Ui) + A (L)) 2) A where A i s the maximum value of A(X) for the hole image and N i s the maximum sizc allowed for the state dictionary

updating procedure by forcing the algorithm to learn not only the concatenations of hlocks hut also shifted versions of those Th is allows the dictionary 2 to grow faster reducing the mean distor- tion Note that the mean rate i s nut significantly affecled since the size N of the state dictionary V3 i s independent o f thc size of D (see equation (2))

To further improve the performance we modilied the dictionary-

4 EXPERIMENTAL RESULTS

We have implemented the original MMP and the SM-MMP We used the programs to lossy compress gray-scale still image data The input images were initially divided in I6 x 16 blocks that were sequentially processcd by the algorithms The initial dictionaly contained only vectors at scale 1 x 1 and was set to Vo = (0 2 253) The vectors at all other scales were oh- tained by the use of a sepuroble two-dimensional scale transfor- mation that was implementcd using classical sampling-rate change operations as described in [IT Since the block size i s 16 x 16 9 different scales have been used (see figure I(a))

Figures 3 4 5 and 6 show the R-D performance of the al- gorithms with the images Lena F-16 PPI205 and PPIZOY all o f size 512 x 512 The results lor the Side-Match vector quantizer described in 131 Gust for Lena and Flh) and for the SPIHT algo- rithm 151 are also shown for comparison Since very similar results are obtained for JPEG2000 [41 and SPlHT algorithms we have not included the results for the first (in fact results Tor JPEC2000 tend to be slightly worse than the results for SPIHT since due to JPEG2000s much larger flexibility i ts headers are much longer what decreases its codingefficiency) The test images Lcna and F- 16 were downloaded from the location httplsipiuscedu The im- ages PPI205 and PPI209 were scanned respectively from pages 1205 and 12W of the IEEE Transucrions on lmupe prucessing volume 9 number 7 July 2000 PPI205 contains only text and formulas while PPI029 i s a compound o f gray-scale (two com- pressed versions oflena) tcxt formulas and graph show us that

07 02 03 04 05 06 07 08 09 1 11

R PPp)

Fig 3 R-D pertormancc with Lena 512 x 512

01 02 03 04 05 06 07 08 09 1 11

R (I

Fig 4 R-D performance with F-I6 512 x 512

i) The SM-MMP algorithm outperformed MMP for almost all images of the test group and presented equivalent results for PP1205

i i) Both SM-MMP and M M P algorithms outperformed SPIWT z 2dB for PPI209 (compound grayscale text and graphs) and hy FZ 4d8 for PPI205 (text and equations)

iii) The SPlHT algorithm outperformed SM-MMP for the image Lena The pertormancc with F-I6 below 05 bpp was equiva- lent for the two algorithms

iv) SM-MMP outperformed the Side-Match VQ (31 for the F16 test image at a l l rates For the image Lena SM-MMP per- formed heller above 04 bpp Note although the rate range provided in reference [3] i s very narrow we can see a clear tendency for the improvement provided by SM-MMP over Side-Malch V Q 10 increase with the rate (we did not have ac- cess for results of Sidc-Match VQ for either other images or other rates) Figure 7 shows a delail ofthe image Lena compressed by SM-

MMP and MMP It i s quire clear the reduction of the blocking effect by the use o f the smoothness criterion in SM-MMP

5 CONCLUSIONS

In this paper we have proposed SM-MMP a method for multi- dimensional signal compression based on MMP We were able to achieve significant gains over the original MMP for smooth images by introducing an underlying statistical model to the image source This was obtained without rxr i f ic ing its performance for images with texts and graphics The adopted model is quite simple based on the side-match vector quantisation approach We expect even better results as we improve the image model using For example Gibbs distribution models

3203

The nrgosirv can he seen as a measure of how different are border pixels from neighboring hlixks As we assume that the image being prnccssed is smooth the first elements in the sfure dicrionuy are the ones that present the smallest nrgosirier

To encode a given block Xi we choose a state dictionary D that i s composed of the N least rugose elements S of the dictio- nary D according to equation ( I ) that is the N elements with smaller values of rij The size N of the state dictionary depends on the level of acriviry or thc blocks Uj and LJ We evaluate the activity of a block X of size 71 x m hy the function 4(X) defined as

The acrivit i s a measurc of the uncertainty about the other pixels that do not helong to the border of the block A high level of uncertainty about them means that we need a bigger srnre d i e riunay for a more precise encoding

The size N of the state dictionary used t o cncodr a block X3 isdetermined by the ratio N ( (A (Ui) + A (L)) 2) A where A i s the maximum value of A(X) for the hole image and N i s the maximum sizc allowed for the state dictionary

updating procedure by forcing the algorithm to learn not only the concatenations of hlocks hut also shifted versions of those Th is allows the dictionary 2 to grow faster reducing the mean distor- tion Note that the mean rate i s nut significantly affecled since the size N of the state dictionary V3 i s independent o f thc size of D (see equation (2))

To further improve the performance we modilied the dictionary-

4 EXPERIMENTAL RESULTS

We have implemented the original MMP and the SM-MMP We used the programs to lossy compress gray-scale still image data The input images were initially divided in I6 x 16 blocks that were sequentially processcd by the algorithms The initial dictionaly contained only vectors at scale 1 x 1 and was set to Vo = (0 2 253) The vectors at all other scales were oh- tained by the use of a sepuroble two-dimensional scale transfor- mation that was implementcd using classical sampling-rate change operations as described in [IT Since the block size i s 16 x 16 9 different scales have been used (see figure I(a))

Figures 3 4 5 and 6 show the R-D performance of the al- gorithms with the images Lena F-16 PPI205 and PPIZOY all o f size 512 x 512 The results lor the Side-Match vector quantizer described in 131 Gust for Lena and Flh) and for the SPIHT algo- rithm 151 are also shown for comparison Since very similar results are obtained for JPEG2000 [41 and SPlHT algorithms we have not included the results for the first (in fact results Tor JPEC2000 tend to be slightly worse than the results for SPIHT since due to JPEG2000s much larger flexibility i ts headers are much longer what decreases its codingefficiency) The test images Lcna and F- 16 were downloaded from the location httplsipiuscedu The im- ages PPI205 and PPI209 were scanned respectively from pages 1205 and 12W of the IEEE Transucrions on lmupe prucessing volume 9 number 7 July 2000 PPI205 contains only text and formulas while PPI029 i s a compound o f gray-scale (two com- pressed versions oflena) tcxt formulas and graph show us that

07 02 03 04 05 06 07 08 09 1 11

R PPp)

Fig 3 R-D pertormancc with Lena 512 x 512

01 02 03 04 05 06 07 08 09 1 11

R (I

Fig 4 R-D performance with F-I6 512 x 512

i) The SM-MMP algorithm outperformed MMP for almost all images of the test group and presented equivalent results for PP1205

i i) Both SM-MMP and M M P algorithms outperformed SPIWT z 2dB for PPI209 (compound grayscale text and graphs) and hy FZ 4d8 for PPI205 (text and equations)

iii) The SPlHT algorithm outperformed SM-MMP for the image Lena The pertormancc with F-I6 below 05 bpp was equiva- lent for the two algorithms

iv) SM-MMP outperformed the Side-Match VQ (31 for the F16 test image at a l l rates For the image Lena SM-MMP per- formed heller above 04 bpp Note although the rate range provided in reference [3] i s very narrow we can see a clear tendency for the improvement provided by SM-MMP over Side-Malch V Q 10 increase with the rate (we did not have ac- cess for results of Sidc-Match VQ for either other images or other rates) Figure 7 shows a delail ofthe image Lena compressed by SM-

MMP and MMP It i s quire clear the reduction of the blocking effect by the use o f the smoothness criterion in SM-MMP

5 CONCLUSIONS

In this paper we have proposed SM-MMP a method for multi- dimensional signal compression based on MMP We were able to achieve significant gains over the original MMP for smooth images by introducing an underlying statistical model to the image source This was obtained without rxr i f ic ing its performance for images with texts and graphics The adopted model is quite simple based on the side-match vector quantisation approach We expect even better results as we improve the image model using For example Gibbs distribution models

3203

Fig 7 Image Lena (detail) (a) MMP at 030 hpp PSNR = 3271 dB (b) SM-MMP at 030 hpp PSNR = 3413 dB

I l l

P I

131 n 02 04 06 08 i 12

R

Fig 5 R-U performance with PPI205 512 x 512

[41

[51

01 02 03 04 05 06 07 08 09 1 11

R PPP)

Fig 6 R-D performance with PP1209 512 x 512

6 REFERENCES

M B de Carvnlho E A B da Silva and W A Finamore ldquoMultidimensional Signal Compression using Multiscale Re- cument Pettemsrsquo Elsevicrlsquos Signal Processing Vol 82 No 11 pp 1559-1580 November 2002 S B Yang and L Y Tseng ldquoSmooth Side-Match Classified Vector Quantizer with Variable Block Sirerdquo IEEE Transac- tions on Image Processing Vol IO No 5 pp 677485 May 2001 S B Yang ldquoGeneral-Trec-Structured Vector Quantizer for Image Progressive Cuding Using thc Smmth Side-Match methodrdquo IEEE Transactions on Circuits and Systems for Video Tcchnology Vol 13 No 2 pp 193-202 February 2003 U S Taubman M W Marcellin ldquoJPECZWX) Image Com- pression Fundamentals Standards and Practicerdquo Kluwer Academic Publishers 2001 ASaid and WA Pexlmon ldquoA new fast and efficient image codec hased on set partitioning in hierarchical treesrsquo IEEE Transactions on Circuits and Systems for Video Technology vol6 pp243-250 June 1996

3204