New high-fidelity medical image compression based on modified set partitioning in hierarchical trees

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1 New high-fidelity medical image compression based on modified set partitioning in hierarchical trees Shen-Chuan Tai Yen-Yu Chen Wen-Chien Yan National Cheng Kung University Institute of Electrical Engineering Number 1, Ta Hsueh Road Tainan, Taiwan Abstract. Medical images must be compressed before transmission due to bandwidth and storage limitations. The set partitioning in hierarchical trees () algorithm is an efficient method for lossy and lossless coding of medical images. We propose some modifications to the algorithm. It is based on the idea of the insignificant correlation of wavelet coefficients among medium- and high-frequency subbands. In this scheme, insignificant wavelet coefficients that correspond to the same spatial location in the medium subbands can be used to reduce the redundancy by a combined function proposed in associated with the modified. In high-frequency subbands, the modified proposes a dictator to reduce the interband redundancy. Experimental results indicate that the proposed technique improves the quality of the reconstructed medical image in terms of both the peak signal-to-noise ratio (PSNR) and the perceptual results over JPEG2000 and the original at the same bit rate Society of Photo-Optical Instrumentation Engineers. [DOI: / ] Subject terms: set partitioning in hierarchical trees; JPEG2000. Paper received Jul. 22, 2002; revised manuscript received Nov. 26, 2002; accepted for publication Dec. 13, Introduction In diagnoses, medical images including those obtained by computer tomography CT, magnetic resonance imaging MRI, ultrasonography US, and x-ray diffraction are an important basis. The modalities provide a flexible means of viewing anatomical cross sections and physiological states, and may reduce patient radiation dosage and examination trauma. However, medical images have large storage requirements. The limits on storage capacity are such that medical image compression techniques must be employed to reduce the storage requirements. In medical applications, large volumes of digitized images are presented, so image compression is indispensable. In recent years, some American industrial standards such as the American College of Radiology/National Electrical Manufacturers Association ACR/NEMA 1 and Digital Imaging and Communication in medicine DICOM, 2 and others have been established. All apply the method of lossy compression. Lossy method compression is the main area of research. The set partition in hierarchical tree algorithm is an efficient method for lossy and lossless coding of still images. 3 The quality of compressed medical images must reach an acceptable level to avoid misdiagnosis. Wavelet transform coding is a special case of subband coding. 4 Subband coding has been generally adopted for compressing medical images. In Ref. 5, wavelet transform coding was suggested to outperform subband coding at the same bit rate. The algorithm represents an advancement over the innovative wavelet-based image coding method, which employs a tree representation of the zero wavelet coefficients. Other wavelet-based image compression methods, such as trellis-coded quantization 6 in place of the scalar quantization of the original algorithm, has much higher complexity and better performance than the original algorithm. Trellis-coded quantization can improve performance, but this advantage does not justify the raising of the time complexity. Therefore, scalar quantization is needed. The modified algorithm employs scalar quantization. We consider the modification of the algorithm. Section 2 reviews the original algorithm of Ref. 6. Section 3 proposes the modification and details the algorithm. Section 4 presents the simulation results and compares the modified with JPEG2000 and the original algorithm for several kinds of medical images. Section 5 draws conclusions. 2 Original Algorithm A wavelet-transformed image typically has a nonuniform distribution of energy within and across subbands. This distribution motivates the partitioning of each subband into regions, and the assigning of each region to a class, based on region energy. This classifying approach has led to very effective image compression algorithms. The algorithm, introduced by Said and Pearlman, is an efficient method for lossy and lossless coding of natural images. The algorithm adopts a hierarchical quadtree 7 data structure on a wavelet-transformed image. The energy of a wavelet-transformed image is centered on the lowfrequency coefficients. The coefficients are ordered in hierarchies. Figure 1 presents the parent-child relationship 1956 Opt. Eng. 42(7) (July 2003) /2003/$ Society of Photo-Optical Instrumentation Engineers

2 1. Thresholding: in each pass, only the wavelet coefficients that exceed the threshold are encoded. The threshold T(u) is computed according to the expression T u 2 P u, 1 where u 0, 1, 2, 3..., and P denotes the pass number. And P log 2 max c i,j, 2 Fig. 1 Parent-child relationship. through the subbands. According to this relationship, the algorithm saves many bits that specify insignificant coefficients. Normally, most of an image s energy is concentrated in the low-frequency components. Therefore, the variance decreases from the highest to the lowest levels of the subband pyramid. Moreover, a spatial self-similarity has been observed to exist between subbands, and the coefficients are expected to be more effectively magnitude ordered as the pyramid descends. For instance, large, low-activity areas are identified in the highest levels of the pyramid, and they are replicated in the lower levels at the same spatial locations. A tree structure, called a spatial orientation tree SOT, naturally defines the spatial relationship on the hierarchical pyramid. Figure 1 shows how a spatial orientation tree is defined in a pyramid constructed with recursive four-subband splitting. Each node of the tree corresponds to a pixel and is identified by the pixel s coordinate. Its direct descendants offspring correspond to the pixels of the same spatial orientation in the next finer level of the pyramid. The tree is defined such that each node has either no offspring the leaves or four offspring, which always form a group of 2 2 adjacent pixels. In Fig. 1, the arrows are oriented from the parent node to its four offspring. The pixels in the highest level of the pyramid are the tree s roots and are also grouped in groups of 2 2 adjacent pixels. In the original algorithm, the set of the root node and corresponding descendents are defined as a SOT, and three lists are used in encoding the list of significant pixels LSP, the list of insignificant pixels LIP, and the list of the insignificant sets LIS. All lists have the queue structure; the order is first-in first-out FIFO. Initially, the LSP is empty, and the LIP is formed by the elements of the lowest-frequency subband, and the LIS is the roots of the SOT. The wavelet coefficients are encoded and transmitted in multiple passes in the algorithm, as follows. where c(i, j) is the coefficient at position (i, j) in the image. The encoder just sends the maximum value to the decoder, and the thresholds can be calculated from Eqs. 1 and Sorting pass: when u n, n is the integer. The pixel that satisfies T(n) c(i, j) 2T(n) is identified as significant. c(i, j) is the coefficient. The pixel s position and sign bit must be encoded. 3. Refinement pass: the pixels that satisfy c(i, j) 2T(n) are refined by encoding the n th most significant bit of those whose coordinates were transmitted in the previous sorting pass. 4. Increase u by 1, and go to step 2. Figure 2 shows the encoding process in the algorithm. If the coefficient is significant T(n) c(i, j) 2T(n), then the coefficient is moved to the LSP, and the sign bit is encoded. For each node in the LIS, if any descendent is found to be significant, D(i, j) T(n), D(i, j) refers to the nodes of the quadtree of which c(i, j) is a root, then the children are moved to the LSP and removed from the LIP. The child nodes are placed at the end of the LIS. The algorithm can define recursively using the sequence of thresholds. 8 3 Proposed Method A method is proposed to modify the original algorithm to make it suitable for medical images. The original algorithm was an efficient method for lossy and lossless coding of natural images. The original algorithm is modified according to the characteristic that the wavelet coefficients of the medical images are more centered on the low frequency than those of natural images. The contours of medical image contents have less edge than those of natural image contents. Additionally, the quality of medical compressed images must support the diagnosis. The original algorithm ignores the correlation within the same level of subbands, such as LH 3, HL 3, and HH 3. For the insignificant coefficients at high frequency, the original algorithm saves space using the quadtree concept. The modified algorithm not only inherits the quadtree concept, but also deals with the correlation within the same level subbands to reduce the bit rate. In Fig. 1, the nodes coordinates in LL 3 ) have no descendent trees; the nodes coordinates in LH 3, HL 3, and HH 3 ) are the roots of the quadtree, and the rest of the nodes coordinates in other subbands are tree nodes. LH 3, HL 3, Optical Engineering, Vol. 42 No. 7, July

3 Fig. 2 Original algorithm flowchart. 6 Table 1 Percentages of important coefficients at corresponding coordinates in LH 3, HL 3, and HH 3 for several kinds of medical images. Same condition (in LH3, HL3, HH3) Different condition (in LH3, HL3, HH3) Xhead1 97.5% 2.5% Angio2 95.0% 5.0% Ctbone2 93.4% 6.6% Ercp2 95.6% 4.4% Utheart3 80.4% 19.6% and HH 3 are strongly correlated. Table 1 shows the correlation at the corresponding coordinate in LH 3, HL 3, and HH 3 in several types of medical images Fig. 3. The same condition implies that the coefficients at corresponding coordinates in LH 3, HL 3, and HH 3 have unimportant values. The different condition implies that the coefficients at corresponding coordinates in LH 3, HL 3, and HH 3 have at least one important value. For example, if the image decomposed into seven subbands and the first threshold value is 32 in Fig. 4, then the coefficients at corresponding coordinates in LH 3, HL 3, and HH 3 subbands and their offspring are all lower than the threshold value 32. In Fig. 4, the coefficients in the ellipse are insignificant, and this situation is defined as the same condition and the value 0 is sent to the decoder. The coefficients at corresponding coordinates in LH 3, HL 3, and HH 3 subbands and their offspring exceed the threshold value 32 by at least one. In Fig. 4, the coefficients in the dotted rectangle are at least significant, and this situation is defined as the different condition, and the value 1 and the position of the significant coefficient are sent to the decoder. In the Xhead1 test image Fig. 3 b, the percentage of insignificant coefficients at corresponding coordinates in LH 3, HL 3, and HH 3 subbands is 97.5%. This statistic shows that the percentage of significant coefficients in subbands not include LL 3 ) is rare. These coefficients are essential in reconstructing image edges. Large redundancies were hidden in these coefficients. In the Utheart3 test image, the percentage associated with the different condition was higher than other test images, primarily because the Utheart3 sample context is more complex than the others. Table 2 indicates the correlation of the corresponding coordinate in LH 1, HL 1, HH 1, LH 2, HL 2, and HH 2 in all recursions in several types of images. The same condition implies that the treenode s coefficients are unimportant on a quadtrees whose roots are at corresponding coordinates in LH 3, HL 3, and HH 3. The different condition implies that the treenode s coefficients are at least important on a quadtree whose roots are at corresponding coordinates in LH 3, HL 3, and HH 3. That is, the medical image encoded by the original algorithm has several redundancies. The same level subband relationship that is ignored by the original algorithm is exploited here to reduce redundancy. After a wavelet is transformed, the energy is centered on the wavelet coefficients in the low-low band. Accordingly, the modified algorithm divides a wavelet-transformed image into three partitions. C x,y x,y in LL 3, C x,y x,y in LH 3,HL 3,HH 3, 1958 Optical Engineering, Vol. 42 No. 7, July 2003

4 Fig. 4 An image decomposed into seven subbands. unbalanced distribution. To prevent this phenomenon, the reconstructive value R should be calculated by R 1 (T 1 R 0 )/2, and the exact value would yield a balanced distribution. The reduction in the recursive number yields a considerable compression advantage, and the bit rate is reduced by 0.05 to 0.1 bpp. Figure 5 shows that the original algorithm needed six recursions, and the modified algorithm needed just three. The exact value is not changed and balanced. Fig. 3 Test image. C x,y x,y in LH 2,HL 2,HH 2,LH 1,HL 1,HH 1. The partitions include as a partition of the low-frequency coefficients, as a partition of the middle-frequency coefficients, and as a partition of the high-frequency coefficients. C determines whether wavelet coefficients are significant, and (x,y) is the coordinate of the image. If the wavelet coefficient c(x, y) exceeds the threshold value T, then C(x,y) is set to 1. If the wavelet coefficient c(x,y) is below the threshold value T, then C(x,y) is set to For C(x,y) (x,y) inll 3 Each recursion in the original algorithm must send a bit map C(x,y) inll 3, from the threshold value at T 0 to T 1 (T 1 T 0 /2), and reduce the reconstructive value from R 0 to R 1 (R 1 R 0 /2). Both the threshold value T and the reconstructive value R follow geometric progressions. The threshold value must be decreased to reduce the number of encoding bits. The threshold value T 1 was changed from T 0 /2 into T 0 /4 in each recursion. It follows also a geometric progression. Meanwhile, if the reconstructive value R is changed from T 0 /2 to T 0 /4, the exact value would yield an 3.2 For C(x,y) (x,y) inlh 3,HL 3,and HH 3 The modified algorithm uses a set w to reduce the redundancy of partition. The original algorithm does not consider the correlation in the same level subbands. The modified algorithm adopts a set w, and w records which subband among LH 3, HL 3, and HH 3 has a significant coefficient. The modified algorithm adopts a set w to eliminate the correlation in the same level subbands. w w x,y LH 3 x,y HL 3 x,y HH 3 x,y. The partition w must be sent to the decoder. If w(x,y) 1, then the values in LH 3 (x,y), HL 3 (x,y), and HH 3 (x,y) would be send to the decoder. If w(x,y) 0, nothing is sent to the decoder, unlike the original, which sends all the bits of LH 3 (x,y), HL 3 (x,y), and HH 3 (x,y) to the decoder. The method can reduce the bit rate about 0.1 to 0.2 bpp at a given peak signal-to-noise ratio PSNR. Table 2 Percentages of important coefficients in treenodes whose roots are at corresponding coordinates in LH 3, HL 3, and HH 3 in all recursions for several kinds of medical images. Same condition Different condition Xhead1 99.7% 0.3% Angio2 97.3% 2.7% Ctbone2 97.6% 2.4% Ercp2 97.4% 2.6% Utheart3 80.3% 19.7% Optical Engineering, Vol. 42 No. 7, July

5 Fig. 5 In each recursion, threshold T and reconstructive value R. 3.3 For C(x,y) (x,y) in LH 2,HL 2,HH 2,LH 1,HL 1,and HH 1 The modified algorithm proposes a method called the dictator to reduce the redundancy in partition. This partition includes few significant coefficients, and the original algorithm suggests the use of one bit to represent whether the significant coefficient is in the quadtree. The fact that a quadtree includes at least one significant coefficient is represented as 1. That all of the nodes in the quadtree are insignificant coefficients is presented as 0. The subbands originally neglected by the algorithm exhibit quite a large correlation among the same level subbands, and the modified algorithm presents the dictator to solve this problem. According to the quadtree concept, a correlation exists between LH 1 and LH 2. Equally the correlation exists between HL 1 and HL 2, and the correction exists between HH 1 and HH 2. Therefore, LH 2, LH 1, HL 2, HL 1, HH 2, and HH 1 are divided into three partitions, Q t, t 1, 2, and 3. Q 1 LH 2 LH 1, Q 2 HL 2 HL 1, d d m,n T 1 m,n T 2 m,n T 3 m,n. Figure 6 shows the concept and framework of the dictator. The oblique-line block is the set S u, u 1, 2, and 3. This saves in the bits required to represent insignificant coefficients. From d, the subband with significant coefficients can be identified. The subbands with significant coefficients are classified into seven types, encoded according to the significant coefficients in the various subbands. Seven types are as follows. (LH means LH 1 or LH 2 ; HL means HL 1 or HL 2 ; and HH means HH 1 or HH 2.) Type 1: the significant coefficients are in LH. Type 2: the significant coefficients are in HL. Type 3: the significant coefficients are in HH. Type 4: the significant coefficients are in LH and HL. Type 5: the significant coefficients are in LH and HH. Type 6: the significant coefficients are in HL and HH. Q 3 HH 2 HH 1. The set S u, u 1, 2, and 3 is defined. The set S u indicates whether the subtree coefficients in Q t are significant. S 1 is modified by the following conditions in the set Q 1 : S 1 I,J 1, if LH 1 x,y 1, I x/4 and J y/4. S 1 I,J 1, if LH 2 x,y 1, I x/2, and J y/2. S 1 I,J 0, otherwise. Q 2 and Q 3 in the same steps result in S 2 and S 3. The correlation among the three sets (S 1,S 2,S 3 ) is greater than, so the modified algorithm creates the dictator that determines what subband has significant coefficients. The dictator d will decide what needs to be sent. Fig. 6 The modified algorithm uses the dictator concept and framework Optical Engineering, Vol. 42 No. 7, July 2003

6 Fig. 7 Modified algorithm flowchart. Type 7: the significant coefficients are in LH, HL, and HH. For example, type 6 refers to significant coefficients in HL and HH. Accordingly, the bitmap of LH and HH is encoded to indicate the positions of significant coefficients. The sign information of significant coefficients is also encoded. The modified algorithm differs from the original algorithm in reducing redundancy of the same level subband. Figure 7 presents the complete block diagram of the encoder for compressing still images. First, we input the test image, and the test image passes through three wavelet transforms. Then, the modified algorithm finds the maximum MAX that is sent to the decoder, and calculates Fig. 9 (a) Sonogram test image; (b) compressed by JPEG2000 with a bit rate 1.4 bpp and PSNR value 40.6 db; (c) compressed by modified with a bit rate 1.4 bpp and PSNR value 43.3 db; (d) difference image. the number of recursions, RUN. Then, the modified algorithm deals with partition in a sorting pass that is the same as that of the original algorithm. The modified algorithm sends Q, which includes the bitmap and the sign information of significant coefficients. Partition is handled by a combined function that reduces the interband redundancy in partition, and then outputs B, which includes information indicating which subband has significant coefficients. The modified algorithm deals with partition using a dictator function and outputs W, which decides what should be sent. According to output W, subbands with significant coefficients are classified into seven types to save bits and output information R. The modified algorithm also uses the refinement pass and sends G that includes the bits to correct the reconstructed value. Finally, entropy coding is used to improve performance. 4 Simulation Result Various kinds of medical images are selected as test data. They include angiogram Fig 8 a, sonogram Fig. 9 a, and x-ray Fig. 10 a images. All are gray-level images with a size of pixels with 8 bpp. The proposed algorithm is compared with JPEG2000, which adopts original and trellis-coded quantization TCQ. The performance is evaluated by PSNR. PSNR is mathematically evaluated as Fig. 8 (a) Angiogram test image; (b) compressed by JPEG2000 with a bit rate 0.1 bpp and PSNR value 44.1 db; (c) compressed by modified with a bit rate 0.1 bpp and PSNR value 45.2 db; and (d) difference image. PSNR 10 log n 1 n 1 j 0 T i 0. x i, j x i, j 2 3 Optical Engineering, Vol. 42 No. 7, July

7 Table 4 PSNR values for the original, JPEG2000, and modified at various bit rates in a sonogram test image. Bit rate (bpp) Original JPEG2K Modified Fig. 10 (a) X-ray test image; (b) compressed by JPEG2000 with a bit rate 0.8 bpp and PSNR value 41.9 db; (c) compressed by modified with a bit rate 0.8 bpp, PSNR value 43.6 db; and (d) difference image. PSNR has been accepted as a widely used measure of quality in the field of image compression. Figures 8, 9, and 10 present test images. Figure 8 a shows the original image. Figure 8 b shows an angiogram image decoded by JPEG2000 at a bit rate of 0.1 bpp with a PSNR of 44.1 db. Figure 8 c shows an angiogram image decoded by modified at a bit rate of 0.1 bpp with a PSNR of 45.2 db. Figure 8 d shows the difference image. The quality of the modified is excellent and the only difference is in the outline of the circular. Table 3 compares the PSNR values at various bit rates obtained using the original, JPEG2000, and the modified. At a given bit rate, the modified PSNR values are absolutely higher than these of JPEG2000 and the original. Figure 9 a is a sonogram image with text to record information about the patient. This property of the image raises difficulties in compression and is not displayed by other images. Even so, the decoded quality of texts and electrocardiogram ECG waveforms is acceptable for diagnosis, as shown in Fig. 9 c, which shows an electrocardiogram image decoded by a modified algorithm at a bit rate of 1.4 bpp with a PSNR value of 43.3 db. Figure 9 b shows an electrocardiogram image decoded by JPEG2000 at a bit rate of 1.4 bpp with a PSNR value of 40.6 db. Figure 9 d shows that the difference has no intention. In Table 4, the PSNR values at different bit rates are compared for the original, JPEG2000, and the modified. At a given bit rate, the modified PSNR values are absolutely higher than those of the original and JPEG2000. Figure 10 a shows an X-ray image. Figure 10 b shows an electrocardiogram image decoded by JPEG2000 at a bit rate of 0.8 bpp with a PSNR of 41.9 db. Figure 10 c shows an electrocardiogram image decoded by modified at a bit rate of 0.8 bpp with a PSNR of 43.6 db. Table 5 compares the PSNR at various bit rates for the original, JPEG2000, and the modified. At a given bit rate, all of the modified PSNR values are higher than those of the original and JPEG Conclusion The challenges posed by medical imaging involve the development of compression algorithms that are nearly lossless for diagnoses, yet support high-compression ratios to reduce storage, transmission, and processing. We state that while the original algorithm was proposed to achieve good performance, the modified algorithm is a modification to suit medical images. This is the first technique that has employed a algorithm to compress medical images. Employing this strategy to compress a huge volume of medical images reduces bandwidth and capacity. Moreover, the high-decoded quality prevents misdiagnosis, making the management of medical images more effective. The main aim of this work is to find a bit-rate-reduced method for saving on storage and achieving fast transmission of a remote diagnosis. The modified algorithm reduces the redundancy more than the original al- Table 3 PSNR for the original, JPEG2000, and modified at various bit rates in an angiogram test image. Table 5 PSNR values for the original, JPEG2000, and modified at various bit rates in an x-ray test image. Bit rate (bpp) Original JPEG2K Modified Bit rate (bpp) Original JPEG2K Modified Optical Engineering, Vol. 42 No. 7, July 2003

8 Fig. 11 PSNR values at various bit rates for several kinds of medical images. gorithm and JPEG2000. The modified algorithm is better suited than JPEG2000 for compressing medical images. The simulation results indicate that the modified algorithm can produce a reconstructed image with better medical images. The PSNR values of our proposed method are better than the PSNR values of JPEG2000, and those of Ref. 9 at a given bit rate. Figure 11 illustrates the PSNR values. References 1. American College of Radiology ACR /National Electrical Manufacturers Association NEMA standards Publication for Data Compression Standards, NEMA Publication PS-2, Washington, D.C Digital Imaging and Communication in Medicine (DICOM), version 3, American College of Radiology ACR /National Electrical Manufacturers Association NEMA standards draft, Dec A. Said and W. A. Pearlman, A new, fast, and efficient image codec based on set partitioning in hierarchical trees, IEEE Trans. Circuits Syst. Video Technol. 7 3, Jun S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans. Pattern Anal. Mach. Intell. 11 7, Jul I. Daubechies, Ten Lectures on Wavelets, Capital City Press, Montpelier, VT B. A. Banister and T. R. Fischer, Quadtree classification and TCQ image coding, IEEE Trans. Circuits Syst. Video Technol. 11 1, 3 8 Jan A. Munteanu, J. Cornelis, G. V. D. Auwera, and P. Cristea, Wavelet image compression The quadtree coding approach, IEEE Trans. Inf. Technol. Biomed. 3 3, Sep D. S. Taubman and M. W. Marcellin, JPEG2000 Image Compression Fundamentals, Standard and Practice, Kluwer Academic Publishers, Norwell, MA Y. G. Wu and S. C. Tai, Medical image compression by discrete cosine transform spectral similarity strategy, IEEE Trans. Info. Technol. Biomed. 5 3, Sep Shen-Chuan Tai received the BS and MS degrees in electrical engineering from the National Taiwan University, Taipei, in 1982 and 1986, respectively, and PhD degree in computer science from the National Tsing Hua University, Hsinchu, Taiwan, in He is currently a professor of electrical engineering at the National Chen Kung University, Tainan, Taiwan. His teaching and research interests include design automation of very large scale integration, data compression, biomedical engineering, and computer algorithms. Yen-Yu Chen received his BS and MS in computer science from Tamkang University, Tamsui, Taiwan, in 1991 and 1993, respectively. He is currently working toward his PhD degree at the Institute of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan. His research interests include biomedical signal processing, image processing, and data compression. Wen-Chien Yan received the BS degree in information and computer engineering from Chung Yung Christian University, Chung Li, Taiwan, in She is now working toward an MS degree in electrical engineering with the Institute of Electrical Engineering at the National Chen Kung University, Tainan, Taiwan. Her current research interests include image processing and data compression. Optical Engineering, Vol. 42 No. 7, July