115 CHAPTER 6 A SECURE FAST 2D-DISCRETE FRACTIONAL FOURIER TRANSFORM BASED MEDICAL IMAGE COMPRESSION USING SPIHT ALGORITHM WITH HUFFMAN ENCODER 6.1. INTRODUCTION Various transforms like DCT, DFT used to achieve image transformation have been described by Ramesh Babu Durai et al (2012). Contourlet based ROI method with wavelet transform is a better method of digital signals and images. By means of expensive calculation, processing of data compression has eased the burden of image transmission and storage as discussed by Tamilarasi & Palanisamy (2009). Data compression attempts to decrease the size of the image by concentrating on the removal of superfluous data. Storage area of the image can be doubled by compressing an image into half its original size as stated by Al-Sammraie & Khamis (2008). Thus, the spatial and spectral redundancies which minimize the number of bits needed to represent an image is eliminated. This facilitates substantial reduction in the bandwidth requirement for transmitting an image over the network. Data storage, archiving and communication of medical images over the internet to the end user have significant applications for data compression as stated by Ghrare et al (2008). In this chapter, a DICOM images are encrypted using Quasigroup Encryption with Hadamard and Number Theoretic ation. For a
116 secure compression Fast Two Dimensional Discrete Fractional Fourier (DFRCT) and a SPIHT Algorithm with Huffman Encoder is used. 6.2. COMPRESSION USING FAST 2D-DISCRETE FRACTIONAL FOURIER TRANSFORM AND SPIHT ALGORITHM WITH HUFFMAN ENCODER This approach comprises of the following phases namely, Encryption, Domain ation, SPIHT algorithm with Huffman Compression; Decoding through SPIHT with Huffman Encoder and Inverse Fast Two Dimensional Fractional Fourier and finally Quasi group decryption with Hadamard and Number Theoretic. Original DICOM Image Quasi Group Encryption with Hadamard and Number Theoretic ation Fast Two- Dimension Discrete Fractional Fourier SPIHT Algorithm with Huffman Encoder Decompressed DICOM Image Quasi Group Decryption with inverse Hadamard and Number Theoretic ation Fast Two- Dimension Discrete Inverse Fractional Fourier Decoding with SPIHT Algorithm with Huffman Encoder Figure 6.1 Overall Flow of the Proposed SPIHT Algorithm with Huffman Coding Image Compression Approach 6.2.1. Quasigroup Encryption with Hadamard and Number Theoretic ation The usage of transforms would effectively diffuse statistics where the security is improved through a variety of them and by transforming them (Reddy 2012). The employment of chained Hadamard transforms and
117 Number Theoretic s (NTT) are investigated in this approach to introduce diffusion together with the Quasigroup transformation. Number Theoretic s are also a certain kind of discrete Fourier transforms. It is based on generalizing the nth primitive root of unity to a quotient ring rather than using complex numbers. Figure 6.2 represents the general architecture of the proposed encryption and hash system scheme. In this approach, the input image will be subjected to different transformations sequentially like Quasigroup transformation, Hadamard transformation and Number theoretic transformation. For Hadamard and Number theoretic transforms, the input data is divided into a definite group of bits in such a manner that each group bit count is the order of the equivalent matrix. Spreading Code Order of matrices Input data Encryption System Output data Figure 6.2 General Architecture of the Proposed Encryption System A. Hadamard s The Hadamard is a generalized class of Discrete Fourier transforms (Ulman 1970; Ce & Bing 2009). It is created either recursively, or through binary representation. All the values in the matrix are non-negative. Each negative number is replaced with equivalent modulo number. For instance, in modulo 7 Hadamard matrixes -1 is replaced with 6 to make the matrix non-binary. Owing to its symmetric form, it can be used in
118 applications such as data encryption and randomness measures Goldburg et al (1993). Only prime modulo operations are carried out since non-prime numbers can be divisible with numbers other than 1 and itself. Recursively, 1 1 Hadamard transform is defined by the identity = 1, and then for m > 0 by, = (6.1) A Hadamard matrix,, is a square matrix of order n = 1, 2 or 4k where k represents a positive integer. The elements of H are either +1 or 1 and. =, where is the transpose of, and is the identity matrix of order n. A Hadamard matrix is said to be normalized if all of the elements of the first row and first column are +1. Some examples of the Hadamard matrices are given below, = + 1 (6.2) = 1 1 1 (6.3) Hadamard matrix of modulo 31 of size 8*8 1 1 1 1 1 1 1 1 1 30 1 30 1 30 1 30 1 1 30 30 1 1 30 30 1 30 30 1 1 30 30 1 1 1 1 1 30 30 30 30 1 30 1 30 30 1 30 1 1 1 30 30 30 30 1 1 1 30 30 1 30 1 1 30
119 Hadamard matrix of modulo 7 of size 4*4 1 1 1 1 1 6 1 6 1 1 6 6 1 6 6 1 The concept of encryption is to multiply the decimated input sequence with the non-binary Hadamard matrix in a chained manner block by block. The block size is based upon the size of the selected Hadamard matrix. Input sequence is taken in the form of column matrix. Figure 3 shows the block diagram of Hadamard Encryption. Block 1 Hadamard Block 2 Hadamard Hadamard Block n Figure 6.3 Hadamard Encryption B. Number Theoretic s Number Theoretic depends on generalizing the nth primitive root of unity to a quotient ring rather than through complex numbers (Kak 1971). 1 1 1 1 1 1 1 (6.4)
120 about = 1 The unit w is exp (2 / n). Number Theoretic is now all NTT matrix of order 6*6 1 1 1 1 1 1 1 3 2 6 4 5 1 2 4 1 2 4 1 6 1 6 1 6 1 4 2 1 4 2 1 5 4 6 2 3 i. NTT Encryption Block 1 NTT Block 2 NTT NTT Block n Figure 6.4 Number Theoretic Encryption Figure 6.4 shows the block diagram for the Number Theoretic Encryption. The notion of encryption is to multiply the decimated input sequence which is the output attained after encryption by means of Hadamard transform with the non-binary Number theoretical matrix in a chained manner block by block. The block size is based upon the size of the selected Number theoretical matrix. The Input sequence is taken in the form of column matrix.
121 C. Encryption Phase1: Encryption of input data using Quasigroup based encryption system. Phase2: Output of Phase1 is given as input to the Phase 2. In phase2 Hadamard transformation of data is carried out. Phase3: Output of Phase2 is given as input to the Phase 3. In phase 3 Number Theoretic is performed. Phase4: Phase2 is repeated with a different order of Hadamard matrix. These four phases are clearly depicted in Figure 6.5. Input integer stream Phase 1 Phase 2 Phase 3 Quasigroup Encryptor (q*q) Hadamard (m1*m1) Number Theoretical (n*n) Hadamard (m2*m2) Encrypted output Phase 4 Figure 6.5 Proposed Quasigroup Encryption System
122 6.2.2. Fast Two-Dimension Discrete Fractional Fourier A. Development Of 1D DFRFT Algorithm In Shih s definition of FRFT, the FRFT is subjective to the weighted composition of the j th order Fourier transforms (j =0, 1, 2, 3) of the original function. Generally the FRFT is written as, [ ( )] = exp 3 4 ) cos 2 cos 4 ( ) (6.5) In the same way, it can be incidental that DFRFT is also subjective to the weighted composition of the first four orders of Discrete Fourier (DFT). Thus, the th order of DFRFT can be implemented by the equation below. [ ( )] = exp 3 4 ) cos 2 cos 4 ( ) (6.6) where ( )( = 0, 1, 2, 3) is the m th order of DFT of the original sequence f (n). DFT is defined here as follows, [ ( )] = 1 ( ) (6.7) where N is length of the sequence.
123 So ( )( = 0, 1, 2, 3) in (6.7) can be obtained by the Fast Fourier (FFT) algorithm. After obtaining the m th (m=0, 1, 2, 3) order of DFT of f (n), the DFRFT of f (n) can then be calculated as a linear combination. Obviously, such an algorithm shares the same level of accuracy and efficiency with FFT, which means a sample of N points, can be computed by ( ) time. B. Generalization to Fast 2D DFRFT Fact that 1D DFRFT can be said as the linear combination of DFT and 2D DFT of a matrix with N rows and M columns can be achieved by implementing M+N times 1D DFT, the 2D DFRFT fast algorithm can be developed on the basis of the 1D DFRFT algorithm. Thus, similar to 2D DFT, 2D DFRFT of a matrix with N rows and M columns can be obtained by carrying out N times of 1D DFRFT row transforms and M times of 1D DFRFT column transforms. For a matrix (, ), (, ) order of 2D DFRFT (, ) can be obtained by the following two steps. I. For each row in matrix (, ) calculate its th order 1D DFRFT, then place the results of the transform as the original row sequence to form a matrix which is marked as (, ). II. For each column in (, ), calculate the th order 1D DFRFT. Later place the results of the transform in the original column sequence, thus the final result (, ) is obtained.
124 As the 2D fast DFRFT algorithm mentioned above is based on the FFT algorithm, its computing efficiency is equal to that of FFT, which means the 2D DFRFT can compute a sample in ( ) time. In Medical image processing, compression plays a very important role. This means minimizing the dimensions of the images to a processing level. Image compression using transform coding provides significant results, with fair image quality. The cut-off of the transform coefficients can be tuned to bring out a negotiation between image quality and compression factor. To use this approach, an image is initially partitioned into non-overlapped (generally taken as 8x8 or 16 16) sub images. A Fast 2D-DFrFT is applied to each block to transform the gray levels of pixels in the spatial domain into coefficients in the frequency domain. The coefficients are normalized by various scales based on the cut-off selected. At Decoder, the process of encoding is simply reversed. C. SPIHT Algorithm With Huffman Encoder For Image Compression According to statistic analysis of the output binary stream of SPIHT encoding, a simple and effective method combined with Huffman encode is proposed for further compression. SPIHT stands for Set Partitioning in Hierarchical Trees, is very fast and effective one. In this method, more (wide-sense) zero trees are efficiently found and represented by separating the tree root from the tree, thereby, making compression more efficient. The image through the fractional transform, the coefficients values in high frequency region are generally small, hence, it will appear as "0" to quantify. SPIHT does not adopt a special method to treat with it, but directly gives the output. A simple and effective method combined with Huffman encode has been proposed in the present research.
125 D. SPIHT Algorithm With Huffman Encoder 1) First divide every output binary stream into 3 bits as a group; 111 000 111 000 100 000 010 101 100 00. In this process, there will be remaining 0, 1, 2 bits that cannot participate. Hence, in the head of the output bit stream of Huffman encoding there are two bits to record the number of bits which do not participate in the group and that remainder bits are direct output in the end. Figure 6.6 shows the bit stream structure of Huffman encoding. Number of remain bits Bits Stream Remaining Bits Figure 6.6 The Bit Stream Structure of Huffman Encoding 2) The emergence of statistical probability of each symbol grouping results as follows, P( 000 )=0.3333 P( 010 )=0.1111 P( 100 )=0.2222 P( 110 )=0 P( 001 )=0 P( 011 )=0 P( 101 )=0.1111 P( 111 )=0.2222 3) According to the above probability results, by applying Huffman encoding the following code word book is obtained as in Table 6.1. is obtained
126 Table 6.1 Code Word Book Table 000 001 01 100000 100 101 11 101 010 1001 110 10001 011 10001 111 00 Through the above code book can get the corresponding output stream; 10 00 01 00 01 11 01 1001 101 11 00, a total of 25 bits. The 10 in the first is binary of remainder bits numbers. The last two bits 00 are the result of direct output remainder bits. Compared with the original, bitstream saves 4 bits. Decoding is the reverse process of the above mentioned process. 6.2.3. Decompression This process is the reverse of the compression technique. After SPIHT, it is necessary to transform the data to the original domain (spatial domain). To do this, the Inverse Fractional Fourier is applied first in the columns and then in the rows. A. Quasigroup Decryption with Hadamard and Number Theoretic ation As the Hadamard matrix operations are invertible, decryption of the data can be performed by generating inverse Hadamard matrix. All the matrices such as the Quasigroup, Hadamard Matrix and Number Theoretic transform matrix have the same orders of matrices. The order used for all Quasigroups, Hadamard and NTT is 16 since the input data stream is 16 bit. Hadamard transforms and Number Theoretic transforms perform as hash functions which produce diverse hash values for different input values as
127 stated by Satti & Kak (2009). There is a huge difference in the generated random sequence if there is a one bit change in the input sequence. 6.3. HYBRID COMBINATION OF DISCRETE COSINE TRANSFORM AND SET PARTITION IN HIERARCHICAL TREE (DCTSPIHT) CODING ALOGRITHM FOR MEDICAL IMAGE COMPRESSION This approach comprises of the following phases namely Encryption, Domain ation, DCTSPIHT algorithm is used for compression and finally Quasigroup decryption with Hadamard and Number Theoretic. Original DICOM Image Quasi Group Encryption with Hadamard and Number Theoretic ation DCTSPIHT Encoding Decompressed DICOM Image Quasi Group Decryption with inverse Hadamard and Number Theoretic ation DCTSPIHT Decoding Figure 6.7 Overall Flow of the Proposed DCTSPIHT Image Compression Approach 6.3.1. DCTSPIHT Algorithm for Image Compression The sensitivity of Human eye to different frequencies is different and especially it is highly sensitive to the image edge features. Thus, the SPIHT algorithm has been used to improve the transformation process and to increase the edge threshold. The human visual characteristics and SPIHT algorithm pay more attention to image edge information. At the same time, the DCT coding and SPIHT algorithm are combined to achieve hybrid DCT and SPIHT coding.
128 Figure 6.8 DCTSPIHT Algorithm Coding /Decoding Diagram This DCTSPIHT algorithm combines two different techniques DCT and SPIHT to achieve better image compression as every image consists of low frequency and high frequency component. It is observed that, DCT is the technique which is more efficient for low frequency component and SPIHT gives a better result for high frequency component. In Figure 6.8, initially, the original image is given through the DCT coding. After that, the wavelet transformation of DCT output is created. This output is then encoded with SPIHT technique, now the overall coded data is to be transmitted. In the receiver side, the received data is to be decoded. 6.4. EXPERIMENTAL RESULTS The same experimental setup used in the previous chapter has been used in this approach. A. Result Analysis for Fast 2D-Discrete Fractional Fourier and SPIHT Algorithm with Huffman Encoder Three DICOM lung images are considered.
129 Lung 1 Lung 2 Lung 3 Figure 6.9 DICOM Lung Test Images Table 6.2 shows the comparison of the encryption and decryption time between traditional RSA approach and the proposed Quasigroup encryption with HTT and NTT approach. It is clearly observed from the table that the proposed Quasigroup approach takes lesser encryption and decryption time than RSA. For all the standard images considered, the proposed Quasigroup attains lesser encryption and decryption time. Table 6.2 Comparison of Quasigroup Encryption and Decryption Time with HT and NTT Standard Images Lung 1 Lung 2 Lung 3 Encryption Time Decryption Time Modulus Quasi Group Quasi Group (bits) RSA Encryption with HT and NTT RSA Encryption with HT and NTT 2048 3.105 1.412 102.54 31.68 1024 1.965 0.631 62.35 19.08 512 1.510 0.42 31.54 05.89 2048 2.901 0.863 90.10 28.39 1024 1.789 0.45 53.47 16.07 512 1.443 0.31 27.51 04.05 2048 2.936 1.255 91.64 29.05 1024 1.839 0.701 56.05 17.11 512 1.493 0.410 29.69 05.68
130 Table 6.3 shows the comparison of the PSNR value comparison of the proposed Fast 2D-DFrFT and DCTSPIHT with the existing approaches such as DFrFT, Wavelet with SPIHT and D2 Modified SPIHT. It is observed that the proposed approach provides better PSNR value when compared with the existing technique. The highest PSNR value obtained is for Lung 3 image in the proposed DCTSPIHT approach and the next higher PSNR value is obtained in proposed Fast 2D-Discrete Fractional Fourier with SPHIT with Huffman Encoder. When considering 2 bpp, the two proposed approaches, DCTSPIHT approach and Fast 2D-Discrete Fractional Fourier with SPHIT with Huffman Encoder attained PSNR of 41.82 and 40.09 respectively. However, the other approaches such as Wavelet with SPIHT and D2 Wavelet with Modified SPIHT attain a much lesser PSNR of 33.11 and 35.76 respectively. Table 6.3 Comparison of PSNR value of the Proposed Hybrid Technique Standard Images Lung 1 Lung 2 Lung 3 Bit Per Pixel (Bpp) Wavelet with SPIHT D2 Wavelet with Modified SPIHT Proposed Fast 2D-Discrete Fractional Fourier with SPHIT with Huffman Encoder Proposed DCTSPIHT algorithm 0.5 29.48 31.89 38.8 39.94 1 33.56 34.33 39.60 40.75 2 36.62 37.29 40.9 41.82 0.5 20.10 21.7 27.08 29.24 1 27.69 29 33.98 35.14 2 33.11 35.76 39.15 40.76 0.5 20.3 21.75 26.30 27.85 1 27.83 29.36 34.90 35.69 2 33.81 35.88 40.12 42.09
131 45 40 35 Wavelet with SPIHT D2 Wavelet with Modified SPIHT Fast 2D-Discrete Fractiona Fourier with SPHIT with Huffman Encoder DCTSPIHT algorithm2 PSNR value (db) 30 25 20 15 10 5 0 Lung 1 Lung 2 Lung 3 Test Images for 0.5 bpp Figure 6.10 PSNR Evaluation of the Proposed Hybrid Compression Technique for DICOM Images for 0.5 (Bpp) Figure 6.10 is drawn for PSNR Evaluation of the Image Compression Techniques for DICOM Images for 0.5 (Bpp). From the figure, it is observed from the figure that the PSNR value of the proposed DCTSPIHT algorithm approach is very high when compared with the existing transformation approaches.
132 45 40 Wavelet with SPIHT D2 Wavelet with Modified SPIHT Fast 2D-Discrete Fractiona Fourier with SPHIT with Huffman Encoder DCTSPIHT algorithm2 35 30 PSNR (db) 25 20 15 10 5 0 Lung 1 Lung 2 Lung 3 Test Images for 1 bpp Figure 6.11 PSNR Evaluation of the Proposed Hybrid Compression Technique for DICOM Images for 1 (Bpp) Figure 6.11 is drawn for PSNR Evaluation of the Image Compression Techniques for DICOM Images for 1(Bpp). From the figure, it is observed that the PSNR value of the proposed DCTSPIHT algorithm approach is much higher than that of the existing transformation approaches.
133 45 40 Wavelet with SPIHT D2 Wavelet with Modified SPIHT Fast 2D-Discrete Fractiona Fourier with SPHIT with Huffman Encoder DCTSPIHT algorithm2 35 30 PSNR (db) 25 20 15 10 5 0 Lung 1 Lung 2 Lung 3 Test Images for 2 bpp Figure 6.12 PSNR Evaluation of the Proposed Hybrid Compression Technique for DICOM Images for 2 (Bpp) Figure 6.12 is drawn for PSNR Evaluation of the Image Compression Techniques for DICOM Images for 2 (Bpp). From the figure, it is observed that the PSNR value of the proposed DCTSPIHT algorithm approach is much higher than that of the existing transformation approaches. The MSE value comparison is shown in Table 6.4.
134 Table 6.4 Comparison of MSE of the Proposed Hybrid Techniques Standard Images Bit Per Pixel (Bpp) Wavelet with SPIHT D2 Wavelet with Modified SPIHT Fast 2D-Discrete Fractional Fourier with SPHIT with Huffman Encoder Proposed DCTSPIHT algorithm Lung 1 2 115.68 35.11 22.85 20.19 Lung 2 2 114.52 33.14 20.16 18.82 Lung 3 2 113.54 32.54 19.02 17.32 140 120 Wavelet with SPIHT D2 Wavelet with Modified SPIHT Fast 2D-Discrete Fractiona Fourier with SPHIT with Huffman Encoder DCTSPIHT algorithm 100 MSE 80 60 40 20 0 Lung 1 Lung 2 Lung 3 Test Images for 2bpp Figure 6.13 MSE Evaluation of the Proposed Hybrid Technique for DICOM Images for 2 (Bpp) Compression From the Figure 6.13, it is observed that the MSE value of the proposed DCTSPIHT algorithm approach is much less than the existing transformation approaches.
135 6.5. SUMMARY This chapter clearly discusses about the proposed Fast 2D-Discrete Fractional Fourier based Medical Image Compression using SPIHT Algorithm with Huffman Encoder. The performance of this proposed approach is compared with that of the various image compression techniques. It is observed from the experimental results that the proposed Fast 2D- Discrete Fractional Fourier and SPIHT Algorithm with Huffman Encoder provides the best results. However, in SPIHT, the image is first converted into its wavelet transform and the wavelet coefficients are then fed to the encoder. In DCTSPIHT, the input image has been subjected to DCT coding. The output is then decomposed using biorthogonal wavelet transform. This decomposed output is further compressed using SPIHT encoding. There is a very wide range of practical uses that have large number of image data to be transmitted. It is observed from the empirical result that the proposed DCTSPHIT approach provides high PSNR values. Moreover, MSE value of the proposed approach is also much lesser than that of the other existing technique.