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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 Transformation. For a

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secure compression Fast Two Dimensional Discrete Fractional Fourier

Transform (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 Transformation, SPIHT algorithm with Huffman

Compression; Decoding through SPIHT with Huffman Encoder and Inverse

Fast Two Dimensional Fractional Fourier Transform and finally Quasi group

decryption with Hadamard and Number Theoretic Transform.

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

Transformation

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

Original DICOM Image

Quasi Group Encryption with Hadamard and Number Theoretic

Transformation

Fast Two-Dimension

Discrete Fractional Fourier

Transform

SPIHT Algorithm with Huffman

Encoder

Decoding with SPIHT Algorithm

with Huffman Encoder

Fast Two-Dimension

Discrete Inverse Fractional Fourier

Transform

Quasi Group Decryption with inverse Hadamard and Number Theoretic

Transformation

Decompressed DICOM Image

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Number Theoretic Transforms (NTT) are investigated in this approach to

introduce diffusion together with the Quasigroup transformation.

Number Theoretic Transforms 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.

Figure 6.2 General Architecture of the Proposed Encryption System

A. Hadamard Transforms

The Hadamard Transform 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

Encryption System Input data Output data

Spreading Code

Order of matrices

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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 11 (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

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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.

Figure 6.3 Hadamard Encryption

B. Number Theoretic Transforms

Number Theoretic Transform depends on generalizing the nth

primitive root of unity to a quotient ring rather than through complex numbers

(Kak 1971).

1 1 1 1111

(6.4)

Block 1 Hadamard Transform Block 2 Hadamard

Transform

Block n Hadamard Transform

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The unit w is exp (2 / n). Number Theoretic Transform is now all

about = 1

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

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.

Block 1 NTT

Transform Block 2 NTTTransform

Block n NTTTransform

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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 Transform is performed.

Phase4: Phase2 is repeated with a different order of Hadamard

matrix.

These four phases are clearly depicted in Figure 6.5.

Figure 6.5 Proposed Quasigroup Encryption System

Phase 1 Phase 2 Phase 3

Phase 4

Quasigroup Encryptor (q*q)

Hadamard Transform (m1*m1)

Number Theoretical

Transform (n*n)

Hadamard Transform (m2*m2)

Input integer stream

Encrypted output

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6.2.2. Fast Two-Dimension Discrete Fractional Fourier Transform

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 34

) cos2

× cos4

( ) (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

Transform (DFT). Thus, the th order of DFRFT can be implemented by the

equation below.

[ ( )] = exp 34

) cos2

× cos4

( ) (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.

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So ( )( = 0, 1, 2, 3) in (6.7) can be obtained by the Fast

Fourier Transform (FFT) algorithm. After obtaining the mth (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.

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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.

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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(‘001’)=0

P(‘010’)=0.1111 P(‘011’)=0

P(‘100’)=0.2222 P(‘101’)=0.1111

P(‘110’)=0 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

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Table 6.1 Code Word Book Table

‘000’

‘001’

‘010’

‘011’

‘100’

‘101’

‘110’

‘111’

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 Transform is applied first

in the columns and then in the rows.

A. Quasigroup Decryption with Hadamard and Number Theoretic

Transformation

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

‘01’

‘100000’

‘1001’

‘10001’

‘11’

‘101

‘10001’

‘00’

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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 Transformation, DCTSPIHT algorithm is used for

compression and finally Quasigroup decryption with Hadamard and Number

Theoretic Transform.

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.

Original DICOM Image

Quasi Group Encryption with Hadamard and Number Theoretic

Transformation DCTSPIHT Encoding

DCTSPIHT Decoding

Quasi Group Decryption with inverse Hadamard and Number

Theoretic Transformation Decompressed DICOM Image

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 Transform

and SPIHT Algorithm with Huffman Encoder

Three DICOM lung images are considered.

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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

Modulus(bits)

Encryption Time Decryption Time

RSAQuasi Group

Encryption with HT and NTT

RSAQuasi Group

Encryption with HT and NTT

Lung 1 2048 3.105 1.412 102.54 31.681024 1.965 0.631 62.35 19.08512 1.510 0.42 31.54 05.89

Lung 2 2048 2.901 0.863 90.10 28.391024 1.789 0.45 53.47 16.07512 1.443 0.31 27.51 04.05

Lung 3 2048 2.936 1.255 91.64 29.051024 1.839 0.701 56.05 17.11512 1.493 0.410 29.69 05.68

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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 Transform with SPIHT and D2 Transform 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 Transform with SPHIT with

Huffman Encoder. When considering 2 bpp, the two proposed approaches,

DCTSPIHT approach and Fast 2D-Discrete Fractional Fourier Transform

with SPHIT with Huffman Encoder attained PSNR of 41.82 and 40.09

respectively. However, the other approaches such as Wavelet Transform with

SPIHT and D2 Wavelet Transform 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

Bit Per Pixel(Bpp)

Wavelet Transform

with SPIHT

D2 Wavelet Transform

withModifiedSPIHT

Proposed Fast2D-Discrete

Fractional Fourier Transform with

SPHIT with Huffman Encoder

ProposedDCTSPIHT algorithm

Lung 1 0.5 29.48 31.89 38.8 39.941 33.56 34.33 39.60 40.752 36.62 37.29 40.9 41.82

Lung 2 0.5 20.10 21.7 27.08 29.241 27.69 29 33.98 35.142 33.11 35.76 39.15 40.76

Lung 3 0.5 20.3 21.75 26.30 27.851 27.83 29.36 34.90 35.692 33.81 35.88 40.12 42.09

131

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.

0

5

10

15

20

25

30

35

40

45

Lung 1 Lung 2 Lung 3

PSN

Rva

lue

(db)

Test Images for 0.5 bpp

Wavelet Transform with SPIHT

D2 Wavelet Transform with Modified SPIHT

Fast 2D-Discrete Fractiona Fourier Transform with SPHIT with Huffman Encoder DCTSPIHT algorithm2

132

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.

0

5

10

15

20

25

30

35

40

45

Lung 1 Lung 2 Lung 3

PSN

R(d

b)

Test Images for 1 bpp

Wavelet Transform with SPIHTD2 Wavelet Transform with Modified SPIHTFast 2D-Discrete Fractiona Fourier Transform with SPHIT with Huffman Encoder DCTSPIHT algorithm2

133

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.

0

5

10

15

20

25

30

35

40

45

Lung 1 Lung 2 Lung 3

PSN

R(d

b)

Test Images for 2 bpp

Wavelet Transform with SPIHTD2 Wavelet Transform with Modified SPIHTFast 2D-Discrete Fractiona Fourier Transform with SPHIT with Huffman Encoder DCTSPIHT algorithm2

134

Table 6.4 Comparison of MSE of the Proposed Hybrid Techniques

Standard Images

Bit Per Pixel(Bpp)

Wavelet Transform

withSPIHT

D2 Wavelet Transform

withModifiedSPIHT

Fast 2D-Discrete Fractional Fourier

Transform with SPHIT with

Huffman Encoder

ProposedDCTSPIHT algorithm

Lung 1 2 115.68 35.11 22.85 20.19Lung 2 2 114.52 33.14 20.16 18.82Lung 3 2 113.54 32.54 19.02 17.32

Figure 6.13 MSE Evaluation of the Proposed Hybrid Compression Technique for DICOM Images for 2 (Bpp)

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.

0

20

40

60

80

100

120

140

Lung 1 Lung 2 Lung 3

MSE

Test Images for 2bpp

Wavelet Transform with SPIHTD2 Wavelet Transform with Modified SPIHTFast 2D-Discrete Fractiona Fourier Transform with SPHIT with Huffman Encoder DCTSPIHT algorithm

135

6.5. SUMMARY

This chapter clearly discusses about the proposed Fast 2D-Discrete

Fractional Fourier Transform 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 Transform 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.