EFFICIENT IMAGE COMPRESSION TECHNIQUE USING FULL, COLUMN AND ROW TRANSFORMS ON COLOUR IMAGE

International Journal of Advances in Engineering & Technology, Mar. 2013.

©IJAET ISSN: 2231-1963

88 Vol. 6, Issue 1, pp. 88-100

EFFICIENT IMAGE COMPRESSION TECHNIQUE USING FULL,

COLUMN AND ROW TRANSFORMS ON COLOUR IMAGE

H. B. Kekre1, Tanuja Sarode2 and Prachi Natu3 1Sr. Professor, MPSTME, Deptt. of Computer Engg., NMIMS University, Mumbai, India 2 Associate Professor Department of Computer Engg., TSEC, Mumbai University, India

3Ph. D. Research Scholar, MPSTME, NMIMS University, Mumbai, India

ABSTRACT

This paper presents image compression technique based on column transform, row transform and full transform

of an image. Different transforms like, DFT, DCT, Walsh, Haar, DST, Kekre’s Transform and Slant transform

are applied on colour images of size 256x256x8 by separating R, G, and B colour planes. These transforms are

applied in three different ways namely: column, row and full transform. From each transformed image, specific

number of low energy coefficients is eliminated and compressed images are reconstructed by applying inverse

transform. Root Mean Square Error (RMSE) between original image and compressed image is calculated in

each case. From the implementation of proposed technique it has been observed that, RMSE values and visual

quality of images obtained by column transform are closer to RMSE values given by full transform of images.

Row transform gives quite high RMSE values as compared to column and full transform at higher compression

ratio. Aim of the proposed technique is to achieve compression with acceptable image quality and lesser

computations by using column transform.

KEYWORDS: Image compression, Full transform, Column transform, Row transform.

I. INTRODUCTION

Rapid increase in multimedia applications has been observed since last few years. It leads to higher

use of images and videos as compared to text data. Use of these applications play important role in

communication, educational tools, gaming applications, entertainment industry and many other areas.

When images and videos come into picture, issue of their storage, processing and transmission cannot

be neglected. Images contain considerable amount of redundancies. Hence storage and transmission

of compressed images instead of uncompressed images has been proved to be advantageous. Image

compression has the added advantage of being tolerant to distortion due to peculiar characteristics of

human visual system [1]. Major aim of image compression is to reduce the storage space or

transmission bandwidth and maintain acceptable image quality simultaneously. Image compression

techniques are broadly classified into two categories: lossless compression and lossy compression. In

lossless image compression exact replica of original image can be obtained from compressed image;

however it gives low compression ratio, which is not the case in lossy image compression. Wide

research has been done in this area and it includes compression using Discrete Fourier Transform

(DFT) [11] and Discrete Cosine Transform (DCT) [2].Compression using warped DCT is proposed in

[16]. Recent work includes wavelet based image compression using orthogonal wavelet transform[12]

and hybrid wavelet transform[17].Fractal image compression is discussed by Veenadevi and Ananth

in [18]. This paper presents transform based image compression that uses column transform, row

transform and full transform of an image.



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II. TRANSFORM BASED IMAGE COMPRESSION

Image compression plays a vital role in several important and diverse applications including tele-

video conferencing, remote sensing, medical imaging [2,3] and magnetic resonance imaging[4].

Transform based coding is major component of image and video processing applications. It is based

on the fact that pixels in an image exhibit a certain level of correlation with their neighbouring pixels.

A transformation is, therefore defined to map this spatial (correlated) data into transformed

(uncorrelated) coefficients [5]. It means that the information content of an individual pixel is

relatively small and to a large extent visual contribution of a pixel can be predicted using its

neighbours [1, 6].Transform based compression techniques use a reversible linear mathematical

transform to map the pixel values onto a set of coefficients which are then quantized and encoded. It

is lossy compression technique. Previously, Discrete Fourier Transform (DFT) is used to change the

pixels in the original image into frequency domain coefficients. Discrete Cosine Transform (DCT) is

most widely used approach in image and video compression, as the performance approaches to that of

Karhunen-Loeve transform (KLT) for first order Morkov process[16].

2.1. Discrete Cosine Transform (DCT)

Discrete Cosine Transform (DCT) is widely used transformation technique for image compression.

Other transforms like Haar, Walsh, Slant, Discrete sine transform (DST) can also be used for image

compression. DCT converts the spatial image representation into frequency components. Low

frequency components appear at the topmost left corner of the block that contains maximum

information of the image.

2.2. Walsh Transform

Walsh transform is non-sinusoidal orthogonal transform that decomposes a signal into a set of

orthogonal rectangular waveforms called Walsh functions. The transformation has no multipliers and

is real because the amplitude of Walsh functions has only two values, +1 or -1. Walsh functions are

rectangular or square waveforms with values of -1 or +1. An important characteristic of Walsh

functions is sequency which is determined from the number of zero-crossings per unit time interval.

Every Walsh function has a unique sequency value. The Walsh-Hadamard transform involves

expansion using a set of rectangular waveforms, so it is useful in applications involving discontinuous

signals that can be readily expressed in terms of Walsh functions.

2.3. Haar Transform

Haar transform was proposed in 1910 by a Hungarian mathematician Alfred HaarError! Reference

source not found.. The Haar transform is one of the earliest transform functions proposed.

Attracting feature of Haar transform is its ability to analyse the local features. This property makes it

applicable in electrical and computer engineering applications. The Haar transform uses Haar function

for its basis. The Haar function is an orthonormal, varies in both scale and position [8]. Haar

transform matrix contains ones and zeros. Hence it requires no multiplications and less number of

additions as compared to Walsh transform which makes it computationally efficient, fast and simple.

2.4. Discrete Sine Transform (DST)

Discrete Sine Transform (DST) is a complementary transform of DCT. DCT is an approximation of

KLT for large correlation coefficients whereas DST performs close to optimum KLT in terms of

energy compaction for small correlation coefficients. DST is used as low-rate image and audio coding

and in compression applications [9,10].

2.5. Fourier Transform

In conventional Fourier transform, it is not easy to detect local properties of the signal. Hence Short

Term Fourier Transform (STFT) was introduced. But it gives local properties at the cost of global

properties [11].



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2.6. Kekre’s Transform

Most of the transform matrices have to be in powers of two. This condition is not required in Kekre

transform [12, 13] matrix. In Kekre transform matrix, all diagonal elements and the upper diagonal

elements are one. Lower diagonal elements except the one exactly below the diagonal are zero.

2.7. Slant Transform

Slant transform matrix is orthogonal with a constant function for the first row. The elements in other

rows are defined by linear functions of the column index. Properties of Slant transform are: It has

orthonormal set of basis vectors. First basis vector is constant basis vector, one slant basis vector, the

sequency property, variable size transformation, fast computational algorithm and high energy

compaction. Definition of slant transform and its properties are given in [14, 15].

III. PROPOSED TECHNIQUE

In proposed compression technique, seven different transforms namely DFT, DCT, DWT, DST, DHT,

DKT and Slant transform are applied on each 256x256 size colour image to obtain transformed image

content. These transforms are applied in three different ways: column transform, row transform and

full transform. Let ‘T’ denotes the transformation matrix, ‘f’ denotes an image and ‘F’ indicates

transformed image. Then,

Column transform of an image ‘f’ is [F] = [T]*[f]

Row transform is written as: [F] = [f]*[T’]

where, T’= Transpose of T

And full transform is given by: [F] = [T]*[f]*[T’]

In each of these three cases, low energy coefficients are eliminated from transformed image content.

Then image is reconstructed by applying inverse transform on it. In column transform, number of

coefficients is reduced by eliminating some rows of transformed image. In row transform, it is done

by eliminating some columns of transformed image whereas in full transform some rows as well as

some columns are eliminated such that number of coefficients reduced is equal as that of column or

row transform. Image is then reconstructed by applying inverse transform on the image which

contains reduced number of coefficients than original image. Root mean square error and compression

ratio is calculated in each case till acceptable image quality is obtained. Average of these RMSE

values and compression ratio is used for performance analysis.

IV. EXPERIMENTAL RESULTS

Experimentation is done on 12 sample colour images. Images of 256x256 sizes from different classes

are selected. Experiments are performed in MATLAB 7.2 on a computer with dual core processor and

4 GB RAM. Test images are shown in figure 1.



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Figure 1: Set of twelve test images of different classes used for experimental purpose namely (from left to right

and top to bottom) Mandrill, Peppers, Lord Ganesha, Flower, Cartoon, dolphin, Birds, Waterlili, Bud, Bear,

Leaves and Lenna

For each transform, comparison of three cases i.e. RMSE in Full, column and row transform is shown

in figure 2 to 8. Figure 2 shows this comparison for DFT. RMSE values for full and column transform

are almost same in this case. But row transform gives slight high values of RMSE.

Figure 2. Performance comparison of Average

RMSE for Full DFT, column DFT and Row DFT

against different Compression Ratios


RMSE for Full Haar, column Haar and Row Haar


Figure 3 shows comparison for Haar transform. Here, up to compression ratio 4, RMSE in full and

column transform are almost same. Afterwards RMSE in column transform is approximately same as

that of full transform.



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RMSE for Full DCT, column DCT and Row DCT



RMSE for Full Walsh, column Walsh and Row

Walsh against different Compression Ratios

As found in figure 4 and 5, RMSE values of column and full transform are closer. Row transform

RMSE values are slightly higher in both DCT and Walsh transform.


RMSE for Full Slant, column Slant and Row

Slant against different Compression Ratios


RMSE for Full Kekre transform, column Kekre

and Row Kekre transform against different

Compression Ratios

Graph plotted in figure 6 and 7 shows RMSE values obtained by applying Slant transform and Kekre

transform respectively. These values are higher than the values obtained in DFT, DCT, Haar and

Walsh. But difference between Full transform values and column transform values is again very

small. Comparison of RMSE values for DST is shown in figure 8. Here also there is slight difference

in column transform RMSE values and the values in Full transform.

Figure 8. Performance comparison of Average RMSE for Full DST, Column DST and Row DST against

different compression ratios



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Figure 9.Performance comparison of Average RMSE for Full DFT, Haar, DCT, Walsh, Slant, Kekre

Transform, and DST against compression ratio 1 to 5

Graph plotted in figure 9 shows comparison of RMSE values for seven different full transforms

namely DFT, Haar, DCT, Walsh, Slant, Kekre Transform and DST. From the graph it can be

observed that, full DFT gives least RMSE value among all other full transforms.

Figure 10.Performance comparison of Average RMSE for Column DFT, Haar, DCT, Walsh, Slant, Kekre

Transform, and DST against compression ratio 1 to 5.

By observing and comparing Figure 10 with Figure 9, it is found that RMSE values of column

transform for compression ratio 1 to 5 are close to the values obtained by full transform. Since in

column transform we use [F] = [T]x[f] and not [F] = [T]x[f]x [T’] like in full transform, it saves half

number of computations.

Figure 11. Performance comparison of Average RMSE for Row DFT, Haar, DCT, Walsh, Slant, Kekre

Transform, DST against compression ratio 1 to 5.

It can be seen from Figure 11 that RMSE values for row transform are slight higher than column and

full transforms.



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Table 1 presents the summary of Average RMSE and PSNR for full transforms. It can be observed

that, good PSNR upto32 dB is obtained by DFT, DCT and DST at compression ratio 2.

Table 1. Summary of Average RMSE and PSNR for various ‘Full Transforms’

Full

Transform

Compression Ratio

2 4 8

AVG

RMSE PSNR

AVG

RMSE PSNR

AVG

RMSE PSNR

DFT 6.325 32.21 10.47 27.73 14.71 24.77

Haar 10.828 27.44 14.932 24.64 18.843 22.62

DCT 6.012 32.55 10.241 27.92 14.665 24.8

Walsh 9.273 28.78 13.195 25.72 16.950 23.54

Slant 33.628 17.59 40.301 16.02 42.536 15.55

Kekre

Transform 28.666 18.98 39.332 16.23 44.867 15.09

DST 6.229 32.24 10.661 27.57 15.135 24.53

Table 2 gives average RMSE and PSNR summary for column transform. Average RMSE in column

transform is closer to that of full transform. Better PSNR is obtained for DFT.

Table 2. Summary of Average RMSE and PSNR for various ‘Column Transforms’

Column

Transform

Compression Ratio

2 4 8

AVG

RMSE PSNR

AVG

RMSE PSNR

AVG

RMSE PSNR

DFT 2.541 40.03 4.4072 35.24 6.288 32.16

Haar 9.728 28.37 15.440 24.35 20.886 21.73

DCT 7.386 30.76 12.915 25.91 18.343 22.86

Walsh 9.728 28.37 15.440 24.35 20.886 21.73

Slant 35.900 17.02 42.512 15.56 44.686 15.12

Kekre

Transform 31.232 18.23 43.213 15.41 47.717 14.55

DST 8.046 30.01 14.770 24.74 21.893 21.32

Table 3 shows performance of different row transforms in terms of RMSE and PSNR. DFT, DCT and

DST show good average RMSE. Better PSNR is obtained for DFT.

Table 3. Summary of Average RMSE and PSNR for various ‘Row Transforms’

Row

Transform

Compression Ratio

2 4 8

AVG

RMSE PSNR

AVG

RMSE PSNR

AVG

RMSE PSNR

DFT 2.559 39.96 4.458 35.14 6.410 31.99

Haar 9.910 28.2 15.869 24.12 21.705 21.4

DCT 7.530 30.59 13.168 25.74 18.874 22.61

Walsh 9.910 28.2 15.869 24.12 21.705 21.4

Slant 36.765 16.82 43.484 15.36 45.761 14.92

Kekre

Transform 32.164 17.98 42.313 15.6 46.788 14.72

DST 8.260 29.79 15.124 24.53 22.458 21.1

From twelve different query images with different colour and texture combination, ‘Mandrill’ image

is selected as representative image for perceptual comparison. It contains different colour



95 Vol. 6, Issue 1, pp. 88-100

combinations and edges. Compressed images obtained by applying full, column and row transforms

are shown below with corresponding compression ratio and RMSE value for each image.

Compression Ratio= 2 Compression Ratio= 4 Compression Ratio= 8

RMSE= 2.685373 RMSE=4.641359 RMSE=6.248745 Figure 12: Compressed ‘Mandrill’ images by applying full DFT


RMSE=3.615713 RMSE=5.167002 RMSE=6.27264 Figure 13: Compressed ‘Mandrill’ images by applying column DFT


RMSE=8.59163 RMSE=13.5814 RMSE=17.0931 Figure 14: Compressed ‘Mandrill’ images by applying Row DFT

Figures 12, 13, 14shows compressed ‘Mandrill’ image using full, column and row DFT respectively.

In each of the three cases compression ratio 2, 4 and 8 is considered. It is observed that RMSE value

of column DFT at compression ratio 8 is very closer to one obtained by total DFT at same

compression ratio.


RMSE=9.224652 RMSE=14.47318 RMSE=18.31172

Figure 15: Compressed ‘Mandrill’ images by applying full DCT



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RMSE= 11.8693776 RMSE= 17.16473991 RMSE= 20.90614907 Figure 16: Compressed ‘Mandrill’ images by applying column DCT


RMSE= 9.88663 RMSE= 16.16006 RMSE= 21.11612 Figure 17: Compressed ‘Mandrill’ images by applying row DCT

Figures 15,16,17 show compressed ‘Mandrill’ image using full, column and row DCT for

compression ratio 2,4 and 8. Again it is observed that RMSE value of column DCT at compression

ratio 8 is very closer to one obtained by total DCT at same compression ratio.


RMSE= 11.5486 RMSE= 16.14666 RMSE= 19.64522 Figure 18: Compressed ‘Mandrill’ images by applying full Haar Transform


RMSE=12.93917685 RMSE= 18.34300842 RMSE= 22.14955424 Figure 19: Compressed ‘Mandrill’ images by applying column Haar Transform


RMSE=11.84323 RMSE= 18.0668 RMSE= 22.9688 Figure 20: Compressed ‘Mandrill’ images by applying row Haar Transform



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Similarly, figures 18, 19, 20 present compressed images for full, column and row Haar transform

respectively. At compression ratio 8, it gives acceptable compressed image but RMSE is higher than

DFT and DCT.


RMSE= 11.38372 RMSE= 16.26495 RMSE= 19.62808 Figure 21: Compressed ‘Mandrill’ images by applying full Walsh Transform


RMSE= 12.93918 RMSE= 18.34301 RMSE= 22.14955 Figure 22: Compressed ‘Mandrill’ images by applying column Walsh Transform


RMSE=11.8432 RMSE= 18.0668 RMSE= 22.9688 Figure 23: Compressed ‘Mandrill’ images by applying row Walsh Transform

Same results regarding RMSE values are observed for Walsh transform in figure 21, 22 and 23. For

full, column and row Walsh transforms, image quality is acceptable but at the cost of higher RMSE

values.


RMSE= 9.331971 RMSE= 14.69161 RMSE= 18.56635 Figure 24: Compressed ‘Mandrill’ images by applying full DST



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RMSE= 12.2453 RMSE= 18.74562 RMSE= 24.4524 Figure 25: Compressed ‘Mandrill’ images by applying column DST


RMSE=10.29123 RMSE= 17.4362 RMSE= 23.74187 Figure 26: Compressed ‘Mandrill’ images by applying row DST

As shown in figures 24, 25 and 26 DST also gives good image quality with less error in three different

cases i.e. full column and row DST. Slant and Kekre’s transform show poor performance in terms of

RMSE for comp ratio greater than two. As compressed image quality is not perceptible, these

transforms are not recommended.

V. CONCLUSIONS

Here performance of column transform, row transform and full transform is compared using Root

Mean Square Error (RMSE) as a performance measure with respect to compression ratio. RMSE

values are calculated for compression ratio 1 to 5. Experimental results prove that RMSE values

obtained for various compression ratios in column transform are closer to those obtained in full

transform of an image. Hence instead of full transform of an image, column transform can be used for

image compression, saving half number of computations. RMSE obtained in row transform is quite

higher than column and full transform at higher values of compression ratio. Hence it is not

recommended. Good PSNR is obtained using column transform. Among all the seven transforms

used, DFT, DCT and DST give better results in terms of RMSE and reconstructed image quality than

other transforms. Walsh and Haar transforms also give acceptable results with an advantage of less

computation whereas Slant and Kekre transform do not give good results. Hence they are not

recommended.

VI. FUTURE WORK

Future work includes application of orthogonal wavelet transforms on colour images. Change in the

RMSE values if any, can be compared. Also PSNR values and quality of reconstructed image can be

studied to compare their performance against the one in this paper.

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AUTHORS

H. B. Kekre has received B.E. (Hons.) in Telecomm. Engg. from Jabalpur University in

1958, M.Tech (Industrial Electronics) from IIT Bombay in 1960, M.S.Engg. (Electrical

Engg.) from University of Ottawa in 1965 and Ph.D. (System Identification) from IIT

Bombay in 1970. He has worked Over 35 years as Faculty of Electrical Engineering and

then HOD Computer Science and Engg. at IIT Bombay. After serving IIT for 35 years, he

retired in 1995. After retirement from IIT, for 13 years he was working as a professor and

head in the department of computer engineering and Vice principal at Thadomal Shahani

Engg. College, Mumbai. Now he is senior professor at MPSTME, SVKM’s NMIMS University. He has guided

17 Ph.Ds., more than 100 M.E./M.Tech and several B.E. / B.Tech projects, while in IIT and TSEC. His areas of

interest are Digital Signal processing, Image Processing and Computer Networking. He has more than 450

papers in National / International Journals and Conferences to his credit. He was Senior Member of IEEE.

Presently He is Fellow of IETE, Life Member of ISTE and Senior Member of International Association of

Computer Science and Information Technology (IACSIT). Recently fifteen students working under his guidance

have received best paper awards. Currently eight research scholars working under his guidance have been

awarded Ph. D. by NMIMS (Deemed to be University). At present seven research scholars are pursuing Ph.D.

program under his guidance.

Tanuja K. Sarode has received M.E. (Computer Engineering) degree from Mumbai

University in 2004, Ph.D. from Mukesh Patel School of Technology, Management and

Engg. SVKM’s NMIMS University, Vile-Parle (W), Mumbai, INDIA. She has more than

11 years of experience in teaching. Currently working as Assistant Professor in Dept. of

Computer Engineering at Thadomal Shahani Engineering College, Mumbai. She is member



100 Vol. 6, Issue 1, pp. 88-100

of International Association of Engineers (IAENG) and International Association of Computer Science and

Information Technology (IACSIT). Her areas of interest are Image Processing, Signal Processing and Computer

Graphics. She has 137 papers in National /International Conferences/journal to her credit.

Prachi Natu has received B.E. (Electronics and Telecommunication) degree from Mumbai

University in 2004. Currently pursuing Ph.D. from NMIMS University. She has 08 years of

experience in teaching. Currently working as Assistant Professor in Department of Computer

Engineering at G. V. Acharya Institute of Engineering and Technology, Shelu (Karjat). Her

areas of interest are Image Processing, Database Management Systems and Operating

Systems. She has 12 papers in International Conferences/journal to her credit.

Design

EFFICIENT IMAGE COMPRESSION TECHNIQUE USING FULL, COLUMN AND ROW TRANSFORMS ON COLOUR IMAGE