In This Study we Introduce the Effective of some Mathematical In This Study we Introduce the Effective of some Mathematical

Transformations such as Fourier Transformation and its variants, Transformations such as Fourier Transformation and its variants,

as well as Wavelet Transformations, to Image Compression.as well as Wavelet Transformations, to Image Compression.

Digital images contain large amount of information that need Digital images contain large amount of information that need

evolving effective techniques for storing and transmitting the ever evolving effective techniques for storing and transmitting the ever

increasing volumes of data. Image compression addresses the increasing volumes of data. Image compression addresses the

problem by reducing the amount of data required to represent a problem by reducing the amount of data required to represent a

digital image. digital image.

The discrete cosine transform (DCT) and the discrete wavelet The discrete cosine transform (DCT) and the discrete wavelet

transform (DWT) are techniques for converting a signal into transform (DWT) are techniques for converting a signal into

elementary frequency components. They are widely used in elementary frequency components. They are widely used in

image compression. These functions illustrate the power of image compression. These functions illustrate the power of

Mathematics in the image compression. In this work, a Mathematics in the image compression. In this work, a

comparison study between discrete wavelet transform and comparison study between discrete wavelet transform and

discrete cosine transform is introduced. Our results show that discrete cosine transform is introduced. Our results show that

the discrete wavelet transform gives better performance than the discrete wavelet transform gives better performance than

the discrete cosine transform in terms of peak signal to noise the discrete cosine transform in terms of peak signal to noise

ratio as a quality measure.ratio as a quality measure.

In the modern digital age, computer storage technology continues at a In the modern digital age, computer storage technology continues at a

rapid pace, a means for reducing the storage requirements of an image is rapid pace, a means for reducing the storage requirements of an image is

still needed in most situations. In images, The common characteristic of still needed in most situations. In images, The common characteristic of

most of the images is that, the neighboring pixels are correlated, and most of the images is that, the neighboring pixels are correlated, and

image contains redundant information. Therefore the most important image contains redundant information. Therefore the most important

task in image compression is to find a less correlated representation of task in image compression is to find a less correlated representation of

the image. The fundamental component of image compression is the image. The fundamental component of image compression is

reduction of redundancy and irrelevancy. Redundancy reduction aims at reduction of redundancy and irrelevancy. Redundancy reduction aims at

removing duplication from image, and irrelevancy reduction omits parts removing duplication from image, and irrelevancy reduction omits parts

of the signal that will not be noticed by Human Visual System (HVS).of the signal that will not be noticed by Human Visual System (HVS).

There are various methods of compressing still Images, one of these There are various methods of compressing still Images, one of these

methods is a transform coding is one of the most popular image methods is a transform coding is one of the most popular image

compression techniques, and use a reversible, linear mathematical compression techniques, and use a reversible, linear mathematical

transform. For image compression, it is desirable that the selection of transform. For image compression, it is desirable that the selection of

transform should reduce the size of resultant data set as compared to transform should reduce the size of resultant data set as compared to

source data set. Some mathematical transformations have been invented source data set. Some mathematical transformations have been invented

for the sole purpose of image compression such as, Discrete Fourier for the sole purpose of image compression such as, Discrete Fourier

Transform (DFT), Discrete Cosine Transform (DCT), Hadamard-Haar Transform (DFT), Discrete Cosine Transform (DCT), Hadamard-Haar

Transform (HHT), Karhune-Loeve Transforms (KLT), Slant-Haar Transform Transform (HHT), Karhune-Loeve Transforms (KLT), Slant-Haar Transform

(SHT), Walsh-Hadamard Transform (WHT), and Wavelet Transforms (SHT), Walsh-Hadamard Transform (WHT), and Wavelet Transforms

(WT). selection of proper transform is one of the important factors in (WT). selection of proper transform is one of the important factors in

data compression scheme.data compression scheme.

DCT converts data (image pixels) into sets of frequencies. DCT-based DCT converts data (image pixels) into sets of frequencies. DCT-based

image compression relies on two techniques to reduce data required image compression relies on two techniques to reduce data required

to represent the image. The first is quantization of the image’s DCT to represent the image. The first is quantization of the image’s DCT

coefficients; the second is entropy coding of the quantized coefficients; the second is entropy coding of the quantized

coefficients. However, Discrete wavelet transformation (DWT) coefficients. However, Discrete wavelet transformation (DWT)

transforms discrete signal from the time domain into time frequency transforms discrete signal from the time domain into time frequency

domain. DWT have higher decorrelation and energy compression domain. DWT have higher decorrelation and energy compression

efficiency, so DWT can provide better image quality on higher efficiency, so DWT can provide better image quality on higher

compression ratios, and have some properties which makes it better compression ratios, and have some properties which makes it better

choice for image compression than DCT.choice for image compression than DCT.

The discrete cosine transforms (DCT) is a technique for converting The discrete cosine transforms (DCT) is a technique for converting a signal into elementary frequency components. It represents an a signal into elementary frequency components. It represents an image as a sum of sinusoids of varying magnitudes and image as a sum of sinusoids of varying magnitudes and frequencies. The DCT has the property that, for a typical image, frequencies. The DCT has the property that, for a typical image, most of the visually significant information about the image is most of the visually significant information about the image is concentrated in just a few coefficients of the DCT. For this reason, concentrated in just a few coefficients of the DCT. For this reason, the DCT is often used in image compression applications.the DCT is often used in image compression applications. For compression, the input image is first divided into blocks, and For compression, the input image is first divided into blocks, and the 2D – DCT is computed for each block. The DCT coefficients are the 2D – DCT is computed for each block. The DCT coefficients are then quantized, coded and transmitted. then quantized, coded and transmitted.

The 1D – DCT is given byThe 1D – DCT is given by:

)2.(..........2

)12cos()(

2)(

)1(...........)(1

)0(

1

0

1

0

N

x

N

x

N

uxxf

NuC

xfN

C

WhereWhere 1-N ...., 2, 1, 0,u ),( uC is DCT ofis DCT of )(xf The inverse transformThe inverse transformof 1D – DCT is given of 1D – DCT is given

by:by:

)3......(....2

)12cos()(

1)0(

1)(

1

1

N

u N

uxuC

NC

Nxf

1.-N ...., 2, 1, ,0xForFor

)4.........(2

)12(cos

2

)12(cos),()()(),(

1

0

1

0

N

vy

N

uxyxfvuvuc

N

x

N

y

The 2D – DCT is a direct extension of the 1 – D case and is given byThe 2D – DCT is a direct extension of the 1 – D case and is given by:

For u,v = 0,1,2,…..,N-1. and α (u) and α (v) are defined as:For u,v = 0,1,2,…..,N-1. and α (u) and α (v) are defined as:

)5.........(

else 1

0k if 2

1

)(

k

To reconstruct the image, receiver decodes the quantized DCT To reconstruct the image, receiver decodes the quantized DCT coefficients, computes the inverse 2D – DCT of each block, and then coefficients, computes the inverse 2D – DCT of each block, and then puts the blocks back together into a single image. The inverse puts the blocks back together into a single image. The inverse transform is defined as:transform is defined as:

)6.........(2

)12(cos

2

)12(cos),()()(),(

1

0

1

0

N

vy

N

uxvuCvuyxf

N

u

N

v

For x, y = 0,1,2,….,N-1.For x, y = 0,1,2,….,N-1.

This transforms illustrate the power of Mathematics in image This transforms illustrate the power of Mathematics in image compression field. Image compression is one of the most compression field. Image compression is one of the most important applications of wavelets. Wavelets are mathematical important applications of wavelets. Wavelets are mathematical functions that satisfy certain properties and can be used to functions that satisfy certain properties and can be used to transform one function representation into another. Wavelet transform one function representation into another. Wavelet transform decomposes an image into a set of band limited transform decomposes an image into a set of band limited components which can be reassembled to reconstruct the components which can be reassembled to reconstruct the original image without error. Wavelet transform (WT) original image without error. Wavelet transform (WT) represents an image as a sum of wavelet functions (wavelets) represents an image as a sum of wavelet functions (wavelets) with different locations and scales. any decomposition of an with different locations and scales. any decomposition of an image into wavelets involves a pair of waveforms: one to image into wavelets involves a pair of waveforms: one to represent the high frequencies corresponding to the detailed represent the high frequencies corresponding to the detailed parts of an image (wavelet function ψ) and one for the low parts of an image (wavelet function ψ) and one for the low frequencies or smooth parts of an image (scaling function Ф) .frequencies or smooth parts of an image (scaling function Ф) .

Discrete wavelet transforms for two – dimensional can be derived from one – Discrete wavelet transforms for two – dimensional can be derived from one –

dimensional DWT. The Easiest way for obtaining scaling and wavelet function dimensional DWT. The Easiest way for obtaining scaling and wavelet function

for two-dimensions is by multiplying two one-dimensional functions. The for two-dimensions is by multiplying two one-dimensional functions. The

scaling function for 2D – DWT can be obtained by multiplying two 1 – D scaling scaling function for 2D – DWT can be obtained by multiplying two 1 – D scaling

functionfunction )()(),( yxyx

Wavelet functions for 2D – DWT can be obtained by multiplying two Wavelet functions for 2D – DWT can be obtained by multiplying two

wavelet functions or wavelet and scaling functions for one – dimensional wavelet functions or wavelet and scaling functions for one – dimensional

analysis. From that follows that for 3D case there exist three wavelet analysis. From that follows that for 3D case there exist three wavelet

functions that analysis details infunctions that analysis details in

)()(),( yxyx verticavertical )()(),( yxyx andand diagonaldiagonal

)()(),( yxyx

.

horizontailhorizontail

Wavelet compression technique uses the wavelet filters for Wavelet compression technique uses the wavelet filters for

image decomposition, image is divided into approximation and image decomposition, image is divided into approximation and

detail sub image. The filter is applied along the row and then detail sub image. The filter is applied along the row and then

along the columns, the filters divide the input image into four along the columns, the filters divide the input image into four

non – overlapping multi-resolution coefficient sets, a lower non – overlapping multi-resolution coefficient sets, a lower

resolution approximation image ( LL1) as well as horizontal (HL1) resolution approximation image ( LL1) as well as horizontal (HL1)

, vertical (LH1) and diagonal (HH1) detail components. The sub – , vertical (LH1) and diagonal (HH1) detail components. The sub –

band LL1 represents the coarse – scale DWT coefficients while band LL1 represents the coarse – scale DWT coefficients while

the coefficient sets LH1, HL1 and HH1 represent the fine – scale the coefficient sets LH1, HL1 and HH1 represent the fine – scale

of DWT coefficients.of DWT coefficients.

JPEG 2000 uses the wavelet transform (WT) to reduce the amount of JPEG 2000 uses the wavelet transform (WT) to reduce the amount of information contained in a picture, while JPEG systems use the discrete information contained in a picture, while JPEG systems use the discrete cosine transform (DCT). It is true that the WT requires more processing cosine transform (DCT). It is true that the WT requires more processing power than the DCT. The DCT, or any type of Fourier transform, power than the DCT. The DCT, or any type of Fourier transform, expresses the signal in terms of frequency and amplitude—but only at a expresses the signal in terms of frequency and amplitude—but only at a single instant in time. The WT transforms a signal into frequency and single instant in time. The WT transforms a signal into frequency and amplitude over time, and is therefore more efficient. Undesirable amplitude over time, and is therefore more efficient. Undesirable blocking artifacts affect the reconstructed images (high compression blocking artifacts affect the reconstructed images (high compression ratios or very low bit rates. DCT function is fixed can not be adapted to ratios or very low bit rates. DCT function is fixed can not be adapted to input data. DWT No need to divide the input image into non-input data. DWT No need to divide the input image into non-overlapping 2-D blocks, it has higher compression ratios avoid blocking overlapping 2-D blocks, it has higher compression ratios avoid blocking artifacts. Disadvantages of DWT the cost of computing DWT as artifacts. Disadvantages of DWT the cost of computing DWT as compared to DCT may be higher. The use of larger DWT basis functions compared to DCT may be higher. The use of larger DWT basis functions or wavelet filters produces blurring and ringing noise near edge regions or wavelet filters produces blurring and ringing noise near edge regions in images. Longer compression time and Lower quality than JPEG at low in images. Longer compression time and Lower quality than JPEG at low compression ratescompression rates

Simulations were carried out to test the effect of some mathematical Simulations were carried out to test the effect of some mathematical

transformations such as DCT as a variant of Fourier transformation, as transformations such as DCT as a variant of Fourier transformation, as

well as wavelets transformation, to image compression. Matlab code was well as wavelets transformation, to image compression. Matlab code was

written for the generation of the studied techniques. The test set used is written for the generation of the studied techniques. The test set used is

four 512×512 monochromatic images of 8-bit intensity (256 grey levels), four 512×512 monochromatic images of 8-bit intensity (256 grey levels),

Lena512, Baboon512, Barbara512 and Peppers512 (as shown in Figure 1)Lena512, Baboon512, Barbara512 and Peppers512 (as shown in Figure 1) . .

As the image content being viewed influences the perception of quality As the image content being viewed influences the perception of quality

irrespective of technical parameters of the system, test images that have irrespective of technical parameters of the system, test images that have

different spatial and frequency characteristics have been selected: different spatial and frequency characteristics have been selected:

Lena512, Baboon512, Barbara512 and Peppers512 (shown in Figure 1).Lena512, Baboon512, Barbara512 and Peppers512 (shown in Figure 1).

Lena (512×512)Baboon (512×512)

Barbara (512×512)Peppers (512×512)

The test image Baboon512 has a lot of details, it contains components in The test image Baboon512 has a lot of details, it contains components in

high frequency area and low predictability, so it presents low redundant high frequency area and low predictability, so it presents low redundant

image and consequently difficult for compression. On the other hand, image and consequently difficult for compression. On the other hand,

the test images Lena512 and Barbara512 are images with less detail than the test images Lena512 and Barbara512 are images with less detail than

Baboon512. The test image Lena512 has higher predictability than the Baboon512. The test image Lena512 has higher predictability than the

image Baboon512 since the latter has components in high frequency image Baboon512 since the latter has components in high frequency

area more than the image Lena512. The performance of these schemes area more than the image Lena512. The performance of these schemes

is usually characterised using the mean square of the error (MSE) and is usually characterised using the mean square of the error (MSE) and

the Peak Signal to Noise Ratio (PSNR) .Table 1 shows the performance the Peak Signal to Noise Ratio (PSNR) .Table 1 shows the performance

of the studied methods while Figure 2 shows the reconstructed of the studied methods while Figure 2 shows the reconstructed

compressed images of the methods, respectively.compressed images of the methods, respectively.

JPEGJPEGJPEG-2000JPEG-2000

ImageImageBit-rate Bit-rate (bpp)(bpp)

PSNR PSNR (dB)(dB)

CompressioCompression ration ratio

Bit-rate Bit-rate (bpp)(bpp)

PSNR PSNR (dB)(dB)

CompressioCompression ration ratio

Lena Lena 512512

0.20980.209834.014834.014838.124538.12450.19390.193934.068334.068341.258441.2584

Baboon Baboon 512512

0.22220.222229.706629.706636.008836.00880.19640.196429.603029.603040.730940.7309

Barbara Barbara 512512

0.23080.230831.629131.629134.656834.6568..0.25930.259331.333631.333630.855030.8550

Peppers Peppers 512512

0.22390.223933.948533.948535.724235.72420.22710.227133.400633.400635.234435.2344

JPEGJPEGJPEG2000JPEG2000

Lena 512Lena 512Lena 512Lena 512

Baboon 512Baboon 512Baboon 512Baboon 512

This paper proposed an efficient implementation of some This paper proposed an efficient implementation of some mathematical functions to Image Compression. DCT is used for mathematical functions to Image Compression. DCT is used for transformation in JPEG standard. DCT performs efficiently at transformation in JPEG standard. DCT performs efficiently at medium bit rates. at higher compression ratios, image quality medium bit rates. at higher compression ratios, image quality degrades because of the artifacts resulting from the block-based degrades because of the artifacts resulting from the block-based DCT scheme. DWT is used as basis for transformation in JPEG 2000 DCT scheme. DWT is used as basis for transformation in JPEG 2000 standard. DWT provides high quality compression at low bit rates standard. DWT provides high quality compression at low bit rates because of overlapping basis functions and better energy because of overlapping basis functions and better energy compaction property of wavelet transforms. DWT performs better compaction property of wavelet transforms. DWT performs better than DCT in the context that it avoids blocking artifacts which than DCT in the context that it avoids blocking artifacts which degrade reconstructed images. However DWT provides lower degrade reconstructed images. However DWT provides lower quality than JPEG at low compression rates. In the future we hope quality than JPEG at low compression rates. In the future we hope to improve the performance of the DCT by emerging other to improve the performance of the DCT by emerging other compression techniques to reduce the staircase problem that compression techniques to reduce the staircase problem that results from using the DCT.results from using the DCT.