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.
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