Survey papper on image processing
A Hybrid Image Compression Technique Based on DWT and DCT Transforms
Salam Benchikh and Michael Corinthios, Life Fellow, IEEE
Department of Electrical Engineering, Ecole Poly technique de Montreal, Montreal, Qc, Canada
Keywords: Image compression, Image coding, Discrete wavelet transform (DWT), Discrete cosine transform (DCT).
Abstract In this paper, a hybrid technique using the discrete cosine transform (DCT) and the discrete wavelet transform (DWT) is presented. We show evaluation and comparative results for DCT, DWT and hybrid DWT-DCT compression techniques. Using the Power Signal to Noise Ratio (PSNR) as a measure of quality, we show that DWT with a two-threshold method named "improved-DWT" provides a better quality of image compared to DCT and to DWT with a one-threshold method. Finally, we show that the combination of the two techniques, named improved-DWT-DCT compression technique, showing that it yields a better performance than DCT -based JPEG in terms of PSNR.
1 Introduction The goal of data compression is to minimize the information redundancy with the objective of reducing archiving costs and data transmission bandwidth [1]-[4]. There exist two types of
compression: Lossless compression where the reconstructed
image is identical to the original one but the compression
ratio is very low and lossy compression where the
compression ratio is very high but incurring appreciable data
loss [2]. In particular, we focus our attention on the lossy compression. The use of transform techniques is
recommended for this type of compression [1]-[3]. For a
given application, the choice of a particular transform
technique is highly dependent on the amount of error that can
be tolerated in the compression and computational complexity [I]. The goal is to find a transform that is able to concentrate
the signal energy in the smallest number of parameters and which does not require excessive computational complexity.
The fast algorithms of these transforms can significantly
reduce the number of arithmetic operations to evaluate the forward transform and its inverse. Several transforms [5] are used in the domain of data compression, discrete Walsh
Hadamard transform (DWHT), discrete Walsh generalized
transform, discrete cosine transform (DCT), discrete Fourier
transform (DFT) and discrete wavelet transform (DWT). All of these transforms are symmetric, unitary and reversible [1][2]. And the total energy before and after transformation
remain the same.
The main objective is to study image compression systems based on transform techniques for data transmission and
efficient data storage, while providing a good tradeoff between the compression rates and signal to noise ratio. The
transform techniques chosen in this paper for image
compression includes discrete cosine transform (DCT) and discrete wavelets transform (DWT) [6], [7], [8]. The DCT
transform is used in image coding (JPEG) and video coding (MPEG2, H263, H264) [4][2]. Moreover, the DWT transform is used in image and video coding (JPEG2000) with
embedded zero-tree wavelet (EZW) and set partitioning in
hierarchical trees (SPIRT) coder [4], [7], [9].
In order to get both advantages of DCT and DWT, a hybrid scheme based on DCT and DWT schemes has been presented
in [10] and [II]. In [10], the application of DWT-DCT compression for medical images such as X-ray and ultra
sound images has been presented and evaluated, showing the
benefits of the hybrid method. The authors considered all the decomposed images obtained from DWT analysis (i.e., approximation and details images) and applied DCT
transform technique on those images. In [11], the authors
considered two iterations for the DWT image analysis stage
and for an efficient compression they applied a two
dimensional DCT of size 4x4 on only the approximation
image, without considering the details images from the
moment that they claim that details coefficients are not
important information.
To develop a compression system with a DWT transform,
four parameters are important to take into the consideration: image test, wavelet function (and the order of filter), number
of iterations, and finally calculation complexity. Image with
high spectral activity are not recommended for test image compression. High number of iterations provides more details
information on the image or signal. However, the calculation
complexity becomes high. Hence, we have fixed the number
of iterations equal to three in our work. Also, unlike the
works in [10] and [11], we have applied a two-dimensional
DCT of size 8x8 on only the third approximation image. And
on all the details images which are obtained from DWT
analysis, we have applied a threshold to keep the important information existing in those details images. In addition, due to the use of different entropy coding methods in DCT and
DWT (DCT uses Huffman coding while DWT uses SPIRT) and in order to obtain a better comparison for efficiency and
the performance of the DCT and DWT transforms, we compare those transform techniques without considering entropy coding.
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This paper is organized as fellows. We first review and analyze the compression technique using DCT and then that
using DWT. We place an emphasis on the performance
criteria for DCT and DWT. In addition, we study the impact
of the spectral distribution of the images on the quality of each compression technique. After comparing the two
techniques, we combine both of them in order to improve the
performance of image compression. The idea of applying two
transforms is based on the fact that combined transforms
could reduce the draw-back existing in each transform.
2 Transformation Techniques In the following, we briefly review the DCT and DWT transforms. We then present our hybrid DCT-DWT algorithm. Discrete cosine transform and discrete wavelet transform have been used in many digital signals processing application and specifically in data compression.
2.1. Discrete Cosine Transform (DCT)
We introduce DCT transform used in JPEG standard [12], as shown in Fig. 1. The JPEG process is as follows: first, the image is broken into 8x8 blocks. Second, DCT is applied on each block from the left to the right and from the top to the bottom. Then, quantization is applied for compression process, and data are stored following a specific process to reduce the information in the memory. And to reconstruct the compressed image we apply IDCT transform.
The coefficients of DCT transform are computed using
1 N-l N-l D(i, j) = .J2N C(i)C(j) � � p(x, y)
xcos( (2 X;�
)ilr ] cos(
(2y ;� )jlr
)
(1)
where, p(x,y) is an input matrix image NxN, (x,y) are the coordinate of matrix elements and (iJ) are the coordinate of coefficients, and
CCU)= j.!n if u=O 1 if u > 0
(2)
The reconstructed image is computed by using the inverse DCT (mCT) according to
1 N-l N-l p(x,y)= J2N ��C(i)C(j)D(i,J)
( (2X + 1 )ilrJ (
(2Y + l)JlZ"J xcos cos 2N 2N
(3)
The pixels in black and white image are arranged from 0 to 255 with the step of 1, where 0 corresponds to a pure black and 255 corresponds to a pure white. Since DCT is used for pixels arranged from -128 to 127, then for each input pixel we subtract 128.
DCT could be accomplished by D = T M Tt, where M is the original matrix levelled, (.y denotes transpose operator, and T
X(iJI y
y X'(iJ)
Fig.l: JPEG diagram block.
with elements T(i, j) is a DCT matrix, which is computed according to
TU, j)
� 1 g ' " . (2j 2 '�)iK;
if i = 0 (4)
if i> 0
Since human eyes are sensible to low frequency therefore, high frequency information is usually removed in compression through quantization. This operation is achieved
by dividing each component of D (i.e., DU, j» by the corresponding element in the quantization matrix according to
B(i,j) == round[D(�'�») (5) Q(z,])
As explained earlier, DCT is applied on each 8x8 block, hence, from each block we get 64 DCT coefficients. Each
element of the matrix B (i.e., BU, j») is coded as a binary data. The first coefficient (i.e., the DC term) is coded differently from the other 63 coefficients (i.e., the AC terms) which follow a zigzag line. Note that the most part of the information of the block image is concentrated in DC term. In this work, we do not consider the entropy coder as explained in Section 1. For the reconstruction of the signal we use,
R(i,j) == Q(i,j)xB(i,j) (6) where, each element of matrix B is multiplied by the corresponding element in quantization matrix Q. Finally, mCT is applied for each block and then each coefficient of mCT is added by 128. Also, to control the quality of DCT compression, we choose a matrix quantization using the following expression,
Q(i,j)==.!..[l-u+(1+i+ j)(lOO-fq)] 4
where, h is quality factor, i,J = 0, ... ,7, and i < U < i+j.
2.2. Discrete Wavelets Transform (DWT)
(7)
As the basis functions of Fourier transform are sinusoids, basis functions of Wavelets transform are small waves (i.e. wavelets), in which the waves are varying in frequency but have limited duration [13]. The implementation of wavelets transform is equivalent to implement a filter with its impulse response that is a wavelet function. For one-dimensional signals and one iteration, two filters bank are required for analysis stage and two other filters bank for synthesis stage.
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Analysis of a signal by DWT for a first iteration provides one High-pass filter
approximation signal (AI) and one detail signal (Dd. If there DD ,(11, 111)
is a need for more precision, the decomposition is used on the approximated signal (AI) to provide A2 and D2 and so on as seen in Fig.2. In this figure, we can see the sequence is VD ,( 11, m ) decomposed into high frequency and low frequency using filters g�(-n) and hl/,{-n) respectively, followed by down-
HD,( l1,m) sampling in the first and the second iterations.
0, A, (n, 111 )
A,
Fig.2: Analysis stage for two iterations.
For signal reconstruction, we apply the up-sampling to all approximations and details signals and we apply a complementary filters h,;{n) and g�(n) as shown in Fig.3.
Fig.3: Synthesis stage for two iterations.
For two dimensional signals (e.g., image), given by a matrix
of size nxm, the processes of analysis and synthesis are the
same as for one dimensional signals but applied first for rows
and then for columns of that matrix. In image analysis and for one iteration, we need three similar banks of two filters. The
image is filtered by a first two filters bank using high filter
hl/,(-n) and a low filter g�(-n), each filter is followed by down
sampled. Each output of this two filters bank is crossing
another filter-bank similar to the first one. The results are four
images, approximation (A), horizontal detail (HD), vertical detail (VD) and diagonal detail (DD), as seen in Fig.4. For
more number of iterations, the same analysis process is applied only on approximation image of the last iteration that is the iteration of level 3.
High·pass filter Diagonal detail coefficients
High·pass filter
VD,(n,m) Horizontal detail coefficients
Fig.4: Image analysis process for one iteration.
Fig.5: Image synthesis process for one iteration.
In image synthesis process, up-sampling is applied for all the outputs of the image analysis process (A, HD, VD and DD)
before filtering operation. For one iteration, three banks of
two filters are applied on synthesis process as seen in Fig 5. We see that the four images are added two by two,
approximation with horizontal detail and diagonal detail with vertical detail. For more number of iterations, we start
applying the same process with images from the same
iteration.
As the number of iterations in DWT algorithm becomes high, the details information becomes high too. However, to avoid high complexity of calculation, we fix a number of iterations equal to 3. Also, to come to a best compromise between Peak Signal to Noise Ratio (PSNR) and compression ratio in DWT algorithm, we select a bi-orthogonal wavelet 'bior4.4' to use in filters analysis and synthesis. Then to improve the DWT compression we choose the threshold equal or less than the smallest coefficient in approximation coefficients. We apply this threshold on all details coefficients resulting from iterations 1 and 2. In order to improve the quality of compression, we use two thresholds in DWT algorithm. The first threshold is applied on details coefficients resulting from iterations 1 and 2, and the second threshold is applied only for details coefficients resulting from iteration 3. We call this method improved DWT.
2.3. Hybrid DCT -DWT Compression Technique As mentioned in Section 1, our goal is to achieve after image compression, a high quality of reconstruction and a low distortion level. To exploit the advantages of both DCT and
DWT transforms and to avoid the drawback of each
transform, we present a hybrid method called improved-DWT
D C T as shown in Fig.6.
In Fig.6, the image X is analysed by DWT for three iterations, as a results, we get one approximation image of size N/8xN/8 and nine details images of different size (3 images of size N/8xN/8, 3 images of size N/4xN/4and 3 images of size Nl2xN/2). The two-dimensional DCT is applied only on approximation image resulting from iteration 3. The quality factor used is equal to 80. For the nine detail images, we apply two thresholds for compression as described in improved-DWT method. The results from the two ways are supposed to be coded and transmitted. For the reason presented in section 1 the entropy coded is not considered in this work.
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x(iJ)
v
Fig.6: Encoder block for hybrid DCT -DWT.
The performance evaluation of DWT, DCT, and the combined Wavelet-DCT compressions are performed using two images of size 256x256: bust and wbarb, as shown in Fig.7. In image compression, we generally use two metrics to evaluate the quality of image compression: objective and subjective evaluations. As an objective metric, we use Peak signal to noise ratio (PSNR) which characterizes the amount of distortion in lossy compression [4], [2]. It is defined by
Pic2 PSNR = 10 loglo -- (8) MSE
WbarbJmage
Fig.7: Test images, bust and wbarb, respectively.
where MSE refers to mean square error between the original image and the reconstructed image and is defined by
MSE =-l-tf(x(i,}) - y(i,}))2 nx m i�1 j�1
(9)
and Pic is equal to 255, the high input pixel in the image of size n x m . As a subjective metric, we use evaluations by human eyes as described in [2].
3 Numerical Results and Discussions In the following, we present numerical results to evaluate and
compare the considered compression techniques.
3.1. Results for DCT compression
Using a method described in section 2.1, and following the block diagram in Fig.l, we evaluate the efficiency of image compression using DCT. The quantization matrices used for evaluation are generated from equation (6). Table.1 shows the changes in the number of coefficients used for image reconstruction depending on the quality factor /g. The smaller the quality factor yields the smaller the number of coefficients used in image reconstruction. So the quality factor and the number of coefficients used in image reconstruction are proportional. Also, the variation of PSNR and quality factor is proportional. It is observed that as the quality factor increases, the PSNR increases.
Table.}: PSNR evaluation for image} and image2 with OCT.
Quality DCT Image 1: bust DCT Image 2: wbarb factor
CoefN�� PSNI�{d� CoefN�%) PSNR(dB)
10 6.18 27.0029 4.84 26.7970
20 6.72 27.4389 5.26 27.0760
30 7.39 27.9533 5.76 27.3918
40 8.25 28.5451 6.46 27.7862
50 9.35 29.2540 7.46 28.3034
70 13.48 31.4066 11.28 30.1256
80 17.88 33.3629 15.96 31.8454
90 27.76 37.2238 28.12 35.5675
95 40.74 41.4938 45.70 39.8848
Fig.8 shows the variation of PSNR as a function of number of coefficients used in image reconstruction. The curves for wbarb and bust clearly indicate the influence of spectral activity on the quality of image. It is shown for the same value of number of coefficients used in image reconstruction; the PSNR of bust image is higher than the PSNR of wbarb image. Also, we notice that when the number of coefficients used in image reconstruction is equal to 38%, the PSNR of wbarb is 38 dB and the PSNR of bust is about 40.5 dB, (a difference of 2.5 dB). Also when the number of coefficients is equal to 9%, the value of PSNR of wbarb and bust is about 29dB, (the gap is zero).
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32
30·
260L-----=---,L-�15o----020=---2c'o-5 --=30--"0350-----=--=---:50. Transform Coefficients non zero (%)
Fig. 8: DCT compression on bust and wbarb images.
Fig.8 clearly shows that when the number of coefficients used in image reconstruction is less or equal to 9%, the PSNR of wbarb is larger than the PSNR of bust. This difference is not too significant. Hence, we can conclude that after a certain threshold reducing, image quality decreases rapidly and the image that has more spectral activity yields a better PSNR.
Fig.9: 15.96 % of transform coefficients used III wbarb reconstruction by OCT -based compression.
Fig.10: 6.46 % of transform coefficients used III wbarb reconstruction by DCT -based compression.
Fig.II: 4.84 % of transform coefficients used III wbarb reconstruction by DCT-based compression.
To compare the quality of image compression using human
eyes, several image compressions using DCT, are given in
Fig.9 to Fig.II. It can be seen that when very few coefficients
(e.g. 4.84%) is used for image reconstruction, there is a slight
loss in the image quality, due to blocking effects. The
blocking effects are well evident in Fig.II.
3.2. Results for DWT compression
Following a method described in section 2.2 and using a
parameter described in section 2.3, we evaluate the efficiency
of image compression using DWT. The threshold used for
DWT compression is applied on images obtained from
iterations one and two.
Table.2 shows the variation of DWT coefficients used in
reconstructed image in function of peak signal to noise ratio
(PSNR). Fig.12 illustrates the variation of PSNR versus the
DWT coefficients. We observe that image bust yields a better
PSNR than image wbarb, This is due to the fact that image
bust has less spectral activity comparing to image wbarb.
30
28
26
8 11 14 17 20 23 26 29 32 35 Number of coefficients (%)
Fig.I2: DWT compression on wbarb and bust images.
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Table 2: PSNR evaluation for wbarb and bust with DWT.
DWTbust DWT wbarb Thre roefficients PSNR Thrc oefficients PSNR
(Threshold) DWT(%) (dB) DWT(%) (dB) Thre 0 8.33 24.7447 Thre 8.33 25.8702
I Thre 0/2 8.66 25.8747 Thrc 8.37 26.0400
1/2 Thre 0/4 9.84 28.0138 Thrc 9.02 27.2794
lI4 Thre 0/8 13.70 31.5310 Thre 12.51 30.3013
1/8 Thre 0/16 21.30 35.9071 Thrc 20.21 34.4 1 08
1116 Thre 0/32 33.39 41.1 103 Thrc 32.49 38.9832
1/32 Thre 0/64 48.66 46.8347 Thrc 52.47 44.9369
1/64
It can be seen that for 27% of DWT coefficients used in
image reconstruction, more than I dB is the difference
between the PSNR of images bust and wbarb. However,
when the number of DWT coefficients used in image
reconstruction is equal or less than 10%, the PSNR of image
wbarb is higher than the PSNR of image bust.
Fig. 13 : 20.21 % of transfonn coefficients used III wbarb reconstruction by DWT-based compression.
Fig.14: 8.37 % of transfonn coefficients used in wbarb reconstruction by DWT-based compression.
We can conclude that the difference of the PSNR of the two images decreases with the decrease of DWT coefficients used in image reconstruction, but it is always in favour of the image that has less spectral activity, i.e. image bust. Nevertheless, from a given number of DWT coefficients this difference disappears and after that it slightly increases in advantage of the image that has more spectral activity, i.e., image wbarb. To compare the quality of image compression using human eyes, some image compression using DWT, are given in Fig.13 and Fig.14. We can see through those figures, the image quality loss when using a few DWT coefficients (e.g. 8.37%) for image reconstruction. The absence of blocking effect is evident however it is replaced by smoothness effect. The smoothness effect is well evident in Fig.14.
3.3. Results for the hybrid method, improved-DWT -DCT
Fig.IS compares the DWT and DCT compression
techniques applied on image bust. It can be seen that for
transform coefficients equal to 33%, the PSNR of image
compressed by DCT is about 39 dB and the PSNR of image
compressed by DWT is higher than 41 dB. The difference
between the PSNR of the two techniques is about 2 dB III
advantage of DWT technique.
39 CD 37· :s- a:: 35 z � 33
31 29 27 2\����������������
Number of coefficients (%)
Fig.lS: Comparison of DCT and DWT techniques for image bust.
45 rr=�====����--��==�� 43 41 rL-"'-=-=-'----_-' 39
CD 37 :s a:: 35 z � 33
31 29 27
5 10 15 20 25 30 35 40 45 50 55 Number of coefficients (%)
Fig.16: Comparison of DCT, DWT and improved-DWT techniques for image wbarb.
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As the transform coefficients used for image reconstruction decreases, this difference decreases and eventually, it becomes zero at 14% coefficients. Nevertheless, when those coefficients continue to decrease this phenomenon is reversed in favour of DCT technique. To improve the compression by DWT, we apply two thresholds instead of one, as described in section 2. The two thresholds should be different and less or equal to the small coefficients in approximation coefficients A3• The resulting algorithm is named improved-DWT.
Fig.16 compares the results of DWT, DCT and improved DWT compression techniques applied on image wbarb. It can be seen that for the coefficients greater than 14%, DWT compression technique provides a better PSNR comparing to DCT compression techniques. In addition, improved-DWT provides a better PSNR comparing to the DCT and DWT compression techniques for all coefficients used for image reconstruction. We also observe from the figure that when the transform coefficients used in image reconstruction is less than 14 %, the PSNR provided by DWT technique is less than that by DCT technique. The evaluation results of reconstructed images using improved-DWT method are shown in Fig.18 and Fig.19.
CD �36 0:: z (J) c..
16 24 56 Number of coefficients (%)
Fig.17: Comparison of improved-DWT and improved-DWT-DCT
techniques for wbarb image.
Fig.18: 10.05 % of transform coefficients used In wbarb reconstruction by improved-DWT-based compression.
Fig.19: 3.57 % of transform coefficients used in wbarb reconstruction by improved-DWT-based compression.
We observe that the quality of image in Fig.19, only with 3.57 % of transform coefficients, is similar to that of Fig.14, which uses 8.37 %. In addition, the image in Fig.18 using 10.05% provides much better quality than that presented in Fig.14.
Now we present the evaluation results for improved-DWT DCT method, as presented in Fig.6 and described in Section 2.3. The image is decomposed by DWT transform with a number of iterations equal to three. Two-dimensional DCT is applied on approximation coefficients resulting from iteration and for quantization the quality factor is chosen equal to 80. For all detail coefficients we use two thresholds. The first one is applied on detail coefficients resulting from iterations I and 2. The second one is applied on detail coefficients resulting from iteration 3. The two thresholds are different and are chosen equal or less than the smallest coefficient in approximation coefficients A2 and A3•
Table.3 shows the variation coefficients percentage used in reconstructed image in function of PSNR for improved DWT-DCT and improved-DWT methods. The comparative results of those methods are shown in Fig.17. When the number of DWT coefficients is higher than 24%, improved DWT technique provides better PSNR than improved-DWT DCT.
Table 3: PSNR versus coefficients percentage for Improved-DWT and Improved-DWT-DCT with wbarb image.
lmproved-DWT Improved-DWT-DCT Two-Thre Coef. PSNR Two-Thre Coef. PSNR
(Threshold) DWT (dB) DWT (dB) (%) (%)
AO, BO 2.32 24.503 CO, DO 2.08 23.291 AO/2, BO/2 2.88 25.455 CO/2, DO/2 2.09 23.465 AO/4, BO/4 3.65 25.713 CO/4, DO/4 2.25 24.204 AO/8, BO/8 3.9 25.875 CO/8, DO/8 2.69 25.103
AO/16,BO/16 5.40 27.223 CO/16,DO/16 3.9 26.708 A0/32,B0/32 10.05 30.276 C0/32,D0/32 8.22 29.871 AO/64,BO/64 19.35 34.410 CO/64,DO/64 16.9 33.877
AO/128,BO/128 32.27 38.982 CO/128,DO/128 30.29 38.022 A0/256,B0/256 52.41 44.937 C0/256,D0/256 51.09 42.205
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Fig.20: 9.38% of transform coefficients used in wbarb reconstruction by improved-DWT -DCT based compression.
Fig.21: 3.42 % of transform coefficients used in wbarb reconstruction by improved-DWT-DCT based compression.
However, from this figure we can see that the improvement results are shown only when the number of coefficients used for image reconstruction is less than 20%. The reconstructed images using improved-DWT-DCT method are shown in Fig.20 and Fig.21. It is observed that using improved-DWT DCT method we can obtain the high quality of images with 9.38%. In addition, compared with the improved-DWT result given in Fig.l9, improved-DWT-DCT method provides a better reconstruction image quality even with less transform coefficients as shown in Fig.21.
4 Conclusion In this paper, we have presented a hybrid DCT-DWT method for image compression technique to exploit the advantages of both DCT and DWT techniques. We have shown that DWT yields a better PSNR compared with that of DCT in image compression, when a number of coefficients used for image reconstruction is high. But when the number of coefficients is very small, DCT compression technique outperforms DWT
technique. In addition, we have shown that the use of two thresholds improves the compression of DWT technique, this technique is named DWT-improved. Improved-DWT presents better PSNR than DCT and DWT. Finally, it is shown that the improved-DWT-DCT technique, a hybrid method, which is the combination of improved-DWT and DCT, presents better PSNR comparing to DCT technique for all transform coefficients used in image reconstruction but comparing to DWT-improved, this technique presents better PSNR only when transform coefficients are below 20%.
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