Mr image compression based on selection of mother wavelet and lifting based wavelet

The International Journal of Multimedia & Its Applications (IJMA) Vol.6, No.2, April 2014
DOI : 10.5121/ijma.2014.6206 59
MR IMAGE COMPRESSION BASED ON
SELECTION OF MOTHER WAVELET AND
LIFTING BASED WAVELET
1
Sheikh Md. Rabiul Islam, 2
Xu Huang, 3
Kim Le
1,2,3
Faculty of Education Science Technology & Mathematics,
University of Canberra, Australia
{Sheikh.Islam, Xu.Huang, Kim.Le}@canberra.edu.au
ABSTRACT
Magnetic Resonance (MR) image is a medical image technique required enormous data to be stored and
transmitted for high quality diagnostic application. Various algorithms have been proposed to improve the
performance of the compression scheme. In this paper we extended the commonly used algorithms to image
compression and compared its performance. For an image compression technique, we have linked different
wavelet techniques using traditional mother wavelets and lifting based Cohen-Daubechies-Feauveau
wavelets with the low-pass filters of the length 9 and 7 (CDF 9/7) wavelet transform with Set Partition in
Hierarchical Trees (SPIHT) algorithm. A novel image quality index with highlighting shape of histogram
of the image targeted is introduced to assess image compression quality. The index will be used in place of
existing traditional Universal Image Quality Index (UIQI) “in one go”. It offers extra information about
the distortion between an original image and a compressed image in comparisons with UIQI. The proposed
index is designed based on modelling image compression as combinations of four major factors: loss of
correlation, luminance distortion, contrast distortion and shape distortion. This index is easy to calculate
and applicable in various image processing applications. One of our contributions is to demonstrate the
choice of mother wavelet is very important for achieving superior wavelet compression performances based
on proposed image quality indexes. Experimental results show that the proposed image quality index plays
a significantly role in the quality evaluation of image compression on the open sources “BrainWeb:
Simulated Brain Database (SBD) ”.
KEYWORDS
CDF 9/7,MRI, Q(Kurtosis),Q(Skewness), SPIHT, UIQI .
1. INTRODUCTION
Wavelet transforms have received significant attentions in the field of signal and image
processing, because of their capability to signify and analyse more efficiently and effectively. For
image compression scheme, data can be compressed and its stored in much less memory space
than in original form.Early 1990’s many researchers have shown energetic interests in adaptive
wavelet image compression. Recently research worked on wavelet construction called lifting
scheme, has been established by Wim Sweldens and Ingrid Daubechies [1]. This construction will
be introduced as part our new algorithm. This method [2] has been shown to be more efficient in
compressing fingerprint images. The properties of wavelets are summarized by Ahuja et al. [3] to
facilitate mother wavelet selection for a chosen application. However, it was very limited in terms
of the relations between mother wavelet and outcomes of wavelet compression, which will be one
of our major contributions in this paper. The JPEG2000 standard[4] presents the result of image
compression for different mother wavelets. It can be concluded that the proper selection of
mother wavelet is one of the very important parameters of image compression. In fact, selection
of mother wavelet can seriously impact on the quality of images[5].

60
In this paper, we have used the technology called lifting based on CDF 9/7 and different mother
wavelet families to compress the test images by using Set Partition in Hierarchical Trees (SPIHT)
algorithm [6]. We have also proposed a new image quality index for selection of mother wavelets
and image compression over BrainWeb: Simulated Brain Database[7]. This new image quality
index with highlighting shape of histogram will be introduced to assess image qualities. The
image intensity histogram expresses a desired shape. In statistics, a histogram is a graphical
representation showing a visual impression of the data distribution. It is used to show the
frequency scattering of measurements. The total area of the histogram is equal to the number of
data. The axis is generally specified as continuous, non-overlapping intervals of brightness values.
The intervals must be adjacent and are chosen to be of the same size. A graphical representation
of image histogram displays the number of pixels for each brightness value in a digital image.
Figure 1 shows the frequency at which each grey-level occurs from 0(black) to
255(white).Histogram is given as ℎሺ‫ݎ‬௞ሻ =
݊௞
ܰൗ , where ‫ݎ‬௞ is the intensity value , ݊௞ is the
number of pixels in image with intensity ‫ݎ‬௞ and N is the total number of samples in the input
image respectively.
In recent years, many IQA methods have been developed. Video quality experts group (VQEG)
and International Telecommunication Union (ITU) are working for standardization [8] [9].
Laboratory for image and video engineering (LIVE) is also working for developing the objective
quality assessment of an image and video [10].
We integrate the shape of histogram into the Universal Image Quality Index metric. The index is
the fourth factor added to existing Universal Image Quality Index (UIQI) to measure the
distortion between original images and distorted images. Hence this new image quality index is a
combination of four factors. The UIQI index approach does not depend on the image being tested
and the viewing conditions of the individual observers. The targeted image is normally a distorted
image with reasonable high resolution. We will consider a large set of images and determine a
quality measurement for each of them. Image quality indexes are used to make an overall quality
assessment via the proposed new image quality index. In this paper the performance evaluation of
the proposed index and compressed image will be tested on open source “BrainWeb: Simulated
Brain Database (SBD)” .The proposed image quality index will be compared with other objective
methods. The image quality assessment has focused on the use of computational models of the
human visual system [11]. Most human vision system (HVS)-based assessment methods
transform the original and distorted images into a “perceptual representation” that takes into
account near-threshold psychophysical properties. Wang et al. [12] and [13] measured structure
based on a spatially localized measure of correlation in pixel values structural similarity (SSIM)
and in wavelet coefficients MS-SSIM. Visual signal to noise ratio (VSNR)[14] is a wavelet based
for quantifying the visual fidelity of distorted images based on recent psychophysical findings
reported by authors involving near –threshold and super threshold distortion.
This paper is structured as follows. Section 2 describes our proposed algorithm, including wavelet
logical transform, CDF 9/7 wavelet transform, the selections of mother wavelet and SPIHT
coding algorithm. Section 3 shows proposed image quality index. Section 4 demonstrates the
simulation results and discussion. Finally in Section 5, a conclusion is presented.
2. PROPOSED ALGORITHM
We first consider the FIR-based discrete transform. The input image x is fed into a ℎ෨(low pass
filter) and ݃෤(high pass filter) separately. The outputs of the two filters are then subsampled. The
resulting low-pass subband ‫ݕ‬௅ and high-pass subband ‫ݕ‬ு are shown in Fig. 1. The original signal
can be reconstructed by synthesis filters h and g which take the upsampled ‫ݕ‬௅ and ‫ݕ‬ு as inputs

61
[15]. An analysis and synthesis system has the perfect reconstruction property if and only if x ' =
x .
The mathematical representations of ‫ݕ‬௅ and ‫ݕ‬ு can be defined as
ቊ
௬ಽሺ௡ሻୀ∑ ௛෩ሺ௜ሻ௫ሺଶ௡ି௜ሻ
ಿಽషభ
೔సబ
௬ಹሺ௡ሻୀ∑ ௚෤ሺ௜ሻ௫ሺଶ௡ି௜ሻ
ಿಹషభ
೔సబ
(1)
where ܰ௅ and ܰு are the lengths of ℎ෨ and ݃෤ respectively. In order to achieve perfect
reconstruction of a signal, the two channel filters shown in Fig. 1 must satisfy the following
conditions[1]:
ቊ
ℎሺ‫ݖ‬ሻℎ෨ሺ‫ݖ‬ିଵሻ + ݃ሺ‫ݖ‬ሻ݃෤ሺ‫ݖ‬ሻ = 2
ℎሺ‫ݖ‬ሻℎ෨ሺ−‫ݖ‬ିଵሻ + ݃ሺ‫ݖ‬ሻ݃෤ሺ−‫ݖ‬ିଵሻ = 0
(2)
)(
~ 1−
zh
)(~ 1−
zg
2↓
2↓
)( 1−
zh
)( 1−
zg
2↑
2↑
Ly
Hy
x 'x
Figure 1: Discrete wavelet transform (or subband transform) analysis and synthesis system: the
forward transform consists of two analysis filters ℎ෨(low pass) and ݃෤(high pass) followed by
subsampling, while the inverse transform first up samples and then uses two synthesis filters ℎ
(low pass) and ݃ (high pass).
In the lifting scheme, the impulse response coefficients h and g are expressed in Laurent
polynomial representation of filter h & g can be defined as
ℎሺ‫ݖ‬ሻ = ∑ ℎ௜‫ݖ‬ି௜௡
௜ୀ௠ (3)
݃ሺ‫ݖ‬ሻ = ∑ ݃௜‫ݖ‬ି௜௡
௜ୀ௠ (4)
where m and n are positive integers. The analysis and synthesis filters as shown in Fig.1 are
further decomposed into the polyphase representations which are expressed as
ℎሺ‫ݖ‬ሻ = ℎ௘ሺ‫ݖ‬ଶሻ + ‫ݖ‬ିଵ
ℎ௢ሺ‫ݖ‬ଶሻ (5)
݃ሺ‫ݖ‬ሻ = ݃௘ሺ‫ݖ‬ଶሻ + ‫ݖ‬ିଵ
݃௢ሺ‫ݖ‬ଶሻ (6)
ℎ෨ሺ‫ݖ‬ሻ = ℎ෨௘ሺ‫ݖ‬ଶሻ + ‫ݖ‬ିଵ
ℎ෨௢ሺ‫ݖ‬ଶሻ (7)
݃෤ሺ‫ݖ‬ሻ = ݃෤௘ሺ‫ݖ‬ଶሻ + ‫ݖ‬ିଵ
݃෤௢ሺ‫ݖ‬ଶሻ (8)
where ℎ௘ሺ‫ݖ‬ሻ = ∑ ℎଶ௞‫ݖ‬ି௞
ܽ݊݀ ℎ௢ሺ‫ݖ‬ሻ = ∑ ℎଶ௞ାଵ‫ݖ‬ି௞
௞௞
ℎ௘ሺ‫ݖ‬ଶሻ =
ℎሺ‫ݖ‬ሻ + ℎሺ−‫ݖ‬ሻ
2
ܽ݊݀ ℎ௢ሺ‫ݖ‬ଶሻ =
ℎሺ‫ݖ‬ሻ − ℎሺ−‫ݖ‬ሻ
2‫ݖ‬ିଵ
݃௘ሺ‫ݖ‬ሻ = ෍ ݃ଶ௞‫ݖ‬ି௞
ܽ݊݀ ݃௢ሺ‫ݖ‬ሻ = ෍ ݃ଶ௞ାଵ‫ݖ‬ି௞
௞௞
݃௘ሺ‫ݖ‬ଶሻ =
݃ሺ‫ݖ‬ሻ + ݃ሺ−‫ݖ‬ሻ
2
ܽ݊݀ ݃௢ሺ‫ݖ‬ଶሻ =
݃ሺ‫ݖ‬ሻ − ݃ሺ−‫ݖ‬ሻ
2‫ݖ‬ିଵ

62
The two polyphase matrices of the filter is defined as
ܲሺ‫ݖ‬ሻ = ቂ௛೐ሺ௭ሻ
௛೚ሺ௭ሻ
௚೐ሺ௭ሻ
௚೚ሺ௭ሻ
ቃ (9)
ܲ෨ሺ‫ݖ‬ሻ = ቂ
௛෩೐ሺ௭ሻ
௛෩೚ሺ௭ሻ
௚෤೐ሺ௭ሻ
௚෤೚ሺ௭ሻ
ቃ (10)
Where ℎ௘ ܽ݊݀ ݃௘ contain even coefficients, and ℎ௢ ܽ݊݀ ݃௢ contain odd coefficients.
The polyphase representations are derived using Z-transform and the subscript e and o denote the
even and odd sub-components of the filters. These are reduced the computation time. The wavelet
transform now is represented schematically in Figure 2 .The perfect reconstruction properties is
given by
ܲሺ‫ݖ‬ሻܲ෨ሺ‫ݖ‬ିଵሻ = ‫ܫ‬ (11)
where I is 2 × 2 identity matrix.
The forward discrete wavelet transform using the polyphase matrix [1] [23] is represented as
ቂ௬ಽሺ௭ሻ
௬ಹሺ௭ሻ
ቃ = ܲ෨ሺ‫ݖ‬ሻ ቂ
௫೐ሺ௭ሻ
௭షభ௫೚ሺ௭ሻ
ቃ (12)
and the inverse discrete wavelet transform
ቂ
௫೐ሺ௭ሻ
௭షభ௫೚ሺ௭ሻ
ቃ = ܲሺ‫ݖ‬ሻ ቂ௬ಽሺ௭ሻ
௬ಹሺ௭ሻ
ቃ (13)
Finally, the lifting sequences are generated by employing Euclidean algorithm which factorizes
the polyphase matrix for a filter pair, reader read reference[16].
1−
z
2↓
2↓
1−
z
)(
~
zP )(zP
2↑
2↑
Ly
Hy
x
ex
ox
Figure 2: Polyphase representation of wavelet transform: first subsample of input signal ‫ݔ‬ into
even as ‫ݔ‬௘ and odd as ‫ݔ‬௢, then apply the dual polyphase matrix. For an inverse transform, apply
the polyphase matrix and then join even and odd.
Cohen-Daubechies-Feauveau 9/7(CDF 9/7) Wavelet Transform is a lifting scheme based a
wavelet transform. The lifting-based WT consists of splitting, lifting, and scaling modules and the
WT itself can be treated as prediction-error decomposition. From Fig.3 we can find that it
provides a complete spatial interpretation of WT. In Fig.3, let ܺ denote the input signal and ܺ௅ଵ
and ܺுଵ be the decomposed output signals where they are obtained through the following three
modules (A, B, and C) of lifting base inverse discrete wavelet transform (IDWT), which can be
described as below:

63
Module Splitting: In this module, the original signal ܺ is divided into two disjoint parts, i.e.,
samples ܺሺ2݊ + 1ሻ and ܺሺ2݊ሻ that denotes all odd-indexed and even-indexed and odd-
indexed samples of ܺ , respectively [17].
Module Lifting: Lifting consist of three basic steps: Split, Predict, and Updating as shown
below.
a) Split -In this stage the input signal is divided in to two disjoint sets, the odd ܺ[2݊ + 1]
and the even samples ܺ[2݊]. This splitting is also called the Lazy Wavelet transform.
b) Predict-In this stage the even samples are used to predict the odd coefficients. This
predicted value, ܲሺܺ [2݊]ሻ, is subtracted from the odd coefficients to give error in the
prediction.
݀[݊] = ܺ[2݊ + 1] − ܲሺܺ[2݊]ሻ (14)
Here ݀[݊] are also called the detailed coefficients.
c) Update-In this stage, the even coefficients are combined with ݀[݊] which are passed
through an update function, U (.) to give
‫]݊[ܥ‬ = ܺ[2݊] + ܷሺ݀[݊]ሻ (15)
Module Scaling: A normalization factor is applied to ݀ሺ݊ሻ and ‫ݏ‬ሺ݊ሻ, respectively. In the
even-indexed part ‫ݏ‬ሺ݊ሻ is multiplied by a normalization factor Ke to produce the wavelet sub
band ܺ௅ଵ. Similarly in the odd-index part the error signal ݀ሺ݊ሻ is multiplied by ‫ܭ‬௢ to obtain
the wavelet sub band ܺுଵ .
Figure 3: The lifting-based WT [17].
The Lifting scheme of the wavelet transform Cohen-Daubechies-Feauveau wavelets with the low-
pass filters of the length 9 and 7 (CDF 9/7) goes through of four steps: two prediction operators
(‘ܽ’ and ‘ܾ’) and two update operators ሺ‘ܿ’ and ‘݀’) as shown it Figure.7.The analysis filter ℎ෨ has 9
coefficients, while the synthesis filter has 7 coefficients. Both high pass filters ݃ , ݃ ̃ have 4
vanishing moments. We chose the filter with 7 coefficients filter because it gives rises to a
smoother scaling function than the 9 coefficients one. In this fact, we run the factoring algorithm
starting from the analysis filter[1]:
ℎ෨௘ሺ‫ݖ‬ሻ = ℎସሺ‫ݖ‬ଶ
+ ‫ݖ‬ିଶ
ሻ + ℎଶሺ‫ݖ‬ + ‫ݖ‬ିଵ
ሻ + ℎ௢ and ℎ෨௢ሺ‫ݖ‬ሻ = ℎଷሺ‫ݖ‬ଶ
+ ‫ݖ‬ିଶ
ሻ + ℎଵሺ‫ݖ‬ + 1ሻ
݃෤௘ሺ‫ݖ‬ሻ = −݃௢ − ݃ଶሺ‫ݖ‬ + ‫ݖ‬ିଵ
ሻ and ݃෤௢ሺ‫ݖ‬ሻ = ݃ଵሺ1 + ‫ݖ‬ିଵ
ሻ + ݃ଷሺ‫ݖ‬ + ‫ݖ‬ିଵ
ሻ
The coefficients of the remainders are computed as:

64
‫ݎ‬௢ = ℎ௢ − 2ℎସℎଵ/ℎଷ
‫ݎ‬ଵ = ℎଶ − ℎସ − ℎସℎଵ/ℎଷ
‫ݏ‬௢ = ℎଵ − ℎଷ − ℎଷ‫ݎ‬௢/‫ݎ‬ଵ
If we now let
ܽ = ℎସ/ℎଷ ≈ −1.58613432,
ܾ = ℎଷ/‫ݎ‬ଵ ≈ −0.05298011854,
ܿ = ‫ݎ‬ଵ/‫ݏ‬௢ ≈ 0.8829110762,
݀ = ‫ݏ‬௢/‫ݎ‬௢ ≈ 0.4435068522,
‫ܭ‬ = ‫ݎ‬௢ − 2‫ݎ‬ଵ ≈ 1.149604398.
Since the 9/7 tape wavelet filter is symmetric we can present h and g in the z-domain .Hence, a
poly-phase matrix ܲ෨ሺ‫ݖ‬ሻ presents the filter pair ሺℎ, ݃ሻ:
ܲ෨ሺ‫ݖ‬ሻ = ቈ
ℎ෨௘ሺ‫ݖ‬ሻ
ℎ෨௢ሺ‫ݖ‬ሻ
݃෤௘ሺ‫ݖ‬ሻ
݃෤௢ሺ‫ݖ‬ሻ
቉ = ቈ
ℎସሺ‫ݖ‬ଶ
+ ‫ݖ‬ିଶ
ሻ + ℎଶሺ‫ݖ‬ + ‫ݖ‬ିଵ
ሻ + ℎ௢
ℎଷሺ‫ݖ‬ଶ + ‫ݖ‬ିଶሻ + ℎଵሺ‫ݖ‬ + 1ሻ
݃ଵሺ1 + ‫ݖ‬ିଵ
ሻ + ݃ଷሺ‫ݖ‬ + ‫ݖ‬ିଵ
ሻ
݃ଵሺ1 + ‫ݖ‬ିଵሻ + ݃ଷሺ‫ݖ‬ + ‫ݖ‬ିଵሻ
቉
then the factorization is given by
ܲ෨ሺܼሻ = ൤1 ܽሺ1 + ܼିଵሻ
0 1
൨ . ൤
1 0ሻ
ܾሺ1 + ܼሻ 1
൨
. ൤1 ܿሺ1 + ܼିଵሻ
0 1
൨ . ൤
1 0ሻ
݀ሺ1 + ܼሻ 1
൨ . ൤
‫ܭ‬ 0
0 1/‫ܭ‬
൨ (16)
We have found four “lifting” steps and the two “scaling” steps from Fig.4 with same parameter as
follows:
‫ە‬
‫۔‬
‫ۓ‬
ܻሺ2݊ + 1ሻ ← ܺሺ2݊ + 1ሻ + ሺa × [Xሺ2nሻ + Xሺ2n + 2ሻ]ሻ
ܻሺ2݊ሻ ← ܺሺ2݊ሻ + ሺb × [Yሺ2n − 1ሻ + Yሺ2n + 1ሻ]ሻ
ܻሺ2݊ + 1ሻ ← ܻሺ2݊ + 1ሻ + ሺc × [Yሺ2nሻ + Yሺ2n + 2ሻ]ሻ
ܻሺ2݊ሻ ← ܻሺ2݊ሻ + ሺd × [Yሺ2n − 1ሻ + Yሺ2n + 1ሻ]ሻ
(17)
ቊ
ܻሺ2݊ + 1ሻ ← ‫ܭ‬ × ܻሺ2݊ + 1ሻ,
ܻሺ2݊ሻ ← ቀ
ଵ
୏
ቁ × ܻሺ2݊ሻ,
(18)
Figure 4: Lifting scheme of the analysis side of the CDF 9/7 filter bank.
The synthesis side of the CDF9/7 filter bank simply inverts the scaling, and reverses the sequence
of the lifting and update steps. Fig.5 shows the synthesis side of the filter bank using lifting
structure to reconstruct of the signal or image.

65
Figure 5: Lifting implementation of the synthesis side of the CDF 9/7 filter bank.
Selection of Mother Wavelets-the choice of wavelets is crucial and determines the image
compression performance. The choice of wavelet functions depends on the contents and
resolution image. The best way for choosing wavelet function is based on the quality of objective
picture and the quality of subjective picture. In this paper, we have considered MR images only
and the major mother wavelets are including most popular wavelets such as Daubechies Wavelet,
Coiflets Wavelet, Biorthogonal Wavelet, Reverse Biorthogonal Wavelet , Symlets Wavelet, Morlet
Wavelet, and Discrete Mayer Wavelet. The different mother wavelets are studied on different
classes of images based on the performance measurements, including novel proposed image
quality index which are normally used for quality of images . These performances are computed
for the cases of above six mother wavelets for compressing the images of different classes.
Set Partition in Hierarchical Trees (SPIHT) [6] is the most popular image compression method.
It provides such kind of features like as the highest image quality, progressive image
transmission, fully embedded coded file, simple quantization algorithm, fast coding/decoding,
completely adaptive, lossless compression, and exact bit rate coding and error protection. It
makes use of three lists: (i) the List of Significant Pixels (LSP), (ii) List of Insignificant Pixels
(LIP) and (iii) List of Insignificant Sets (LIS). These are coefficient location lists that contain
their coordinates in this algorithm. After the initialization, the algorithm takes two stages for each
level of threshold – the sorting pass (in which lists are organized) and the refinement pass (which
does the actual progressive coding transmission). The result is showed in the form of a bit stream.
It is capable of recovering the image perfectly by coding all bits of the transform.
3. PROPOSED IMAGE QUALITY INDEX
The quality index proposed by Wang-Bovik [18] has been proven very efficient on image
distortion performance evaluation. It considers three factors for image quality measurement. They
consider two pixel gray level real-value sequences ‫ݔ‬ = ሼ‫ݔ‬ଵ, … … … ‫ݔ‬௡ሽ and ‫ݕ‬ = ሼ‫ݕ‬ଵ, … … … ‫ݕ‬௡ሽ.
They are obtained by the following expressions:
ߪ௫
ଶ
=
ଵ
௡ିଵ
∑ ሺ‫ݔ‬௜ − ‫̅ݔ‬ሻଶ௡
௜ୀଵ , ߪ௬
ଶ
=
ଵ
௡ିଵ
∑ ሺ‫ݕ‬௜ − ‫ݕ‬തሻଶ௡
௜ୀଵ , ߪ௫௬ =
ଵ
௡ିଵ
∑ ሺ‫ݔ‬௜ − ‫̅ݔ‬ሻ௡
௜ୀଵ ሺ‫ݕ‬௜ − ‫ݕ‬തሻ
Where ‫̅ݔ‬ is the mean of ‫,ݔ‬ ‫ݕ‬ത is the mean of ‫ݕ‬ ߪ௫
ଶ
is variance of x, ߪ௬
ଶ
is variance of y and ߪ௫௬ is the
covariance of x, y
Then, we can compute a quality factor, Q:
ܳ =
ସఙೣ೤௫̅௬ത
ሺ௫̅మା௬തమሻ൫ఙೣ
మାఙ೤
మ൯
(19)
Q can be decomposed into three components as
ܳ =
ఙೣ೤
ఙೣఙ೤
∙
ଶ௫̅௬ത
ሺ௫̅మା௬തమሻ
∙
ଶఙೣఙ೤
൫ఙೣ
మାఙ೤
మ൯
(20)

66
In equation (20), the first component is the correlation coefficient between ‫ݔ‬ and ‫ݕ‬, which
measures the degree of linear correlation between ‫ݔ‬ and ‫.ݕ‬ The second component measures how
close the mean luminance is between ‫ݔ‬ and ‫.ݕ‬ The third component measures how similar the
contrasts of the images are as ߪ௫ and ߪ௬ can be viewed as estimation of the contrasts of ‫ݔ‬ and ‫.ݕ‬
Hence, we have three components for a quality factor, Q which can be rewrite as:
ܳ = ܿ‫݊݋݅ݐ݈ܽ݁ݎݎ݋‬ ∙ ݈‫݁ܿ݊ܽ݊݅݉ݑ‬ ∙ ܿ‫ݐݏܽݎݐ݊݋‬
The values of the three components are in the range of [0, 1]. Therefore, the quality metric is
normalized between[0, 1].
Besides these three factors, many studies show that in human visual system (HVS), histogram of
image information plays a very important role, when human subjective judges the quality of an
image. It does work by redistributing the gray-levels of the input image by using its probability
distribution function. Although its preserves the brightness in the output image with a significant
contrast enhancement. It may produce images which do not look as natural as the input ones. To
take the advantages of known characteristics of human perception, we introduce the shape of
histogram with the Universal Image Quality Index metric. This proposed image quality of index
will be tested throughout standard database of images.
We use the statistical differences to develop a novel image quality index. To find out the shape of
histogram from distorted image, we computed its kurtosis and skewness. Skewness, indicating a
degree of asymmetry of a histogram, is given by the following equation:
‫ܭ‬ௌ௞௘௪௡௘௦௦ =
∑ ሺ௬ି௬തሻయ೙
೔సభ
ሺ௡ିଵሻ௦య ` (21)
Kurtosis quantifies a degree of histogram peakiness and tail weight. That is, data sets with high
kurtosis tend to have a distinct peak near the mean and have a heavy tail. Data sets with low
kurtosis tend to have a flat top near the mean. It can be described by the following equation
‫ܭ‬௄௨௥௧௢௦௜௦ =
∑ ሺ௬೔ି௬തሻర೙
೔సభ
ሺ௡ିଵሻ௦ర (22)
The formula for modified skewness is:
‫ܭ‬ெ௢ௗ௜௙௬ ௌ௞௘௪௡௘௦௦ =
∑ |௬೔ି௬ത|య೙
೔సభ
|௡ିଵ|௦య (23)
where ݊ is the number of pixels at image distortion value ‫ݕ‬௜, ‫ݕ‬ത is the mean value of image
distortion, ‫ݏ‬ is the standard deviation.
The proposed new image quality index Q can be expressed by the four components as below:
ܳ =
ఙೣ೤
ఙೣఙ೤
∙
ଶ௫̅௬ത
ሺ௫̅మା௬തమሻ
∙
ଶఙೣఙ೤
൫ఙೣ
మାఙ೤
మ൯
.
ଶ௄ೣ௄೤
௄ೣ
మା௄೤
మ (24)
The 4th
component in the above equation measures how similar the shape of histogram of the
images is. As ‫ܭ‬௫ and ‫ܭ‬௬ can be viewed as estimation of the shape of ‫ݔ‬ and ‫,ݕ‬ the values of the
four components is normalized so this is still in the range of [0, 1]. The value of ‫ܭ‬௫ and ‫ܭ‬௬ are
computed into three different way using equation (21), (22) & (23). Therefore the Q can be
expressed by the following four factors:

67
ܳ = ܿ‫݊݋݅ݐ݈ܽ݁ݎݎ݋‬ ∙ ݈‫݁ܿ݊ܽ݊݅݉ݑ‬ ∙ ܿ‫ݐݏܽݎݐ݊݋‬ ∙ ‫ݏ‬ℎܽ‫݁݌‬
The new quality index will be applied to local regions using a sliding window for objective image
quality analysis. For example starting from the top-left corner of the image, a sliding window
with the size of ‫ܤ‬ × ‫ܤ‬ is moving pixel by pixel horizontally and then vertically through all
pixels of the image. We assume that at the position of ሺ݅, ݆ሻ in the target image, the local quality
index ܳ௜௝ can be computed as equation (25). Here, the row number and column number of the
image are ݊ and ݉, then the overall normalized quality index is:
ܳ =
ଵ
௡×௠
∑ ∑ ܳ௜௝
௠
௝ୀଵ
௡
௜ୀଵ (25)
The overall performance of the proposed image quality index is based on shape of histogram
with UIQI [18] can be further described in Figure 6. In order to show the efficiency of this image
quality index, the open source “BrainWeb: Simulated Brain Database (SBD) [7] is used for
testing, which will be discussed in the next section.
Start
Study of current metrics
Create a novel image
quality index
Simulate the proposed image
quality index and others
metrics
Apply the metrics on some
standard images (such as open
sauces standard database)
To evaluate and compare
the performance of
proposed quality index and
different metrics
End
Quality measures
Figure 6: Flow chart of propose image quality index.
4. RESULTS AND DISCUSSION
For the simplicity, we have chosen MR images size 512x512 (grayscale) from BrainWeb:
Simulated Brain Database (SBD) (Fig.17(a) as an example with specifications T1 AI msles2
1mm pn3 rf20). The algorithms and proposed image quality index are implemented in MATLAB.
In our experiment, we have used mother wavelet families: Haar Wavelet (HW), Daubechies (db),
Symlets (Sym), Coiflets (Coif), Biorthogonal (bior), Reverse Biorthogonal (rbior) & discrete
Meyer (dmey). The quality of compressed image depends on the number of decompositions. We
have used 3rd
level decomposition for these experiments.
The proposed image quality index metric is generally competitive with the image compression
over the Simulated Brain Database (SBD). Here, the four metrics, peak signal to noise ratio
(PSNR), VSNR [14], UIQI [18], structural similarly(SSIM) [8], , were applied .The results of
PSNR, VSNR, SSIM , UIQI were computed using their default implementation. Table 1-5, have
shown the simulation results of our proposed image quality index and other IQA methods. We
consider three different cases such as image quality index Q (Skewness) for shape of histogram
using skewness equation (21), image quality index Q (Abs-Skewness) for shape of histogram
using absolute skewness equation (23), and image quality index Q (Kurtosis) for shape of
histogram using kurtosis equation (22). It can be seen that the proposed method performs quite
well for image compression with selection of mother wavelet by image quality indexes.

68
4.1 Daubechies family and CDF 9/7 wavelet with SPIHT-Based on the Table 1 and Fig.7-8
shows that the CDF 9/7 wavelet has a highest PSNR value approximately 97 dB and Compression
ratio (CR) 87.5 % and proposed image quality index ,Q (Kurtosis) value 0.9887, Q (Skewness)
value 0.9887 & Q (Abs-Skewness) value 0.9827. On the other hand the wavelets, Daubechies 2,
has the highest PSNR values as well as proposed image quality index, Q (Kurtosis) value 0.9325,
Q (Skewness) value 0.9325& Q (Abs-Skewness) value 0.9225. However when the comparing the
proposed image quality index and other IQA aspects, the top results are of CDF9/7, Daubechies 2
which is satisfy the best suitable wavelet image compression with SPIHT.
Table 1 Wavelet Family: Daubechies, Discrete Meyer, Haar & CDF9/7 wavelet transform with
SPIHT
Wavelet
Family
PSNR(dB) VSNR SSIM Q(Kurtosis) Q(Skewness) Q(Abs-
Skewness)
UIQI CR(%)
CDF9/7 96.7885 97.3671 0.9995 0.9887 0.9887 0.9827 0.7563 87.5
Haar 47.9890 51.6388 0.8265 0.9320 0.9320 0.9222 0.8691 31.28
dmey 48.0520 51.2418 0.8603 0.9311 0.9311 0.9200 0 .8404 40.17
Db1 47.9890 51.6388 0.8603 0.9320 0.9320 0.9222 0.8691 40.17
Db2 48.3710 51.7339 0.8512 0.9325 0.9325 0.9225 0.8622 34.78
Db3 48.3830 50.9332 0.8543 0.9318 0.9318 0.9217 0.8634 33.03
Db4 47.7516 51.2418 0.8372 0.9265 0.9265 0.9151 0.8534 29.49
Db5 47.0383 51.2418 0.8308 0.9307 0.9307 0.9189 0.8423 29.78
Db6 46.5373 51.2418 0.8354 0.9294 0.9294 0.9178 0.8464 29.36
Db7 47.5299 51.2418 0.8471 0.9308 0.9308 0.9208 0.8698 29.23
Db8 47.4315 51.2418 0.8423 0.9311 0.9311 0.9207 0.8698 29.53
Db9 47.9616 51.2418 0.8461 0.9319 0.9319 0.9215 0.8545 29.38
Db10 47.4385 51.2418 0.8313 0.9313 0.9313 0.9212 0.8544 29.52
Figure 7: Wavelet versus proposed Q and other IQA methods (Daubechies family and
CDF9/7).

69
Figure 8: Wavelet versus PSNR, VSNR and Compression ratio (CR) (Daubechies family and
CDF9/7).
4.2 Coiflet Family with SPIHT-From Table 2 and Fig.9&10, it appears that the wavelet Coiflet 1
has the highest compression ratio(CR) and Coiflet 5 has the highest PSNR 49.13dB and % and
proposed image quality index ,Q (Kurtosis) value 0.9323, Q (Skewness) value 0.9323& Q (Abs-
Skewness) value 0.9220 for Coiflet 2. However, when comparing the proposed index and other
IQA aspects, the top results are of Coiflet 2 in this family.
Table 2 Wavelet Family: Coiflet
Wavelet
Family
Skewness)
UIQI CR(%)
Coif1 47.3489 46.4317 0.8515 0.9279 0.9279 0.9174 0.8654 34.39
Coif2 47.7963 50.9647 0.8541 0.9323 0.9323 0.9220 0.8707 31.79
Coif3 47.5209 46.3979 0.8303 0.9273 0.9273 0.9161 0.8469 31.16
Coif4 47.7428 48.9957 0.8524 0.9320 0.9320 0.9218 0.8659 30.91
Coif5 48.1693 50.3344 0.8462 0.9314 0.9314 0.9199 0.9065 30.83
Figure 9: Wavelet versus proposed Q and other QA methods (Coiflet family).

70
4.3 Symlet Family with SPIHT-The result shown in Table 3 and Fig.11&12, it appears that the
wavelet Symlet 3 has the highest PSNR value as 48.3830 dB and Symlet 2 has the highest
Compression ratio(CR) 34.78% and. and proposed image quality index ,Q (Kurtosis) value
0.9323, Q (Skewness) value 0.9323& Q (Abs-Skewness) value 0.9220 for Coiflet 2. However,
when comparing the proposed index and other IQA aspects, the top results are of Symlet 8 in this
family.
Figure 10: Wavelet versus PSNR,VSNR and Compression ratio (CR) (Coiflet family).
Table 3 Wavelet Family: Symlet with SPIHT
Wavelet
Family
PSNR(dB) VSNR SSIM Q(Kurtosis) Q(Skewnes
s)
Q(Abs-
Skewness)
UIQI CR(%)
Sym2 48.3710 51.7339 0.8512 0.9325 0.9325 0.9225 0.8622 34.78
Sym3 48.3830 50.9332 0.8543 0.9318 0.9318 0.9217 0.8634 33.03
Sym4 48.1998 51.0804 0.8490 0.9298 0.9298 0.9169 0.8639 31.66
Sym5 48.1827 49.8120 0.8431 0.9273 0.9273 0.9162 0.8566 30.72
Sym6 48.2877 49.8413 0.8404 0.9322 0.9322 0.9219 0.8536 30.88
Sym7 47.2442 51.3857 0.8462 0 .9310 0.9310 0.9203 0.8579 31.12
Sym8 47.1674 43.7892 0.8526 0.9331 0.9331 0.9227 0.8652 30.63
4.4 Biorthogonal Family with SPIHT-Table 4 and Figure.16 &17, it appears that the wavelet
Biorthogonal 1.1 has the highest Compression ratio(CR) 40.17% and Biorthogonal 3.3 has the
highest PSNR value is 48.1569 dB and proposed image quality index ,Q (Kurtosis) value 0.9343,
Q (Skewness) value 0.9343& Q (Abs-Skewness) value 0.9239 for Coiflet 2 Biorthogonal 3.1.
However, when comparing the proposed index and other IQA aspects, the top results are of
Biorthogonal 3.1 in this family.

71
Figure 11: Wavelet versus proposed Q and other QA methods (Symlet family).
Figure 12: Wavelet versus PSNR,VSNR and Compression ratio(CR) (Symlet family).
Table 4 Wavelet Family: Biorthogonal with SPIHT
Wavelet
Family
Skewness)
UIQI CR(%)
Bior1.1 47.9890 51.6388 0.8603 0.9320 0.9320 0.9222 0 .8691 40.17
Bior1.3 47.1478 50.7201 0.8604 0.9335 0.9335 0.9236 0.8843 39.01
Bior1.5 47.8690 51.6038 0.8536 0.9303 0.9303 0.9176 0.8693 39.1
Bior2.2 47.7850 50.7439 0.8528 0.9301 0.9301 0.9198 0.8693 29.05
Bior2.4 47.2538 49.2861 0.8567 0.9310 0.9310 0.9206 0.8733 29.39
Bior2.6 47.6562 48.8268 0.8575 0.9331 0.9331 0.9231 0.8702 29.66
Bior2.8 47.3795 48.5517 0.8529 0.9328 0.9328 0.9225 0.8718 29.85
Bior3.1 46.4341 46.1780 0.8455 0.9343 0.9343 0.9239 0.8639 38.58
Bior3.3 48.1569 49.5595 0.8562 0.9331 0.9331 0.9230 0.9178 27.33
Bior3.5 47.8803 51.0062 0.8463 0.9324 0.9324 0.9223 0.8695 27.33
Bior3.7 47.0345 48.5183 0.8398 0.9317 0.9317 0.9213 0.8605 27.46
Bior3.9 47.9802 51.7129 0.8307 0.9341 0.9341 0.9239 0.8529 27.61
Bior4.4 48.0226 50.5716 0.8551 0.9327 0.9327 0.9225 0.8392 29.74
Bior5.5 47.3967 49.4120 0.8467 0.9311 0.9311 0.9198 0.8686 30.44
Bior6.8 47.2664 45.8918 0.8349 0.9312 0.9312 0.9198 0.8623 27.87

Figure 16: Wavelet versus proposed Q and other IQA methods (Biorthogonal family).
Figure 17: Wavelet versus PSNR,
4.5 Reverse Biorthogonal Family with SPIHT
16, it appears that the wavelet Reverse Biorthogonal 1.3
dB and Compression ratio(CR)
0.9343, Q (Skewness) value 0.9343
3.1. However, when comparing the proposed index and other IQA aspects, the top results are of
Reverse Biorthogonal 1.3 in this family.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Bior1.1
Bior1.3
Bior1.5
Bior2.2
Bior2.4
0
10
20
30
40
50
60
Bior1.1
Bior1.3
Bior1.5
Bior2.2
Wavelet versus proposed Q and other IQA methods (Biorthogonal family).
Wavelet versus PSNR, VSNR and Compression ratio(CR).
Reverse Biorthogonal Family with SPIHT-The results were shown in Table 5
Reverse Biorthogonal 1.3 has the highest PSNR value as
and Compression ratio(CR) 39.93 % and proposed image quality index ,Q (Kurtosis) value
0.9343& Q (Abs-Skewness) value 0.9239 for Coiflet 2 Biorthogonal
in this family.
Bior2.4
Bior2.6
Bior2.8
Bior3.1
Bior3.3
Bior3.5
Bior3.7
Bior3.9
Bior4.4
Bior5.5
Bior6.8
SSIM
Q(Kurtosis)
Q(Skewness)
Q(Abs-Skewness)
UIQI
Bior2.2
Bior2.4
Bior2.6
Bior2.8
Bior3.1
Bior3.3
Bior3.5
Bior3.7
Bior3.9
Bior4.4
Bior5.5
Bior6.8
PSNR(dB)
VSNR
CR(%)
72
Wavelet versus proposed Q and other IQA methods (Biorthogonal family).
and Fig.15 &
has the highest PSNR value as 48.0356
image quality index ,Q (Kurtosis) value
Coiflet 2 Biorthogonal
Skewness)
PSNR(dB)

73
Table 5 Wavelet Family: Reverse Biorthogonal with SPIHT
Wavelet
Family
Skewness)
UIQI CR(%)
rbio1.1 47.9890 51.6388 0.8603 0.9320 0.9320 0.9222 0.8691 40.17
rbio1.3 48.0356 51.2309 0.8544 0.9326 0.9326 0.9224 0.8946 31.07
rbio1.5 47.9093 50.8185 0.8530 0.9329 0.9329 0.9225 0.8673 39.93
rbio2.2 47.2276 45.6882 0.8536 0.9296 0.9296 0.9194 0.8673 40.36
rbio2.4 47.4499 46.5788 0.8515 0.9312 0.9312 0.9208 0.8624 35.06
rbio2.6 47.6458 46.5177 0.8508 0.9319 0.9319 0.9216 0.8784 33.53
rbio2.8 47.5385 46.4243 0.8526 0.9319 0.9319 0.9216 0.8661 32.96
rbio3.1 41.0837 33.1195 0.7688 0.9214 0.9215 0.9103 0.8672 57.89
rbio3.3 44.1834 43.3229 0.8137 0.9229 0.9229 0.9120 0.8439 42.84
rbio3.5 45.7957 48.5076 0.8324 0.9290 0.9290 0.9181 0.8350 38.77
rbio3.7 45.9650 47.7605 0.8315 0.9291 0.9291 0.9178 0.8547 37.14
rbio3.9 46.0123 49.1438 0.8284 0.9273 0.9273 0.9164 0.8558 36.53
rbio4.4 47.6831 48.8808 0.8554 0.9327 0.9327 0.9227 0.8445 31.87
rbio5.5 47.3042 48.6219 0.8562 0.9321 0.9321 0.9221 0.8696 31.87
rbio6.8 47.2914 44.7912 0.8507 0.9317 0.9317 0.9213 0.8700 31.15
Figure 15: Wavelet versus proposed Q and other IQA methods (Coiflet family).
Figure 16: Wavelet versus PSNR, MSSIM and Compression ratio

74
To show the performance for the better selection of wavelet with SPIHT, we made a comparison
in terms of highest values of image quality by proposed image quality indexes and other IQA
methods represented in Fig.17. We have seen our compression technique found good result
compressed image quality and also achieve higher compression ratio for MR images. We are
deeply investigated into the Table 1-5 and Figures 7-16, the lifting based wavelet transforms
produced high PSNR around 97dB and high compression ratio 88% and proposed image quality
index ,Q (Kurtosis) value 0.9887, Q (Skewness) value 0.9887 & Q (Abs-Skewness) value 0.9827
which keeps the image quality well. The overall performance of different mother wavelets has
shown in this experiment produced highest PSNR around 46.4341dB and compression ratio 38.58
% produced by Biorthogonal 3.1. We have seen lifting based CDF 9/7 coupled with SPIHT is a
better choice for wavelet image compression.CDF9/7 has the highest PSNR, VSNR SSIM value
and highest value of proposed image quality index because of the filters was subsampled and thus
avoids computing samples that was subsampled immediately .Lifting is only one idea is a whole
tool bag of methods to improve the speed of fast wavelet transform. From the above discussion,
it is evident the lifting based wavelets outperform the traditional or mother wavelets. This
compression technique can save time in medical image transmission and achieving process. So
this simple and efficient compression technique an proposed image quality index can very useful
in the field of medical image processing and transmission.
Figure.17 Compressing of MR image with different mother wavelet and SPIHT coding compare
the highest value of PSNR, Q(Kurtosis), Q(Skewness) & Q(Abs-Skewness) and Compression
ratio as seen in Table 1,2,3,4,5 and show the best wavelet for image compression in Table 6 .
Also shown (a) Original image(MR).The best mother wavelet with SPIHT as (b) CDF
9/7 ,(c)Daubechies Wavelet (db2),(d) Coiflets Wavelet(Coif 2),(e) Symlets Wavelet(Sym 8),(f)
Biorthogonal Wavelet(Bior 3.1), and (g) Reverse Biorthogonal Wavelet(rbio 1.5).

75
5. CONCLUSIONS
In this paper, a comparative study of different wavelet families and lifting based CDF 9/7 with
SPIHT on the basis of MR images has been done with new image quality index and other IQAs,
as well as compression ratio. The simulated results have given the choice of optimal wavelet for
image compression. The effects of lifting based CDF 9/7 wavelet transform and traditional
mother wavelets, Haar, Daubechies, Symlets, Coiflets, Biorthogonal and Reverse Biorthogonal
wavelet, Discrete meyer wavelet families together with SPIHT on BrainWeb: Simulated Brain
Database (SBD) . We comprehensively analysed the effects for a wide range of different mother
wavelets family and lifting based CDF9/7. We found that lifting based CDF 9/7 wavelet
provided better compression performance for the BrainWeb: Simulated Brain Database (SBD). It
has produced as high PSNR as around 97dB, as high compression ratio as 88%, and and proposed
image quality index ,Q (Kurtosis) value 0.9887, Q (Skewness) value 0.9887 & Q (Abs-Skewness)
value 0.9827 which keeps the image quality quite well. The wavelet Biorthogonal 3.1 with
SPIHT also provided competitive compression performance with and proposed image quality
index ,Q (Kurtosis) value 0.9343, Q (Skewness) value 0.9343& Q (Abs-Skewness) value 0.9239.
Thus, we conclude that the “best wavelet” choice of wavelet in image compression depend on the
image content and satisfactory results of proposed image quality index ,Q (Kurtosis) , Q
(Skewness) & Q (Abs-Skewness) as well as compressed image quality.
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AUTHORS
Sheikh Md. Rabiul Islam received the B.Sc.in Engg. (ECE) from Khulna University,
Khulna, Bangladesh in December 2003, and M.Sc. in Telecommunication Engineering
from the University of Trento, Italy, in October 2009 and currently doing an PhD by
Research under Faculty of Education Science Technology & Mathematics at University
of Canberra, Australia. He joined as a Lecturer in the department of Electronics and
Communication Engineering of Khulna University of Engineering & Technology,
Khulna, in 2004, where he is joined an Assistant Professor in the same department in the
effect of 2008. He has published 20 Journal and 15 International conferences. His research interests include
VLSI, Wireless communications, signal & image processing, and biomedical engineering.
Professor (Dr) Xu Huang has received the B.E. and M.E. degrees and Ph.D. in
Electrical Engineering and Optical Engineering prior to 1989 and the second Ph.D. in
Experimental Physics in the University of New South Wales, Australia in 1992. He has
earned the Graduate Certificate in Higher Education in 2004 at the University of
Canberra, Australia. He has been working on the areas of the telecommunications,
cognitive radio, networking engineering, wireless communications, optical
communications, and digital signal processing more than 30 years. Currently he is the
Professor at the Faculty of Education Science Technology & Mathematics. He has been a
senior member of IEEE in Electronics and in Computer Society since 1989 and a Fellow of Institution of
Engineering Australian (FIEAust), Chartered Professional Engineering (CPEng), a Member of Australian
Institute of Physics. He is a member of the Executive Committee of the Australian and New Zealand
Association for Engineering Education, a member of Committee of the Institution of Engineering Australia
at Canberra Branch. Professor Huang is Committee Panel Member for various IEEE International
Conferences such as IEEE IC3PP, IEEE NSS, etc. and he has published about two hundred papers in high
level of the IEEE and other Journals and international conference; he has been awarded 9 patents in
Australia.
Professor (Dr) Kim Le has received the B.E. Electrical Engineering School, National
Institute of Technology, Saigon, Vietnam, 1973 and M.E. Electrical & Computer
Engineering Dept., University of Newcastle, NSW, 1986 and Ph.D. in Computer
Engineering 1991 in the University of Sydney, Australia in 1992. He has been working on
the areas of the telecommunications, cognitive radio, wireless communications, and digital
signal & image processing, VLSI more than 30 years. Currently he is assistant professor at
the Faculty of Education Science Technology & Mathematics, University of Canberra,
Australia. He has published about above hundred papers in high level of the IEEE and other Journals and
international conference.

Mr image compression based on selection of mother wavelet and lifting based wavelet

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