Statistical Modelling of Wavelet Coefficients of CT Scan ... - IEEE Xplore

2009 International Conference on Emerging Trends in Electronic and Photonic Devices & Systems (ELECTRO-2009)

Statistical Modelling of Wavelet Coefficients of CT Scan Image Basant Kumar Motilal Nehru National Institute of Technology, Allahabad- 211004, India Email: [email protected]

S.P. Singh, Anand Mohan, Animesh Anand Institute of Technology, Banaras Hindu UniversityVaranasi-221005, India

[8]. Within the same sub-band, the statistical distribution of the wavelet coefficients has also received the attention of researchers. The key issue of any wavelet-based image compression algorithm is the modelling of subband statistics and strategy used for quantizing the wavelet coefficients. The statistics of medical images are quite different from those of natural images owing to imaging equipment characteristics, resulting into different types of noise contents in the image. The presence of inherent characteristic texture called speckle and desire to preserve it are the issues that make the Ultra sound (US) image compression problem different from the compression of natural images [9]. X-ray images suffer from quantum noise which is due to more quantum of x-ray absorbed. Noise in CT scan images is primarily due to the quantum noise inherent in photon detection and electronic noise in the projection of data.

Abstract— Prior knowledge of wavelet coefficient statistics is a key issue in the development of better quantization strategy for enhancing compression efficiency of digital images. Since statistics of medical images are quite different from those of natural images, there is a need for statistical modelling of wavelet coefficients in different subbands. This paper examines the suitability of Student-t, Pareto Weibull and Gaussian distributions for modelling the wavelet coefficients of various subbands in a CT scan image to improve the compression efficiency. It has been found that the statistics of wavelet coefficients in the CT scan images can be better approximated by the generalized Student-t distribution for negative wavelet coefficients whereas generalized Pareto distribution provides better fit for the non-negative coefficients. The results can be potentially useful in designing adaptive quantizer for achieving improved compression gain and reducing computational complexity for medical image coders. Keywords-wavelet coefficients;generalized Student-t distribution; generalized Pareto distribution;Weibull distribution; shape parameter; chi-square test

I.

Harsh Vikram Singh Kamla Nehru Institute of Technology, Sultanpur-228118, India

In recent studies, many researchers have shown that, in the subband representation of medical images, the histograms of wavelet coefficients have heavier tails and are more sharply peaked at zero which can not be obtained using Gaussian distribution [9, 10]. To take into account the heavier-tailed nature of wavelet coefficients, the generalized Student tdistribution was chosen to model the statistics of subband coefficients of a US image by Kaur et al [11]. In this paper, we investigate the suitability of long-tail distributions like generalized Pareto, Weibull and generalized Student -t for statistical modelling of subband wavelet coefficients at different decomposition levels in a CT scan image. The chisquare goodness-of-fit test [12] is performed to identify the most appropriate fit. Section II describes various distribution functions and their performance parameters followed by section III which provides the statistical modelling of wavelet coefficients in different subbands using various statistical distributions and section IV contains the conclusion.

INTRODUCTION

The demands on image compression in radiology are increasing with increase in number of digital imaging modalities and therefore, the management of digital imaging is becoming an important issue. Applying image compression reduces the storage requirements and the network traffic, resulting in the reduction of the image transfer time and the cost Several lossless and lossy techniques for data compression have been proposed [1]-[4]. A lossy medical image compression can be used only to the extent it does not compromise with the diagnostically relevant image details and the image degradation remains imperceptible. Currently, Wavelet coding has proved very effective for achieving higher compression of medical images than JPEG algorithm with comparable computational efficiency [5]-[6]. This is because wavelet coding avoids the presence of blocking artifacts as no image partitioning is required and it also supports progressive transmission capability required for telemedicine applications. Compression is primarily achieved by concentrating the energy of the image into a few wavelet coefficients, while their statistics are also taken into account for obtaining enhanced compression gain.

II.

MODELLING OF SUBBAND WAVELET COEFFICIENTS

Many applications in image processing such as image compression, denoising or retrieval can benefit from a statistical model to characterize the image in the transform domain. The statistical model, even when partially captures the variations in data dependencies and appearance, can substantially benefit image compression by designing model dependant quantization scheme and consequently encoding procedure. For the peaky and heavy-tailed non-Gaussian statistics of typical medical image wavelet decomposition,

Two kinds of wavelet statistics are found useful [7]. The interscale dependencies have been effectively employed in a variety of tree-structured coding techniques, such as SPIHT

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2009 International Conference on Emerging Trends in Electronic and Photonic Devices & Systems (ELECTRO-2009)

The pdf of Weibull distribution is given below:

Student-t, Pareto and Weibull heavy tailed distributions can provide better results for telemedicine applications.

f ( x) = (a b )( x b ) exp− ( x b ) , x ≥ 0 (4) A. Distribution Functions The Gaussian (normal) distribution describes the random f ( x ) =0 , x < 0 variation that occurs in the data taken from many scientific disciplines. The graph of the associated probability density function (pdf) is bell-shaped, with a peak at the mean, and is where a>0 is the shape parameter and b >0 is the scale known as Gaussian function or bell curve. Normal distribution is often mentioned as the short-tailed distribution model. The parameter of the distribution pdf. pdf of the distribution is given by the formula A mirror image distribution function of Weibull has been a −1

1

f ( x) = for -∞‹ x ‹ ∞

2πσ

2

defined to model negative data samples (x< 0). The pdf equation

1 x−μ 2 ( ) ) 2 σ

exp( −

a

(1)

is obtained by replacing x with –x in equation (4). The generalized Student t-distribution is defined as [13]

The parameters μ and σ are the mean and variance of the variable x respectively. The μ determines the location of the peak of the density function. Generalized Pareto, Weibull and generalized Student-t distributions provide a rich and flexible modeling tool for long-tailed data. These distributions have been investigated for modelling of the wavelet coefficients of the radiological images in different subbands. The Generalized Pareto distribution (GP) was developed as a distribution that can model tails of a wide variety of distributions. The probability density function for the generalized Pareto distribution with shape parameter k ≠ 0, scale parameter σ, and threshold parameter θ, is given by 2

⎛ k(x −θ ) ⎞ f (x) = 1 ⎜1+ ⎟ σ⎝ σ ⎠

( )

for

θ


0 and

β απ Γ(α / 2)

(1 +

where Г(x) is the gamma function. The model parameters,

x , when k > 0, or for θ < x < - σ / k when k < 0

If k =0 and

Γ((α + 1) / 2)

= σ / k , the generalized Pareto distribution is

equivalent to the Pareto distribution

χ2 =

N

A mirror image distribution function of the generalized Pareto distribution has been defined to model negative data samples (x