Elsevier

Pattern Recognition

Volume 34, Issue 11, November 2001, Pages 2145-2154
Pattern Recognition

The image moment method for the limited range CT image reconstruction and pattern recognition

https://doi.org/10.1016/S0031-3203(00)00151-5Get rights and content

Abstract

Moment properties of the Radon transform have been discussed. A new concept of the image moment in the Radon transform has been introduced and described. A new approach to reconstruct the image from the projections within a limited range, called the image moment method, has been proposed. The new method has been validated through computer simulations.

Introduction

In the past three decades, computerized tomography (CT) has attracted a great deal of attention in the digital imaging processing field. Along with the maturity of practical methods for image reconstruction from projections, its theoretical basis, Radon transformation received renewed interest. As a linear transform, Radon transform is based on the solutions of the multi-variable integral equation. Because of the explicit geometrical meaning of the Radon transform, it has shown many interesting properties and established varied forms of connections among the space, Frequency and Radon domains.

A number of studies have been carried out on this subject. For example, Lewitt has summarized a series of projection theorems and their corollaries [1]. Deans has collected many materials and provided a detailed description about the properties of Radon transform and its relationship to other transforms [2]. In this paper, a new property of the Radon transform, moment property, is presented. Three new moment theorems are introduced. An new method, the image moment method for image reconstruction, is proposed. The new method is applied to the problem of inverse Radon transform from limited range projections and evaluated by the simulation results.

Section snippets

Definition of the Radon transform

Assume f(x,y) is a two dimensional (2-D) real function in the xy coordinate system, where f(x,y) has a compact support. Assume a ray L(s,θ) is defined by two parameters, s and θ, where s is the distance from the origin to the ray, θ is the angle between the y-axis and the ray. Hence, the ray is described by the following equation (Fig. 1):xcosθ+ysinθ=s.Since s and θ can be any real values, the integral of f(x,y) along the ray L(s,θ) defines a 2-D function, denoted as P(s,θ). We call P(s,θ) the

The projection moment and the projection moment theorem

The moments of Pθ(s) are called projection moments in the Radon domain [1]. The nth projection moment of Pθ(s) is defined asMθ(n)=−∞Pθ(s)snds.

The widely known Uniqueness Theorem of the moments [8] assures that the moments of all orders, Mθ(n), are uniquely determined by Pθ(s). Conversely, Mθ(n) of all orders uniquely determine Pθ(s).

Substituting Eq. (6) into Eq. (9), we haveMθ(n)=−∞−∞−∞f(x,y)δ(xcosθ+ysinθ−s)sndxdyds.

Notice that the δ function has the following identity:f(x)δ(x−y)dx=f

Image Moment and Radon Moment Theorems

Enlightened by the projection moment and the Projection Moment Theorem, the terminology of the image moment of the space domain is introduced into the Radon transformation [9]. Let p and q be any positive integers with k=p+q. It is known that the kth-order image moment of f(x,y) is defined as [10]Mp,q=−∞−∞f(x,y)xpyqdxdy,where f(x,y) has a compact support.

The following table shows the image moment notations of the first three orders.


The orderThe notation of Mp,q
0M0,0
1M1,0,M0,1
2M2,0,M1,1,M0,2

Radon inverse formula with limited range projections

The Radon inverse formula gives a closed-form formula for finding f(x,y) from P(s,θ). This formula needs projections, P(s,θ), for all s and all θ between 0 and π [13]. In many practical applications, it is desired to reconstruct f(x,y) from P(s,θ) with θ in a limited range. It is known that “A compactly supported function f is determined by the function P(s,θ) for any infinite set of θ” [14]. This relation between the image and its limited range projections has been summarized by A.K. Louis and

The image moment method and the simulation verification

The moment properties of the Radon transform described in Section 4 can be used to estimate unknown projections from limited range projections. Two concepts, projection moment and image moment, will play a central role to establish connections between unknown projections and given projections from limited range.

Eq. (34) shows how to find image moments from given projection in a limited range. The following shows calculations of unknown projections from the image moments.

Assume that Pθ(s) is

The image moment method and the limited range projection problem

The convolution-backprojection method is the most widely used algorithm for CT image reconstruction from projections. However, this algorithm needs the projection, P(s,θ), for all s and all θ between 0 and π [13]. In order to use the convolution-backprojection method for the problem of image reconstruction from limited range projections, we use the image moment method to estimate the unknown projections based upon the known projection from the limited range so that the

Conclusion

Three Radon moment theorems introduced in this paper set up two bridges through the projection moments and the image moments: one between Radon transform, P(s,θ), and the image, f(x,y); another between the projections from different views. The use of the moment concepts achieves a better comprehension of the Radon transform and gives a great deal of flexibility when dealing with the problem of inverse Radon transform from limited range projections.

The concept of the image moment and the image

Summary

Radon transformation has been receiving increased attention along with the maturity of the computerized tomography (CT). Because of its explicit geometrical meaning, Radon transformation has many interesting properties and varied forms of connections among the space, frequency and Radon domains. This paper investigates the moment property of the Radon transformation.

There are two types of moments in the Radon transform: projection moments in Radon domain, and image moments in space domain.

Acknowledgements

The authors would like to thank the following people for their invaluable discussions and suggestions in this work: Dr. G. Herman, Dr. R. Lewitt, and Dr. W. Cheung from the Medical Imaging Processing Group at the University of Pennsylvania. Dr. C.C. Li from the Department of Electrical Engineering of University of Pittsburgh. Mr. C. Hoffman from Bethlehem Homer Research Laboratory.

About the Author—TIAN J. WANG completed his M.S. program in Electrical Engineering from Huazhong University of Science and Technology, China in 1980. Received his M.S. and Ph.D. degrees in Electrical Engineering from the University of Pittsburgh, PA in 1981 and 1986, respectively. His research area is in the field of the digital signal processing, digital imaging processing, and digital communication.

From 1980 to 1986, he was with the Department of Electrical Engineering, University of

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About the Author—TIAN J. WANG completed his M.S. program in Electrical Engineering from Huazhong University of Science and Technology, China in 1980. Received his M.S. and Ph.D. degrees in Electrical Engineering from the University of Pittsburgh, PA in 1981 and 1986, respectively. His research area is in the field of the digital signal processing, digital imaging processing, and digital communication.

From 1980 to 1986, he was with the Department of Electrical Engineering, University of Pittsburgh, as research assistant and teaching fellow. From 1986 to 1990, he worked at Analogic, General Datacomm and Philips Laboratories in the areas of spectrum analysis, modem equalization, digital video signal processing. From 1991 to 1998, he was a senior member of corporate research of Thomson Consumer Electronics, Indianapolis, IN, working in the areas of digital communication system and ASIC development for digital Television. Since 1998, he is the chief engineer and manager of Thomson's communication research lab at Guangzhou, China. He is a guest professor of Huazhong University of Science and Technology and a guest professor of Zhejiang University.

About the Author—T.W. SZE received his B.S.E.E from University of Missouri, 1948; M.S.E.E from Purdue University, 1950; Ph.D. from Northwestern University, 1954. His field of specialization is in Digital Image Processing, Pattern Recognition, and Computer Vision.

He holds the positions of Professor Emeritus, University of Pittsburgh; Adjunct Professor, Shanghai Jiaotong University, China; Adjunct Professor, Xian Jiaotong University, China; Adjunct Professor, Northern Jiaotong University, China. He is a senior member of IEEE; Senior fellow of NATO; Member of the Association of Computing Machinery. Members of the honor societies: Sigma Xi, Eta Kappa Nu, Tau Beta Pi, Phi Tau Phi.

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