Reconstruction from projections based on detection and estimation of objects

This paper considers the problem of observing a 2D led to suggestions to apply reconstruction techniques to function via its ID projections (Radon transform); it a variety of novel and technologically demanding tasks, presents a framework for detecting, locating and e.g. real-time monitoring of high production rate describing objects contained within a 2D cross-section manufacturing processes, quality control nondestructive by using noisy measurements of the Radon transform testing, "stop action" internal imaging of very rapidly directly, rather than post-processing a reconstructed changing media, and mesoscale imaging of oceanoimage. This framework offers the potential for graphic regions hundreds of kilometers on a side [1,2]. significant improvements in applications where In such applications of reconstruction techniques, high (1) attempts to perform an initial inversion with resolution cross-sectional imaging may be neither possiinsufficient measurement data result in severely ble nor the ultimate goal: degraded reconstructions, and (2) the ultimate goal of the process is to obtain several specific pieces of infor1. For a variety of reasons (e.g. time, economic, mation about the cross-section. To illustrate this perenvironmental or physical constraints that limit spective, we focus our attention on the problem of either the total measurement viewing angle or obtaining maximum-likelihood (ML) estimates of the time, or limit the number or sensitivity of measparameters characterizing a single random object urement transducers) it may be impossible to situated within a deterministic background medium, obtain a full set of low noise measurements over a and we investigate the performance, robustness, and wide viewing angle. computational structure of the ML estimation procomputational structure of the ML estimation pro-re. 2. In many applications, abundent a priori information about the cross-section is available, and the ultimate goal is not necessarily to obtain an image, but Introduction rather to extract specific information about the cross-section. In oceanographic and nondestructive The problem of reconstructing a multi-dimensional testing applications, for example, projection measfunction from its projections arises, typically in imaging urements are processed in order to determine the applications, in a diversity of disciplines, including location of objects such as oceanographic cold-core oceanography, medicine, and nondestructive testing rings [1], or to detect and locate cracks or flaws [1,2]. In the two-dimensional version of this problem, within a homogeneous material [2]. a 2D function f(x) is observed via noisy samples of its Radon transform When projection measurements are incomplete or of g(t, 0) = f(x)ds (1) low quality due to the factors in (1), reconstruction x'0 leads to images that have artifacts, poor resolution and/or high noise levels; attempted interpretation of where 0 is the unit vector (cos0 sin0)'. The major such imagery may result in unreliable or inconsistent such imagery may result in unreliable or inconsistent emphasis of research and applications in this area has evaluation. been on developing exact and approximate solutions to this integral equation in order to produce highIn this paper, we consider processing incomplete resolution cross-sectional imagery (this approach to the noisy projection measurements when the ultimate goal inverse problem requires a large number of high is to obtain very specific information about a crosssignal-to-noise ratio measurements taken over a wide section. In particular, an approach to such problems is viewing angle [31). Perhaps the most popular example presented, along with the associated analysis, and is of reconstruction from projections is medical computerillustrated via a simple problem of locating an object ized axial tomography (CAT). Initial success with contained within a known cross-sectional background. reconstruction from projections in CAT scanning, as well as in radio astronomy and electron microscopy, has Cross-section and Measurement Model ML Object Localization Consider a 2D cross-section In this section, we consider the special case of full-view f(x) = fb(x) + d fo(x-c;y) (2) measurements (i.e., Sy = S), with w(t,o) a 2D zeromean Gaussian noise process with covariance where fb(x) is a known background and d fo(x-c;y) is a ] = randomly located object having known density or conThe maximum-2 trast d [where fo(x,y)=l] and unknown location cER 2 ; likelihood (ML) location estimate CML is that value of y is a known vector of parameters characterizing, for the parameter c that maximizes the log likelihood funcexample, the size, shape and/or orientation of the tion [6] object. By the linearity of (1), the Radon transform of 7 f(x) is the sum of two components, I (c;Y)= f y(t,o )s(t,O;c)dtdO g(t,O) = f fb(x)ds + df fo(x-c;y)ds 1 c)t (5) x'O=t x'=t . (t;c)dtd gb(t,0) + d'g,(t-c'_I,O;y) (3) The first term in (5) corresponds to a matched filtering where go(t,o;y) is the Radon transform of fo(x,y), the operation in Radon space (this operation maps the unit-contrast object located at the origin. Note that Radon-space measurements into a function on R); the because the location of the object cE R is random, the second term in (5) involves the energy in the Radon second term in the Radon transform (the component space matched filtering template. Since s(t,o;c) due to the object) is characterized by a random depends on c only via a shift in the t variable, the sinusoidalshift c'O in the t variable. second term in (5) is c-independent and can be Let the noisy projection measurements be given dopped, as can the N scaling factor, to yield by' N L(c;Y) = j f y(t,O)s(t-c'O,O;O)dtdO (6) y(t,0) = [dgo(t-cI0,0;y)*h(t) + w(t,o) 0The log likelihood function for this problem is seen to = s(t,0;c) + w(t,0) (4) be obtained by a convolution back-projection (CBP) operation (such as that used in full image reconstruction [3]), where the generally 0-dependent and non(t,) E Sy C S= (t,0 ) : -oo> R. For this full-view circularly-symmetric case, the 1. By the stated assumptions, the Radon transform of the background, gb(t,O), is known and its effect has been subtracted from the measurements. 2. The problem where y(t,O) is a counting process with rate that is a function of g(t,o) may also be considered; such a model is appropriate for very low-dose x-ray problems. 3. Incomplete measurement cases, in which views are available over only a limited view angle or at afinite number of views, are treated similarly. error covariance matrix is cr2 times a 2x2 identityReferencesmatrix. Figure 1 is an illustration of ((r/T)2, theinverse of the normalized error variance, versus the 1. B. Cornuelle, "Acoustic Tomography," IEEE Trans.normalized object size R/T, for several values of theGeoscience and Remote Sensing. Vol. GE-20, pp.ratio of contrast squared to noise level. This Figure326-332, 1982.indicates a definite threshold behavior-for givenvalues of the object contrast d and measurement noise 2. V. Kyuev, E. Vaberg, I. Kazek, V. Kurozae,level No, there exists a smallest object size for reliable"Computational Tomography A New Radiational~~~~~~~~localization~Methodof Nondestructive Testing," Soviet J. Non-destructive Testing. Vol. 16, pp. 180-193, 1980.Summary3. R. Mersereau, A. Oppenheim, "Digital Reconstruc-A framework has been presented for detecting, locatingtion of Multidimensional Signals from Their Pro-and characterizing objects in a cross-section by usingjections," Proc. IEEE. Vol. 62, pp. 1319-1338, 1974.noisy projection measurements directly, rather thannpost-processing a reconstructed image. Within this4. D. Rossi, Reconstruction from Projections Based onpost-processing a reconstructed image. Within thisframework, the performance, robustness and computa-Detection and Estimation of Objects, Ph.D. Thesis,tional structure of ML estimation procedures have beenDepartment of Electrical Engineering and Com-investigated for both object location and geometryparameters [4,51.5. D. Rossi, A. Willsky, "Reconstruction From Pro-jections Based on Detection and Estimation ofObjects," to be submitted to IEEE Trans. Acoust.,Speech, Signal Processing. llJ:. 6. H. Van Trees, Detection, Estimation, and ModulationOTheory, Part I, John Wiley and Sons, New York,