Maximum likelihood estimation of object size and orientation from projection data

zero), and a single object (N=1) is situated at a known objThe problem of detecting, locating and charaprojectioerizing location; see [1-3] for a discussion of ML localization of objects in a 2D cross-section from noisy projection data an object from noisy projection measurements. has been considered recently [1-3], in which objects are characterized by a finite number of parameters, which In the present analysis, a specific parameterization of are estimated directly from noisy projection object size and shape is chosen, and the performance of measurements. In this paper, the problem of maximum ML estimation of the geometry parameters in y is likelihood (ML) estimation of those parameters evaluated. To begin, consider a circularly-symmetric characterizing the geometry of an object (e.g. size and normalized (i.e. unit-sized) object located at the origin, orientation) is considered, and estimation performance denoted fo(x) (f (r) as a function of the radial polar is investigated. coordinate r). Denote the object Radon transform as go (t), which is independent of the projection angle 0, and its Radon transform energy as Introduction or The problem of reconstructing a multi-dimensionalo = g2 (t) dt d function from its projections arises, typically in imaging applications, in a diversity of disciplines, including The object whose projections are measured is modeled oceanography, medicine, and nondestructive evaluation. as arising from the object f, (x) by density scaling with a In the two-dimensional version of this problem, a 2D factor d, and by a series of coordinate transformations, function f(x) is observed via noisy samples of its Radon specifically, isotropic scaling of the x coordinate system transform by a factor R, and/or stretching of the coordinate system g (t, 0) f r f(X)ds by a factor X, and possible rotation by an angle 4). Said 0g (t, 0) j f x ds another way, the object whose projections are measured is modeled as belonging to the class of objects where 0 is the unit vector (cosO sinO)'. The problem d-f(x) = d-fo (x) where of locating and characterizing one or more objects in a cross-section from projection measurements has been cosq sin k 0 O considered recently [1-3], in which the 2D cross-section x -sin cos 0 R x (1) is represented as the superposition of a background field and N objects, f(x)= fb(x) + Y dk f(x--ck ;k) O<R Y 0<X 2 2 kIThroughout this discussion, the subscript "o" refers to Here, the kth object is located at the point ck and has the original unit-sized, circularly-symmetric object, and contrast or density dk (f(0;vk)=l); Yk is a finiteunsubscripted functions refer to the object after the dimensional vector of parameters characterizing the transformation in (1) has been applied. The Radon density fluctuations of the kth object, it contains, for transform of the object d-f(x;R, X, 4) resulting from this example, information about the object's size, shape and transformation will be denoted d-g (t, O;R, X, 4)), and the orientation. In this paper, we consider the special case Radon transform energy [1] is (d,R, X) where the background is known (and assumed to equal d2R3 q (X) o, where * This work was supported by the National Science Foundation under Grant ECS8012668. 1.0 \L(R,';Y)=2 o -f y(t,9)g(t,O;R, X,) dtdo 0 0 -co 0.8 0.6 1 f g2 (t,O;R,Xa ,q)dtdO 0.6 No 0 The ambiguity function, or expected value of the log 0.4 likelihood function, is given by a (R, O;Ra, Xa, a ) 0.2 2 r 0.0 (t, O;Ra , l-= ,f2 j g (t, aa,,)g(tO;R, a, c) dt dO 0 5 10 15 20 -i--£ g (t, O;R, X, 4) dt dO Figure 1. Radon Transform energy dependence on N0 (td eccentricity X. In the following two sections, expressions are presented 2/2 for the ambiguity function and Cramer-Rao lower bound q (A) 2 f [X A cos2 + A-1 sin ,]-V'i dp (CRLB) [4] for the individual problems of size and q o7r f cos~q, + X orientation estimation. For purposes of illustration, the ambiguity function and CRLB will also be evaluated in Note that q (X) q (X1) and q (1) = 1; a plot of q (X) each section for the special case of the class of objects on the interval [1,201 is shown in Figure 1. Let the which arise when the coordinate transformation in (1) is noisy projection measurements be given by applied to the circularly-symmetric Gaussian object y(t, 0) -= d-g (t, O;R, X, ) + w (t, ) (2) f,(r) = exp (-r2 ). This object has Hankel transform (central section of the 2D Fourier transform) F, (p) = (t, 0) E Sy cS = I (t, 0): -oo<t< oo, 0-0<7r 7r exp(-.Tr2p 2), and Radon transform energy o = where w(t, 0) is a zero-mean white Gaussian noise process with spectral level No /2. In terms of the present notation, the problem of characterizing the object Object Size Estimation geometry may be stated as: given noisy incomplete gome asurements of the Radon transform as shownisy incomplete (2), Consider first the problem of using noisy full-view measurements of the Radon transform as shown in (2), estimate the object density d, size R, eccentricity X, and proection measurements to estimate the size of an orientation +. It should be noted that, with the object that results from the isotropic coordinate scaling transformation in (1). In particular, begining with a exception of the density factor d, these parameters enter the problem nonlinearly, and lead to a nonlinear unit-sized circularly-symmetric object fo (r) with Hankel estimation problem of small dimensionality. This is in transform F (p) and Radon transform energy {,0 the contrast to full image reconstruction, in which a linear size estimation ambiguity function is a (R,Ra) = estimation problem of high dimensionality is solved. In (aa/No) a*(R/Ra ), where {a = d 2 R3 ~o is the energy the present analysis, maximum likelihood (ML) in the actual object Radon transform, and a*(.) is the estimation qf object size R and orientation 4 will be normalized ambiguity function considered assuming that the object eccentricity X is a* (R /Ra) known; see [2] for a discussion of estimating object eccentricity. 4± (R/R) fFo (p)Fo (pR /Ra ) dp -R /Ra (3) ML Parameter Estimation 0 Let Ra, Xa, qba, and g (t, O;Ra , Xa , qa ) be the actual For the special case where the original object is the object parameters and Radon transform, respectively, Gaussian object fo (r) = exp (-r 2), the normalized and consider the special case of a full-view set of ambiguity function in (3) is plotted in Figure 2, along measurements Y, that is, Sy = S. ML estimates of with the normalized ambiguity function for the disk object size and orientation correspond to those values of object (indicator function on a disk of radius Ra ). the parameters R and 4 that maximize the log Qualitatively, since these ambiguity functions have their likelihood function [41 peak at R =Ra and decrease rapidly as R moves away