Polyenergetic Model Development. Assume that the 3D object is discretized into a regular grid of voxels and that each of the 2D projection images is discretized into a regular grid of pixels. Specifically, let N represent the number of voxels in the discretized 3D object and let M be the number of pixels in a discretized 2D projection image. In practice N is on the order of a few billion and M is the order of a few million, depending on the size of the imaging detector. The energy- dependent linear attenuation coefficient for voxel j = 1, 2, ..., N in the breast can be represented as true µ(e)(j) = s(e)x(j) + z(e), true where x(j) represents the percentage glandular fraction in voxel j of the “true” object, and s(e) and z(e) are known energy-dependent linear fit coef- ficients. This type of decomposition to reduce the number of degrees of free- dom is similar to an approach used by De Man et. al. [29] for CT, in which they express the energy dependent linear attenuation coefficient in terms of its photoelectric component and Xxxxxxx scatter component. However, their model is not optimal for our particular application. In tomosynthesis, a limited number of projections are taken from various angles in a predetermined angular range, and the photon energies are dis- cretized into a fixed number of levels. Let there be nθ angular projections and assume the incident x-ray has been discretized into ne photon energy levels. In practice, a typical scan may have nθ = 21 and ne = 43. We would like to formulate a mathematical representation for the θth projection image. For a particular projection angle, we first compute a monochromatic ray trace for one energy level and then sum over all energies. Let a(ij) represent the length of the ray that passes through voxel j, contributing to pixel i. Then the discrete monochromatic ray trace for pixel i can be represented by N N N true Σ µ(e)(j)a(ij) = s(e) Σ x(j) a(ij) + z(e) Σ a(ij) . (4.1) j=1 j=1 j=1 Using the standard mathematical model for transmission radiography, the ith pixel value for the θth noise-free projection image, incorporating all photon energies present in the incident x-ray spectrum, can be written as θ Σ N b(i) = ne e=1 q(e) exp .− Σj=1 µ(e)(j)a(ij)Σ , (4.2) where q(e) is a product of the current energy with the number of incident photons at that energy. To simplify notation, let’s define Aθ to be an M × N matrix with entries a(ij). Then equation (4.1) gives the ith entry of vector s(e)Aθx true + z(e)Aθ1 , true where x true is a vector w...
