DBT is a type of X-ray tomographic imaging technique. By taking use the penetrating power of X-ray, it allows the user to see inside the scanned object. In the DBT procedure, cone-shaped x-ray beams are generated by the x-ray source to penetrate the scanned object from different angles. The attenuated x-ray beams are measured by a detector to create 2-D projection images or projection views (PV) of the scanned object at different angles. By processing the PVs, the distribution of the x-ray attenuation coefficient can be estimated in the 3-D volume. This 3-D distribution is the reconstructed image that reveals the structure inside the scanned object.
The procedure of estimating the 3-D distribution of the x-ray attenuation coefficient from the PVs is called tomographic reconstruction. Mathematically, tomographic reconstruction is a type of inverse problem. Let 𝑓(𝑥, 𝑦, 𝑧) denote the 3-D distribution of the x-ray attenuation coefficient. If we neglect the fluctuation of x-ray radiation and the scattered radiation, the attenuation of the x-ray beam inside the breast follows the Lambert-Beer law:
𝑌(𝑡, 𝑠; 𝑖) = 𝑌0(𝑡, 𝑠; 𝑖) exp (− ∫ 𝑓(𝑥, 𝑦, 𝑧)d𝑙
𝐿(𝑡,𝑠;𝑖)
), (2.1)
where (t, s) denotes a location on the detector, (𝑥, 𝑦, 𝑧) denotes a location in the 3-D volume, 𝑖 denotes the index of the projection angle, 𝐿(𝑡, 𝑠; 𝑖) denotes the line from the x-ray source to the location (t, s) for the ith projection angle, 𝑌(𝑡, 𝑠; 𝑖) denotes the projection value at (t, s) and
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𝑌0(𝑡, 𝑠; 𝑖) denotes the projection value if there is no object is in the 3-D volume. 𝑌0(𝑡, 𝑠; 𝑖) can be
measured with an air scan and is usually considered known. In this dissertation we focus on the case where this (mono-energetic) Lambert-Beer law (Equation 2.1) holds for the measured or simulated data from a poly-energetic x-ray source with beam hardening [46].
By taking log transform to both sides of Equation 2.1, we have: 𝑦(𝑡, 𝑠; 𝑖) = ∫ 𝑓(𝑥, 𝑦, 𝑧)𝑑𝑙 𝐿(𝑡,𝑠;𝑖) , (2.2) where 𝑦(𝑡, 𝑠; 𝑖) = log (𝑌0(𝑡, 𝑠; 𝑖) 𝑌(𝑡, 𝑠; 𝑖)). (2.3)
The right-hand side of Equation 2.2 is known as the Radon transform of 𝑓(𝑥, 𝑦, 𝑧), which is a linear transform [47]. Since both 𝑌(𝑡, 𝑠; 𝑖) and 𝑌0(𝑡, 𝑠; 𝑖) are known, 𝑦(𝑡, 𝑠; 𝑖) can be easily calculated with Equation 2.3. Therefore the problem of tomographic reconstruction is to estimate the unknown distribution of 𝑓(𝑥, 𝑦, 𝑧) from the known values of 𝑦(𝑡, 𝑠; 𝑖) at different locations measured from different projection angles.
Equation 2.1-2.3 explains the concept of image reconstruction for a continuous 3-D image and continuous projections. In the practical implementation of image reconstruction, both the PVs and the reconstructed image are digitalized. We denote the discrete array of x-ray attenuation coefficients as f. Although f is a 3D distribution, we write it as a column vector such that a linear operation on f can be written as multiplying it by a matrix. Denoting the number of voxels in the imaged volume to be N, f is a length-N column vector. Assuming the number of pixels of one PV to be M, which is the same for all projection angles, we can use a length-M column vector 𝐘𝑖 to denote the ith measured projection image. Let 𝑁p denote the total number of
projection angles; we write all the projection images as one column vector, denoted as 𝐘: 𝐘 = (
𝐘1 ⋮
𝐘𝑁p). (2.4)
The log-transformed PVs are written as: 𝐲 = log (𝐼0
𝐘) = ( 𝐲1
⋮
𝐲𝑁p), (2.5)
where the length-M column vector 𝐲𝑖 denotes the log-transformed PV at the ith projection angle. For simplicity we assume that 𝑌0(𝑡, 𝑠; 𝑖) = 𝐼0, that 𝐼0 is a constant. In fact due to the heel effect
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of the anode and the non-uniform incident angle of x-rays, 𝑌0(𝑡, 𝑠; 𝑖) is usually non-uniform. To consider the non-uniformity of 𝑌0(𝑡, 𝑠; 𝑖), one can simply replace the scalar 𝐼0 with the measured air scans or an estimated distribution of the air scans.
With the digitalized 𝐲𝑖 and f, the linear Radon transform in Equation 2.2 can be written in the
form of matrix multiplication:
𝐲𝑖 = 𝐀𝑖𝐟. (2.6)
𝐀𝑖 is called the system matrix at the ith projection angle. The size of 𝐀𝑖 is M×N. Since 𝐲𝑖 is the
2-D projection generated by the 3-D image 𝐟, the operation 𝐀𝑖𝐟 is called the forward projection.
𝐀𝑖 is also known as the forward projector in tomographic reconstruction. We call 𝐀𝑖𝐟 ‘forward projection’ instead of just ‘projection’ to differentiate it from the backward projection. The backward projection applied to a PV 𝐲𝑖 can be written as:
𝐟𝑖 = 𝐀𝑖′𝐲𝑖, (2.7)
where 𝐀𝑖′ is the transpose of the matrix 𝐀𝑖. 𝐟𝑖 is the result of the backward projection, which is a
length-N vector (same size as 𝐟). 𝐀𝑖′ is called the backward projector. Obviously Equation 2.7 is not the inverse operation of Equation 2.6. In fact, 𝐀𝑖 is usually not a square matrix and its inverse
does not exist. The backward projection is commonly used in tomographic reconstruction to update the values of 𝐟 from the measured data. The details will be discussed in the next section.
The value of the matrix element 𝐀𝑖(𝑚, 𝑛) represents the contribution of the nth voxel of the image 𝐟 to the mth pixel of the PV 𝐲𝑖. In many studies on DBT reconstruction, 𝐀𝑖(𝑚, 𝑛) is
assumed to be the intersection length of the x-ray from the source to the center of the mth detector element with the nth voxel of the 3-D image. In Chapter III, we will show the derivation of 𝐀𝑖(𝑚, 𝑛). The derivation indicates that assuming 𝐀𝑖(𝑚, 𝑛) to be the intersection length is
equivalent to using the simplified detection model that each detector element has infinite sensitivity at the center and zero sensitivity at all other locations. We will also introduce our new calculation of 𝐀𝑖(𝑚, 𝑛) that improves the accuracy and speed of both the forward and backward projections in iterative DBT reconstruction.
According to Equation 2.6, the image reconstruction problem in DBT is to find a solution to the following set of linear equations:
𝐲 = 𝐀𝐟, (2.8)
10 𝐀 = ( 𝐀1 ⋮ 𝐀𝑁p ). (2.9)
Due to the huge total number of voxels in DBT reconstruction (𝑁~108), Equation 2.8 is a
large-scale inverse problem. Multiple methods to solve Equation 2.8 have been proposed and studied, as introduced in the next section.