C.20100209440000818398-20135171466AGRICOLA SAN JUAN BAUTISTA, SOCIEDAD

The image reconstruction in Equation 2.8 is a simplified form of the problem. In practical DBT, the projection images are highly noisy due to the limited x-ray dose of each projection. In this section, we will discuss three categories of image reconstruction algorithms that have been studied or are practically used in commercial DBT systems. These three categories of algorithms are: (1) filtered back-projection (FBP); (2) the algebraic reconstruction algorithms; (3) the statistical reconstruction algorithms.

The FBP algorithm is a type of Fourier-domain reconstruction algorithm [45, 48-52]. Simply speaking, the FBP method is a frequency-domain-based transform applied to 𝐲 that can be briefly written as:

𝐟̂ = FBP(𝐲), (2.10)

where the operation FBP(∙) consists of high-pass filters and backward projections.

The FBP reconstruction has been used in commercial DBT systems by Hologic and Siemens [44]. The high-pass filters needs to be carefully designed to preserve image quality without severely amplifying the noise. The details of the filters used in commercial DBT systems cannot be found in literatures. In the following chapters of this dissertation, we mainly focus on iterative reconstruction methods for DBT. As result, we will not discuss the FBP algorithm in detail.

Different from the FBP method that directly applies a transform to the projection images, the algebraic reconstruction algorithms [53-55] try to solve Equation 2.8 with iterative methods. The original algebraic reconstruction algorithm is known as the algebraic reconstruction technique (ART). It updates the distribution of the x-ray attenuation coefficients (f) ray by ray to satisfy one row of the Equation 2.8 at a time. Due to the existence of noise, Equation 2.8 does not have a precise solution and the ART algorithm is known to generate highly noisy reconstructed images [31]. Several modified versions of the ART algorithm have been proposed such as the

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simultaneous iterative reconstruction technique (SIRT) [53, 55] and the simultaneous algebraic reconstruction technique (SART) [56, 57]. The comparative study previously conducted by our lab demonstrated the effectiveness of SART in DBT reconstruction [31]. SART is an iterative reconstruction method that the update of the image 𝐟 can be written as:

𝐟(𝑖SART,𝑖+1)_{= 𝐟}(𝑖SART,𝑖)_{+ 𝜆}

𝑖SART(𝐀𝑖′((𝐲𝑖− 𝐀𝑖𝐟)⊘(𝐀𝑖𝟏𝑁)))⊘(𝐀𝑖′𝟏𝑀), (2.11) where 𝑖 is the index of the projection angle and 𝑖SART is the index of the SART iteration. 𝟏𝑁 and

𝟏_𝑀 denote length-N and length-M all-one vectors. The symbol ⊘ denotes the element-wise division. 𝑖_SART is increased by one after all PVs (𝑖 = 1,2, … , 𝑁_p) have been used once. The forward projector 𝐀𝑖 and its transpose, the backward projector 𝐀𝑖′, were introduced in the

previous section. 𝜆𝑖SART is the relaxation factor that is equal to or smaller than 1 to control the amplification of the noise.

We use SART to reconstruct the simulated digital phantoms in Chapter III and VI. SART is also used as a reference algorithm in Chapter IV to compare with our new reconstruction method. Although not state-of-the-art, SART has been shown to provide good image quality for reconstructing DBT acquired with our prototype DBT system [31] and has been evaluated by other investigators [44, 58]. SART will also be used to test our artifact removal methods in Chapter V.

One disadvantage of algebraic methods is the lack of statistical model of the noise. When using algebraic methods, it is also difficult to apply regularization to control the amplification of noise when using the iterations. For this reason, in the image reconstruction for our experimental DBT system, we usually do only 1 or 2 SART iterations [31]. The statistical reconstruction algorithms, on the other hand, allow the probability model of the noise to be considered and flexible choice of regularizations. For statistical image reconstruction algorithms, we include a noise vector 𝛜_𝑖 in Equation 2.6 for the ith projection angle. Therefore we have the following

expression for the measurement model:

𝐲_𝑖 = 𝐀_𝑖𝐟 + 𝛜_𝑖. (2.12)

Here, the noise 𝛜𝑖 has the same size as 𝐲𝑖. It is a realization of a random vector with a known

probability distribution function (PDF) that represents the statistical model of the noise. Therefore 𝐲_𝑖 is also a realization of a random vector. The PDF of this random vector can be

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represents our prior knowledge or expectation for the image to be reconstructed. The statistical image reconstruction tries to find 𝐟̂ such that:

𝐟̂ = argmax

𝐟 𝑃(𝐟|𝐲) = argmax𝐟 𝑃𝐲|𝐟(𝐲|𝐟)𝑃𝐟(𝐟), (2.13)

where 𝑃_𝐲|𝐟(𝐲|𝐟) is the PDF of 𝐲 given 𝐟. The function 𝑃𝐟(𝐟) serves as a regularization term when

we formulate the optimization problem.

A common choice of the probability model for 𝛜_𝑖 is the multivariate Gaussian distribution with zero mean. Let 𝐊𝑖 denote the covariance matrix of this distribution. Assume that 𝑃𝐟(𝐟) can

be written as the exponent of a function: 𝑃𝐟(𝐟) = exp(−𝑅(𝐟)), then Equation 2.13 simplifies to

be a regularized quadratic optimization problem: 𝐟̂ = argmin 𝐟 ∑ 1 2‖𝐲𝑖 − 𝐀𝑖𝐟‖𝐊2_𝑖−1 𝑁𝐩 𝑖=1 + 𝑅(𝐟), (2.14)

This is the commonly used form of the optimization problem in statistical image reconstruction. Similar to the algebraic image reconstruction, Equation 2.14 must be solved with iterative methods.

Compared with Equation 2.7, the expression of Equation 2.14 allows us to include a covariance matrix for the noise, which can be estimated from the measured data given the assumptions on the correlation of the noise. Equation 2.14 also allows us to choose 𝑅(𝐟) based on our expectation of the image 𝐟. The linear operation 𝐀𝑖 can include the modeling of the

physics of the imaging system, such as the blurring of the detector. In summary, Equation 2.14 allows us to incorporate different model components into the image reconstruction. When these model components are incorporated based on the realistic system physics, the resulting reconstruction algorithm is called a model-based image reconstruction (MBIR) algorithm. Modeling system physics in image reconstruction has been investigated in CT and other 3D modalities and improvement in image quality has been observed [37-46, 59-62]. The idea of MBIR is a major inspiration of many studies introduced in this dissertation. The connection between the studies and the idea of MBIR will be discussed in details in each of the following four chapters.

The regularization term 𝑅(𝐟) in Equation 2.14 is especially important for DBT reconstruction. The system matrix 𝐀 usually has more columns than rows in DBT, making Equation 2.14 an underdetermined problem without regularization. Due to the narrow scan angle of DBT, the unregularized image reconstruction is an ill-posed inverse problem that small measurement

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fluctuation such as noise could cause large perturbations of the reconstructed. Several studies have been conducted on regularization of limited-angle reconstruction. For example, the total variation (TV) method was applied to DBT reconstruction [32, 63]. We have also proposed a spatially weighted non-convex (SWNC) regularization for enhancing microcalcifications (MC) in DBT reconstruction; this work was published as an SPIE proceedings paper [64]. Considering the limited practical value of this method, it is not included in this dissertation. In Chapter III and VI, we use noiseless simulated projections so unregularized SART provides satisfactory reconstruction results. In Chapter IV where we use the experimental data, the edge-preserving regularization has been demonstrated to play an important role in our new reconstruction method.

Before ending this section, we introduce a more general form for statistical image reconstruction. In Equation 2.12, the noise is considered to be additive to the log-transformed projection 𝐲_𝑖. In fact, a more general form of Equation 2.12 considers the noise to be additive to the originally measured projection images 𝐘 before the log transform. This will give us the following expression for statistical image reconstruction:

𝐟̂ = argmax

𝐟 𝑃(𝐟|𝐘) = argmax𝐟 𝑃𝐘|𝐟(𝐘|𝐟)𝑃𝐟(𝐟), (2.15)

where 𝑃_𝐘|𝐟(𝐘|𝐟) is the PDF of Y given 𝐟. 𝑃_𝐲|𝐟(𝐲|𝐟) in Equation 2.14 sometimes can be derived from 𝑃𝐘|𝐟(𝐘|𝐟) with approximations, as shown in Chapter IV. Chapter IV starts from Equation

2.15 to derive our cost function. With approximations, the optimization problem formulated is similar to Equation 2.14.