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DLCT-Toolbox, a Matlab package for the

Dictionary Learning Approach to Tomographic

Image Reconstruction

Sara Soltani

∗

Department of Applied Mathematics and Computer Science,

Technical University of Denmark, DK-2800 Kgs. Lyngby,

Denmark.

August 25, 2015

Contents

1 Introduction 2

1.1 Overall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Toolbox and Software Dependencies . . . . . . . . . . . . . . . . 3

1.3 Test Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.4 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Package Details 4

2.1 Dictionary Learning Problem . . . . . . . . . . . . . . . . . . . . 4

2.1.1 DL_demo.m . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 DicLear_M.m . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.3 Lea_Algorithm_D2.m . . . . . . . . . . . . . . . . . . . . 6

2.1.4 Lea_Algorithm_Dinf.m . . . . . . . . . . . . . . . . . . . 6

2.2 Mean Approximation Error . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 MAEM_demo.m . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.2 MAE_M.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 The Reconstruction Problem . . . . . . . . . . . . . . . . . . . . 8

2.3.1 RecM_demo.m . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3.2 RecM_Algorithm.m . . . . . . . . . . . . . . . . . . . . . . 9

2.4 The Tensor Dictionary Learning problem . . . . . . . . . . . . . 11

2.4.1 DL_Tensor_demo.m . . . . . . . . . . . . . . . . . . . . . . 11

2.4.2 DicLear_T.m . . . . . . . . . . . . . . . . . . . . . . . . . 11

∗

ssol@dtu.dk, sarahsoltani@gmail.com

This work is part of the project HD-Tomo funded by Advanced Grant No. 291405 from the

European Research Council.

2010 Mathematics Subject Classiﬁcation: 65F22, 65K10, 15A69.

Key words and Phrases: Tomography, Dictionary learning, Tensor decomposition, Inverse

problem, Regularization, Sparse representation, Image reconstruction.

S. Soltani A Matlab Package for DL Approach to CT

2.4.3 Lea_Algorithm_T_D2.m . . . . . . . . . . . . . . . . . . . 13

2.4.4 Lea_Algorithm_T_Dinf.m . . . . . . . . . . . . . . . . . . 13

2.5 The Approximation Error by the Tensor Dictionary . . . . . . . 14

2.5.1 MAET_demo.m . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5.2 MAE_T.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Tomographic Reconstruction with Tensor Dictionary . . . . . . . 16

2.6.1 RecT_demo.m . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.6.2 RecM_Algorithm.m . . . . . . . . . . . . . . . . . . . . . . 16

3 Alphabetic List of Package Functions 19

1 Introduction

1.1 Overall

In our recent works [5, 6] we have developed a two-stage algorithm using a dictio-

nary learning approach in a discrete tomographic reconstruction problem both

with a matrix and a tensor formulation. We ﬁrst build a training data base from

given training images and then construct a dictionary purely from the training

data by looking for a number of basis elements in terms of the dictionary and the

coeﬃcients/representations of the training data in the dictionary. In these two

works ([5, 6]) dictionary learning is formulated as a non-negative matrix/tensor

formulation with sparsity constraints on the representation, and then the recon-

struction problem is formulated as a sparse approximation problem in a convex

optimization framework. We optimized the dictionary learning problem locally

by using the Alternating Direction Method of Multipliers (ADMM), see e.g., [2].

The tomographic reconstruction problems are solved using the software package

TFOCS (Templates for First-Order Conic Solvers) [1].

This documentation describes the implementations details of the proposed

algorithms for solving the tomographic image reconstruction problem in Mat-

lab. This package is designed for a discretized computed tomography problem

formulation, i.e., we use a matrix formulation of the reconstruction problem

(Ax ≈ b), where b ∈ R

contains the noisy measurement data and A is the

system matrix. The vector x ∈ R

represents an M × N image, with n = MN,

of absorption coeﬃcients. The total number of tomographic measurement is

m. The system matrix A ∈ R

m×n

is available. We use the AIR Tools [4] to

compute the system matrix A. The discretized tomographic problem is a large

sparse and often ill-posed system and incorporating a priori information about

the solution is a necessity to improve the reconstruction and regularize the solu-

tion. This package aims at providing a computational framework for the use of

training images as priors for the solution in tomographic image reconstruction

in the scheme described above. The algorithms work for any size of the sys-

tem matrix A; however, we are more concerned with underdetermined problems

where m < n, because the need for regularization is even more pronounced in

that scenarios.

This package of Matlab routines provides the user with easy-to-use routines

and demo scripts, based on formulation and numerical algorithms described in

[5, 6]. With the demo script ﬁles, the user can easily speciﬁes a few values of

problem parameters to obtain a solution to the dictionary learning problem,

S. Soltani A Matlab Package for DL Approach to CT

the tomographic reconstruction problem and compute the best approximation

error of a given test image by the obtained dictionary. For an advanced use

of the routines, we invite the reader to carefully read [5, 6], as well as this

documentation and the descriptions in the .m ﬁles provided in this package.

1.2 Toolbox and Software Dependencies

The following toolboxes and software routines are required for this package to

be run smoothly on a computer system.

• Matlab : The package has been implemented in Matlab version 2014a.

• AIR Tools [4] is a Matlab software package for tomographic reconstruc-

tion consisting of tomographic test problems and a number of algebraic

iterative reconstruction methods. The user can download AIR Tools in a

zip format ﬁle from http://www2.compute.dtu.dk/~pcha/AIRtools/.

• TFOCS [1] is a suite of programs and routines designed to help construct-

ing of ﬁrst-order methods for a variety of convex optimization problems

arise in compressed sensing, sparse recovery, and low rank matrix com-

pletion e.g. (BPDN and Lasso). The user can download TFOCS in a zip

format ﬁle from http://cvxr.com/tfocs/download/.

The user may download and unpack the software routines in a separate folder

while the paths to those folders can be added to the Matlab’s current path by

including addpath statements.

1.3 Test Images

A collection of gray scale test images has been provided in the package, lo-

cated in the folder “/TestImages”. There are 6 built-in test problems in this

version namely: “Peppers, Matches, Binary, D53, Zirconium and Steel”.

The high-resolution photos of Peppers and Matches are provided by profes-

sor Samuli Siltanen from University of Helsinki. The Zicronium test image is

courtesy of Dr. Hamidreza Abdolvand from University of Oxford. The steel

microstructure image is from [8] and the D53 test image is chosen from the

normalized brodatz texture database [7]. The binary test image is generated by

means of phantomgallery.m function from AIR Tools. The user should update

Matlab’s path with links to the subdirectories /TestImage and /TestResults.

1.4 Notation

The linear tomographic reconstruction problem is formulated as Ax ≈ b where

the vector x ∈ R

represents the unknown image of size M × N , the vector

b ∈ R

is the given noisy data, and the matrix A ∈ R

m×n

represents the

forward model. We use the following notation, where B is an arbitrary matrix:

kBk





1/2

, kBk

sum

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