The Fourier transform in image processing is an important tool that allows
observation of an image in the frequency-domain and due to which several difficult problems become very simple to analyze. The transform is used in a wide range of applications, such as image analysis, image filtering, image compression,
etc. On the other hand, with the recent trend for data-driven approaches, deep-learning
has arisen as a promising learning method that has had success in several
computer vision areas, such as image classification, object detection, pedestrian
detection, etc. Deep learning algorithms train a deep neural network on a large
set of images to learn the parameters instead of using hand-tuned filters.

In this thesis, we propose to design a framework, inspired by both the Fourie...[

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The Fourier transform in image processing is an important tool that allows
observation of an image in the frequency-domain and due to which several difficult problems become very simple to analyze. The transform is used in a wide range of applications, such as image analysis, image filtering, image compression,
etc. On the other hand, with the recent trend for data-driven approaches, deep-learning
has arisen as a promising learning method that has had success in several
computer vision areas, such as image classification, object detection, pedestrian
detection, etc. Deep learning algorithms train a deep neural network on a large
set of images to learn the parameters instead of using hand-tuned filters.

In this thesis, we propose to design a framework, inspired by both the Fourier
transform and deep-learning methodology, to address two image reconstruction
problems: multispectral demosaicking and low-dose computed tomography (LDCT)
artifact reduction. In multispectral demosaicking, we need to reconstruct the
original image from an overly downsampled image, whereas in LDCT artifact
reduction, we need to reconstruct an artifact-free image from an artifact-induced
image. Basically, in both these problems, we must restore the original component
from a given noisy observation and thus require a proper investigation to understand
the fundamental aspects of both problems. In view of this observation,
in this thesis, we first analyze the both problems in the frequency-domain and
then propose a solution that addresses both problems. Specifically, our proposed
framework for each problem can be divided into two phases:

1. Frequency-domain analysis phase: With the help of frequency-domain analysis, we are able to identify the potential issues in the assumptions of each
problem and then give a systematic analysis of why existing methods cannot
produce a result with a sufficient level of quality.

2. Reconstruction phase: Based on the frequency-domain analysis, we first
propose an algorithm based on an image-driven approach, which extracts
information from the target image itself to address the problems. Later, a
data-driven approach is incorporated into the proposed algorithm to further
enhance its performance. In the data-driven approach, we first train a deep
neural network on a large set of images and then address the problems with
the learned parameters.

We first consider the problem of multispectral demosaicking in multispectral
imaging, which is an extension of 3-band (or color) demosaicking. Color demosaicking
is a process of interpolating the missing color samples in each band to
reconstruct a full-color image, but in multispectral demosaicking each band is
significantly undersampled due to the increment in the number of bands. Specifically, in the proposed framework, we demonstrate a frequency-domain analysis of
the subsampled color-difference signal and observe that the conventional assumption
of highly correlated spectral bands for estimating undersampled components
is not precise. Instead, such a spectral correlation assumption is image dependent
and rests on the aliasing interferences among the various color-difference spectra.
To address this problem, we propose an image-driven approach, called the
adaptive spectral-correlation-based demosaicking (ASCD) algorithm, that uses
a novel anti-aliasing filter to suppress these interferences and then integrates it
with an intra-prediction scheme to generate a more accurate prediction for the
reconstructed image. The weights for the integration are obtained in the linear
minimum mean square error (LMMSE) sense to produce an accurate reconstruction.
Further, to enhance the accuracy of the reconstruction, we obtain the
weights by a data-driven approach, namely the deep-learning approach. In this
approach, we first train a deep neural network on a large set of images and then
learn the weights accordingly.

In the second problem, we address the removal of artifacts in LDCT images.
Computed tomography (CT) examination with a reduced radiation dose, known
as low-dose CT (LDCT), has gained increasing attention from the medical community
because it decreases the radiation exposure of patients. However, images
obtained from LDCT reconstruction consist of severe artifacts that degrade the
quality of the images for diagnostic tasks. Recently, a frequency-split technique,
has been proposed to address a different issue of CT reconstruction by extracting
the high-frequency components from both uncorrected and corrected images (obtained
by an existing method). Given this, we propose an optimized frequency-split
algorithm that first investigates the high-frequency and low-frequency component
of both the uncorrected and corrected image and then integrates them
optimally to generate a reduced artifact image. Our algorithm is optimal in the
linear minimum mean square error (LMMSE) sense, and the optimal weights
store the information on whether a pixel or block of pixels in the uncorrected
image belongs to a desired or undesired edge and thus optimally reduces the artifacts.
To further improve the accuracy of the proposed algorithm, we propose
to include a clean prior image in the optimization framework, thus reducing the
artifacts significantly. Experimental results demonstrate that the proposed algorithm,
based on an optimized frequency-split-based technique, drastically reduces
artifacts. Similar to our approach for multispectral demosaicking, we later propose
to use a data-driven approach to learn the weights by training a deep neural
network on a large set of images.

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