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\chapter{INTRODUCTION} % The first chapter
\label{cha:intro} 

\vspace{-12pt}
\section{First Section}
The significance of being able to classify strongly regular graphs under certain parameters has not be lost on researchers the last 50 years. For instance, under what parameters do we have a strongly regular Cayley graph? The generating set for such a graph is called a partial difference set. It is known that there exist partial difference sets for groups of exponent 4 or less. In \cite{DP}, Jim Davis and John Polhill describe how they are able to construct difference sets in groups  $C_2^r$ and $C_4^r$ for all $r\geq 2$, which can then be used to construct partial difference sets. They also prove a new product construction for such sets. Through this method, they are able to show that there do not exist partial difference sets in groups of exponent 16. However, a question raised by Davis and Polhill is whether or not partial difference sets exist in groups of exponent 8. In particular, do they exist in the group $C_8\times C_8$?

In this paper, the study of strongly regular graphs and that of Cayley graphs is combined to find all partial difference sets given specific sets of parameters. In particular, Cayley graphs on finite abelian groups $G$ are sought. This allows for the use of character theory of the groups and rational idempotents in the group ring $\mathbb{C}[G]$ to simplify computations. Given parameter sets, partial difference sets are found through a collapsing process on our unknown generating set $S$ of the Cayley graph, working with smaller cases to build sets $S$ that yield strongly regular Cayley graphs which are thus partial difference sets. Through this method, we answer the existence question posed by Davis and Polhill.
The significance of being able to classify strongly regular graphs under certain parameters has not be lost on researchers the last 50 years. For instance, under what parameters do we have a strongly regular Cayley graph? The generating set for such a graph is called a partial difference set. It is known that there exist partial difference sets for groups of exponent 4 or less. In \cite{DP}, Jim Davis and John Polhill describe how they are able to construct difference sets in groups  $C_2^r$ and $C_4^r$ for all $r\geq 2$, which can then be used to construct partial difference sets. They also prove a new product construction for such sets. Through this method, they are able to show that there do not exist partial difference sets in groups of exponent 16. However, a question raised by Davis and Polhill is whether or not partial difference sets exist in groups of exponent 8. In particular, do they exist in the group $C_8\times C_8$?

In this paper, the study of strongly regular graphs and that of Cayley graphs is combined to find all partial difference sets given specific sets of parameters. In particular, Cayley graphs on finite abelian groups $G$ are sought. This allows for the use of character theory of the groups and rational idempotents in the group ring $\mathbb{C}[G]$ to simplify computations. Given parameter sets, partial difference sets are found through a collapsing process on our unknown generating set $S$ of the Cayley graph, working with smaller cases to build sets $S$ that yield strongly regular Cayley graphs which are thus partial difference sets. Through this method, we answer the existence question posed by Davis and Polhill.

\section{Second Section}
The significance of being able to classify strongly regular graphs under certain parameters has not be lost on researchers the last 50 years. For instance, under what parameters do we have a strongly regular Cayley graph? The generating set for such a graph is called a partial difference set. It is known that there exist partial difference sets for groups of exponent 4 or less. In [6], Jim Davis and John Polhill describe how they are able to construct difference sets in groups  $C_2^r$ and $C_4^r$ for all $r\geq 2$, which can then be used to construct partial difference sets. They also prove a new product construction for such sets. Through this method, they are able to show that there do not exist partial difference sets in groups of exponent 16. However, a question raised by Davis and Polhill is whether or not partial difference sets exist in groups of exponent 8. In particular, do they exist in the group $C_8\times C_8$?
The significance of being able to classify strongly regular graphs under certain parameters has not be lost on researchers the last 50 years. For instance, under what parameters do we have a strongly regular Cayley graph? The generating set for such a graph is called a partial difference set. It is known that there exist partial difference sets for groups of exponent 4 or less. In [6], Jim Davis and John Polhill describe how they are able to construct difference sets in groups  $C_2^r$ and $C_4^r$ for all $r\geq 2$, which can then be used to construct partial difference sets. They also prove a new product construction for such sets. Through this method, they are able to show that there do not exist partial difference sets in groups of exponent 16. However, a question raised by Davis and Polhill is whether or not partial difference sets exist in groups of exponent 8. In particular, do they exist in the group $C_8\times C_8$?

In this paper, the study of strongly regular graphs and that of Cayley graphs is combined to find all partial difference sets given specific sets of parameters. In particular, Cayley graphs on finite abelian groups $G$ are sought. This allows for the use of character theory of the groups and rational idempotents in the group ring $\mathbb{C}[G]$ to simplify computations. Given parameter sets, partial difference sets are found through a collapsing process on our unknown generating set $S$ of the Cayley graph, working with smaller cases to build sets $S$ that yield strongly regular Cayley graphs which are thus partial difference sets. Through this method, we answer the existence question posed by Davis and Polhill.

In this paper, the study of strongly regular graphs and that of Cayley graphs is combined to find all partial difference sets given specific sets of parameters. In particular, Cayley graphs on finite abelian groups $G$ are sought. This allows for the use of character theory of the groups and rational idempotents in the group ring $\mathbb{C}[G]$ to simplify computations. Given parameter sets, partial difference sets are found through a collapsing process on our unknown generating set $S$ of the Cayley graph, working with smaller cases to build sets $S$ that yield strongly regular Cayley graphs which are thus partial difference sets. Through this method, we answer the existence question posed by Davis and Polhill.
The significance of being able to classify strongly regular graphs under certain parameters has not be lost on researchers the last 50 years. For instance, under what parameters do we have a strongly regular Cayley graph? The generating set for such a graph is called a partial difference set. It is known that there exist partial difference sets for groups of exponent 4 or less. In [6], Jim Davis and John Polhill describe how they are able to construct difference sets in groups  $C_2^r$ and $C_4^r$ for all $r\geq 2$, which can then be used to construct partial difference sets. They also prove a new product construction for such sets. Through this method, they are able to show that there do not exist partial difference sets in groups of exponent 16. However, a question raised by Davis and Polhill is whether or not partial difference sets exist in groups of exponent 8. In particular, do they exist in the group $C_8\times C_8$?

In this paper, the study of strongly regular graphs and that of Cayley graphs is combined to find all partial difference sets given specific sets of parameters. In particular, Cayley graphs on finite abelian groups $G$ are sought. This allows for the use of character theory of the groups and rational idempotents in the group ring $\mathbb{C}[G]$ to simplify computations. Given parameter sets, partial difference sets are found through a collapsing process on our unknown generating set $S$ of the Cayley graph, working with smaller cases to build sets $S$ that yield strongly regular Cayley graphs which are thus partial difference sets. Through this method, we answer the existence question posed by Davis and Polhill.

It turns out that such Cayley graphs with our chosen set of parameters are negative Latin Square graphs. To discuss the significance of this, in Section II we begin by reviewing Cayley graphs and strongly regular graphs. This section is meant as a brief overview of the specific topics we need to utilize in this paper. Further study of these can be found in \cite{GS} and \cite{BCN}.

Section III describes how we use the rational idempotents in $\mathbb{C}[G]$ to find partial difference sets in $G$, throughout which we discuss the dual group $G^*$ and properties of the characters in $G^*$. An example of this method is thoroughly investigated in Section IV, where we consider the group $C_8\times C_8$ and parameter set $(64,18,2,6)$. From this construction we can now generate all partial differences sets in groups of exponent 8. Our method can also be extended to groups $C_{2^n}\times C_{2^n}$ with parameter set $(64,9r,r^2+3r-8,r^2+r)$ for positive integer values of $r$. We discuss the more general case and further topics of investigation in Section V.

Filtering is one of the most popular methods in image restoration. The local
filtering restores an image by its local average such that the restored
value at a pixel is obtained as a (weighted) average of its neighboring
pixels. People show favor {\ towards} local filters for they provide fast
and real-time algorithms. The local linear filtering algorithms, such as the
well-known Gaussian filtering and Wiener filtering, are {\ fast} but tend to
blur image. To avoid the blurring effect, many nonlinear filters have been
constructed, for example, the Yaroslavsky neighborhood filter \cite%
{Yaroslavsky} (or sigma filter \cite{Lee}) and the bilateral filter \cite%
{TomasiM} (or SUSAN filter \cite{SmithB}). These filters, whose weights {\
favor} the pixels having the similar gray level values, create faster
diffusion inside a homogeneous region while slower diffusion across the
boundary of the region so that the image edge can be preserved. Although
nonlinear filtering algorithms do not take the benefit of linear
convolution, they can still be implemented very fast using very {\ little
read-only memory (ROM)}. Besides, being easily integrated into hardware,
they are attracting more and more industrial attentions {\ (see \cite%
{ChuiW2, DevaneyGL, Maurer, MaurerB, ZhangM})}.

Unfortunately, all of these local nonlinear filters create artificial
shocks, showing a staircase effect. That is, artificial boundaries are
created in the image of flat region. {\ The authors of \cite{BuadesCM3}
showed that the Yaroslavsky neighborhood filter asymptotically behaves like
a Perona-Malik model \cite{PeronaM}, creating large flat zones and spurious
contours inside smooth regions. They also illustrated that this enhancing
character of the neighborhood filter is due to its failure in reproducing
linear functions (see Section IV in \cite{BuadesCM3}). The similar behavior
of the bilateral filter was also observed by Barash in \cite{Barash} and
Chui and Wang in \cite{ChuiW}. }

To eliminate the staircase effect caused by the above nonlinear filters, the
authors of \cite{BuadesCM3} applied linear regression to a filter {\ and
called the corrected one linear regression neighborhood filter (LRNF), which}
is a sequel of a nonlinear filtering and its linear regression. {\ Thanks to
linear regression correction, LRNF no longer has enhancing character.
However, the correction needs much more computing time, making LRNF} slower
than a standard filtering algorithm.

The purpose of this paper is to construct a new local nonlinear filter, {\
called directional diffusion filter (DDF)}, which still preserves image
edge, but does not show staircase effect. Because DDF reproduces linear
functions, it has no need of linear regression correction and, therefore, is
much faster than LRNF.

The paper is organized as follows. In the next section, we briefly review
the linear regression correction method and analyze its necessity for the
Yaroslavsky filter and the bilateral filter in eliminating staircase effect.
In Section 3, we introduce the directional diffusion filter, reveal the
relation between it and the improved TV model \cite{MarquinaO}, and explore
its ability to reproduce linear functions. In Section 4, we develop its
numerical algorithm {\ and discuss how to set the parameters in the filter}.
In the last section, we implement the algorithm in several experiments.

\section{Review Of Linear Regression Correction For Nonlinear Filters}

Let $u(x),x\in\Omega\subset\mathbb{R}^{2},$ be an image. As usual, we assume 
$\Omega=[0,1]^{2}.$ The Yaroslavsky neighborhood filter \cite{Yaroslavsky}
first defines a neighborhood $N(x):=G_{h}(x)\cap B_{\rho}(x)$ for pixel $x,$
where $G_{h}(x)=\left\{ y\in\Omega;\quad\left\vert u(y)-u(x)\right\vert
<h\right\}$ and $B_{\rho}(x) =\left\{ y\in\Omega;\quad\left\vert \left\vert
y-x\right\vert \right\vert <\rho\right\},$ then sets the updated value at $x$
as the average of the pixels in $N(x):$%
\begin{equation*}
YNF_{h,\rho}u(x)=\frac{1}{\left\vert N(x)\right\vert }\int_{N(x)}u(y)dy,
\end{equation*}
where $\left\vert N(x)\right\vert $ denotes the Lebegues measure of $N(x).$
This filter is often rewritten in a more continuous form as%
\begin{equation*}
YNF_{h,\rho}u(x)=\frac{1}{C(x)}\int_{B_{\rho}(x)}u(y)e^{-\frac{\left\vert
u(y)-u(x)\right\vert ^{2}}{h^{2}}}dy,
\end{equation*}
where $C(x)=\int_{B_{\rho}(x)}e^{-\frac{\left\vert u(y)-u(x)\right\vert ^{2}%
}{h^{2}}}dy$ is the normalization factor. If we also smooth the Yaroslavsky
filter in the spatial domain, then it becomes the bilateral filter \cite%
{TomasiM} (or SUSAN filter \cite{SmithB}):%
\begin{equation}
BNF_{h,\rho}u(x)=\frac{1}{C(x)}\int_{\Omega}u(y)e^{-\frac{c\left\Vert
x-y\right\Vert ^{2}}{\rho^{2}}-\frac{\left\vert u(y)-u(x)\right\vert ^{2}}{%
h^{2}}}dy,  \label{BNF}
\end{equation}
where $C(x)$ is again the normalization factor in the form as 
\begin{equation*}
C(x)=\int_{\Omega}e^{-\frac{c\left\Vert x-y\right\Vert ^{2}}{\rho^{2}}-\frac{%
\left\vert u(y)-u(x)\right\vert ^{2}}{h^{2}}}dy.
\end{equation*}
Note that if we choose $c$ such that $e^{-c}\approx0,$ then 
\begin{equation*}
BNF_{h,\rho}u(x)\approx\frac{1}{C(x)}\int_{B_{\rho}(x)}u(y)e^{-\frac{%
c\left\Vert x-y\right\Vert ^{2}}{\rho^{2}}-\frac{\left\vert
u(y)-u(x)\right\vert ^{2}}{h^{2}}}dy.
\end{equation*}
Hence, no essential difference exists between the bilateral filter and the
Yaroslavsky one.

\endinput

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