arithmetic.tex

%!TEX root = cv-sheet.tex
\section{Arithmetic Operations}
\subsection{Addition}
$I(x,y) = I_1(x,y) + I_2(x,y)$\\
OR\\
$I(x,y) = I_1(x,y) + C$\\
\subsubsection{Overflow}
$I(x,y) > max\ (255)?$
\begin{enumerate}
  \item Wrapping: $I'(x,y) = I(x,y) - (max+1)$
  \item Saturation: $ I'(x,y) = max$
\end{enumerate}

\subsection{Subtraction}
usage: detect changes between 2 images.\\
$I(x,y) = I_1(x,y) - I_2(x,y)$\\
OR\\
$I(x,y) = I_1(x,y) - C$\\
\subsubsection{Underflow}
$I(x,y) < 0?$
\begin{enumerate}
  \item Wrapping: $I'(x,y) = I(x,y) + (max+1)$
  \item Saturation: $ I'(x,y) = 0$
  \item Absolute: $ I'(x,y) = |I(x,y)|$
\end{enumerate}
\subsection{Multiplication}
usage: enhance contrast
$I(x,y) = I_1(x,y) * I_2(x,y)$\\
OR\\
$I(x,y) = I_1(x,y) * C$\\
\subsection{Division}
$I(x,y) = I_1(x,y) \div I_2(x,y)$\\
OR\\
$I(x,y) = I_1(x,y) \div C$\\
\subsection{Blending}
$I(x,y) = I_1(x,y) * C +  I_2(x,y) * (1-C)$\\
\subsection{Logical Operations}
\textsc{NOT, AND/NAND, OR/NOR, XOR/XNOR}
\subsubsection{Conversion}
\begin{enumerate}
  \item Bitwise: convert values to binary base and apply operators bitwise.
  \item Thresholding: convert each pixel value to 1 bit:
$
  I_{new} =
  \begin{cases}
    0, & I > 127,\\
    1, & I \leq 127.
  \end{cases}
$
\end{enumerate}
\subsubsection{Applications}
\begin{itemize}
  \item \textbf{AND/NAND}: intersection bet. 2 images.
  \item \textbf{OR/NOR}: union bet. 2 images.
  \item \textbf{NOT}: negative of input image.
\end{itemize}
\subsubsection{Logical NOT}
Ways to apply NOT:
\begin{itemize}
  \item normal boolean NOT.
  \item grayscale: $I'(x,y) = 255 - I(x,y)$.
  \item float pixel format: $I'(x,y) = - I(x,y)$, followed by normalization.
\end{itemize}
\subsection{Bitshift Operations}
usage: fast multiplication and division
\begin{itemize}
  \item \textbf{Shift left} $i$ bits = $* 2^i$ (multiplication)
  \item \textbf{Shift right} $i$ bits = $\div 2^i$ (division)
\end{itemize}
\subsubsection{Empty Places}
\begin{enumerate}
  \item fill with zeros.
  \item fill with ones.
  \item fill with bits from other side (rotate).
\end{enumerate}
%end of section
\hrule