contour.tex

图像中非刚性曲线的蛇形检测算法

\rhead{class CONTOUR}

\section{CONTOUR : A deformable template object}

{\tt CONTOUR} is a collection of {\tt SNAXEL} objects forming a deformable template. It has the following structure :

\begin{verbatim}
class CONTOUR {

    protected :
        SNAXEL *head;           /* head of snake */
        SNAXEL *tail;           /* tail of snake */
        SNAKEMODE mode;         /* opened or closed snake */
        short numsnaxel;        /* number of snaxel */
        double cg_row, cg_col;  /* center of gravity */
        double avglen;          /* ave. power (length) of snake */
        double sig_nu_sqr;      /* white noise variance on gradient power */
        double Z;               /* normalizing constant */
        short direction;        /* [0/-1/+1] preset gradient direction */
};
\end{verbatim}

{\tt CONTOUR} is a linked list of {\tt SNAXELS} originated at image origin or the reference point ({\tt cg\_col}, {\tt cg\_row}). Each contour can have a gradient direction ({\tt direction}) that pointing outward (+1) or inward (-1). {\tt avglen} refers to $l(U)$ in equation (3.9), {\tt sig\_nu\_sqr} refers to $\sigma_{\eta}^{2}$ in equation (3.24), and {\tt Z} refers to $Z_{i}$ in equation (4.35) of \cite{kn:thesis}.

%
\subsection{{\tt CONTOUR} constructor}

\subsubsection*{Synopsis}
\begin{verbatim}
CONTOUR(void)
\end{verbatim}

\subsubsection*{Description}
The constructor initializes the number of snaxels, white noise variance ($\sigma_{\eta}^{2}$) and contour direction to 0, sets normalizing constant to 1. and directs the head and tail to NULL.

%
\subsection{{\tt CONTOUR} destructor}

\subsubsection*{Synopsis}
\begin{verbatim}
~CONTOUR(void)
\end{verbatim}
 
\subsubsection*{Description}
The destructor frees the memory allocated to snaxels which form a complete chain of contour. 

% 
\subsection{Resetting a {\tt CONTOUR} }

\subsubsection*{Synopsis}
\begin{verbatim}
void reset()	
\end{verbatim}

\subsubsection*{Description}
{\tt reset} allows the reuse of an {\tt CONTOUR} object by freeing up the memory allocated to snaxels and setting the head and tail to NULL.

%
\subsection{Automatic initialization of closed contour}

\subsubsection*{Synopsis}
\begin{verbatim}
int init( short _cg_row, short _cg_col, 
          double radius, short numpts = 16)
\end{verbatim}

\subsubsection*{Arguments}
\tb
	{\tt \_cg\_row, \_cg\_col} & Centre of circle. \\
	{\tt radius} & Radius of circle. \\
	{\tt numpts} & Number of snaxels.
\te

\subsubsection*{Returns}
\tb	
	{\tt NOERROR} & Successful operation.\\
	{\tt MEMORYERROR} & Memory allocation failure.
\te

\subsubsection*{Description}
{\tt init} creates a circle with {\tt numpts} snaxels and sets its {\tt mode} to \_CLOSED, that is, a contour with its head and tail internally connected.


%
\subsection{Automatic initialization of open contour}
 
\subsubsection*{Synopsis}
\begin{verbatim}
int init( short sx, short sy, short ex, short ey, 
          short numpts = 16 )
\end{verbatim}

\subsubsection*{Arguments}
\tb
	{\tt sx, sy} & The starting point of a line. \\
	{\tt ex, ey} & The ending point of a line. \\
	{\tt numpts} & Number of snaxels.
\te

\subsubsection*{Returns}
\tb	
	{\tt NOERROR} &  Successful operation. \\
	{\tt MEMORYERROR} & Memory allocation failure.
\te 
	
\subsubsection*{Description}
{\tt init} creates a line with {\tt numpts} snaxels and sets its {\tt mode} to \_OPENED, that is, a contour with unconnected head and tail.


%
\subsection{Manual initialization of a contour }

\subsubsection*{Synopsis}
\begin{verbatim}	
int init( IMAGE *ximg, unsigned char blowup = 1, 
          INITMODE  theINIT = _CLICKMOUSE, 
          SNAKEMODE _mode = _CLOSED,int spacing)
\end{verbatim}

\subsubsection*{Arguments}
\tb
	{\tt *ximg} & Reference image. \\
	{\tt blowup} & Image magnification factor. \\
	{\tt theINIT} & Contour initialization method, 
			\_DRAGMOUSE or \_CLICKMOUSE \\ 	
	{\tt \_mode} & Snake mode of a contour, \_OPENED or \_CLOSED.
\te		

\subsubsection*{Returns}
\tb	
	{\tt NOERROR} & Successful operation. \\
	{\tt MEMORYERROR} & Memory allocation error.
\te 
	
\subsubsection*{Description}
{\tt init} creates a contour of either through clicking ({\tt \_CLICKMOUSE}) or dragging ({\tt \_DRAGMOUSE}) by mouse. By displaying the reference image on an X window, the user then specifies the desired snaxels. Since the snaxels are joined sequentially, the selection of the snaxels must be in order.

%
\subsection{File interface methods}
	
\subsubsection*{Synopsis}
\begin{verbatim}
	int read( char *filename )
	int write( char *filename )
\end{verbatim}

\subsubsection*{Arguments} 
\tb
	{\tt *filename} &  Contour file.
\te

\subsubsection*{Returns}	
\tb
	{\tt NOERROR} & Successful read or write operation. \\
	{\tt MEMORYERROR} & Memory allocation failure. \\
	{\tt FILEIOERROR} & Unable to read from or write to file.
\te

\subsubsection*{Description}
{\tt read} and {\tt write} provide file manipulation routines for accessing a contour file. The format, which is in text form, is as following :

\begin{quote}

\begin{tabular}{ll}
	$\underline{Field}$ & $\underline{Data type}$ \\
	Magic number & 1010 (hex) \\
	Mode & \_OPENED (0) or \_CLOSED (1) \\
	Number of snaxels & Integer number \\
	White noise variance & Exponential notation \\
	Snaxel coordinates (column, row) & Double \\
	Shape matrix ($\alpha$, $\beta$) & Double \\
	Regularization parameter ($\lambda$) & Double \\
	Normalizing constant & Exponential notation.
\end{tabular}
\end{quote}

	
%
\subsection{Learning shape matrix}

\subsubsection*{Synopsis}
\begin{verbatim}
	void computeShape()																																																																																																																																										\end{verbatim}
	
\subsubsection*{Description}
{\tt computeShape} models and calculates the {\em shape matrix} of a contour as in (Eqn 3.3 of \cite{kn:thesis}). $\alpha$ and $\beta$ are computed as following :

\eq
        \left[
        \begin{array}{c}
                {\bf \alpha_{i}} \\
                {\bf \beta_{i}}
        \end{array}
        \right] =
        \left[
        \begin{array}{cc}
                x_{i \alpha} & x_{i \beta} \\
                y_{i \beta}  & y_{i \beta}
        \end{array}
        \right]^{-1}
	 \left[
        \begin{array}{c}
                {\bf x_{i}} \\
                {\bf y_{i}}
        \end{array}
        \right]
\en

% 
\subsection{Computing normalizing constant for classification purpose}

\subsubsection*{Synopsis}
\begin{verbatim}
	void computeZ()
\end{verbatim}

\subsubsection*{Description}
{\tt computeZ} compute normaling constant $Z_{i}$ by Monte Carlo estimation. {$Z_{i}$ must be calculated during the training stage of classification. Supposes a family of contours $U \in \Omega_{i}$, then
\eq
        Z_{i} = \sum_{ U \in \Omega_{i}} \exp(-E_{int}(U))
\en


% 
\subsection{Computing the average vector distance l(U) between snaxels}

\subsubsection*{Synopsis}
\begin{verbatim}
	double computeAvgLength()
\end{verbatim}

\subsubsection*{Returns}
Average distance of snaxels.

\subsubsection*{Description}
{\tt computeAvgLength} calculates the average distance between snaxels, $l(U)$, as following :
\eq
	l(U) = \frac{1}{n}\sum_{i=1}^{n}\|u_{i+1}-u_{i}\|^{2}
\en
$l(U)$ is also the normalising constant used for internal energy calculation.

%
\subsection{Regenerating shape matrix}

\subsubsection*{Synopsis}
\begin{verbatim}
	int regenerate()
\end{verbatim}

\subsubsection*{Returns}
\tb
	{\tt NOERROR} & Successful operation. \\
	{\tt MEMORYERROR} & Memory allocation failure. \\
	{\tt PARAMERROR} & Parameter error.
\te

\subsubsection*{Description}
{\tt regenerate} generate the complete chain $U$ from {\em shape matrix} $A$ by matrix inversion if the last two snaxels on $U$ is available. This is possible by decomposing $A$ into an $(n-2)\times(n-2)$ invertible submatrix $A_{r}$ such that,
\eq
	A_{r} U_{r} + b = 0 \\
	\label{eqn:regenerate1}
\en
where
\eq
	b = \left[ \begin{array}{c}
			\ldots \\
			0 \\
			\ldots \\
			-\beta_{n-2} u_{n-1} \\
			u_{n-1}-\beta_{n}u_{n}
		  \end{array}
	    \right]
	\label{eqn:regenerate2}
\en

Thus the regenerative contour $U_{r}$ is,
\eq
	U_{r} = -A_{r}^{-1}b
\en


%
\subsection{Conversion of Coordinates}

\subsubsection*{Synopsis}
\begin{verbatim}
	void imageCentered()
	void contourCentered()
\end{verbatim}

\subsubsection*{Description}
{\tt imageCentered} defines a contour as the vector containing an ordered set of points, $V = [v_{1}, v_{2}, \ldots, v_{n}]$, where each $v_{i}$ has its origin at (0, 0) of an image. In contrast, {\tt contourCentered} defines a contour as $U = [u_{1}, u_{2}, \ldots, u_{n}]$, where each $u_{i} = v_{i} - g$ represents the displacement from the center of gravity $g$ of a contour.


%
\subsection{Calculating the internal energy of an individual snaxel}

\subsubsection*{Synopsis}
\begin{verbatim}
	double EInternal (SNAXEL *sxptr)
\end{verbatim}

\subsubsection*{Arguments}
\tb
	sxptr & Targeted snaxel.
\te

\subsubsection*{Returns}
Internal energy of the targeted snaxel.

\subsubsection*{Description}
{\tt EInternal} computes the internal energy of a snaxel as (Eqn 3.21 of \cite{kn:thesis}).

%
\subsection{Affine transformations of contour} 

\subsubsection*{Synopsis}
\begin{verbatim}
	void rotate (double angle)
	void translate (int dx, int dy)
	void scale (double sx, double sy)
	void dilate (double idx, double idy)
	void affineTransform(double sx, double sy, double rt, 
                double tx, double ty, double idx, double idy)
\end{verbatim}

\subsubsection*{Arguments}
\tb
	{\tt angle} & Rotation angle in degrees. \\
	{\tt dx, dy} & Translation vector. \\
	{\tt sx, sy} & Scaling factor. \\
	{\tt idx, idy} & Dilation factor.
\te

\subsubsection*{Description}
These methods provides routines for global deformations of a contour so that user can observe the effects of rigid motion on the {\em shape matrix}.