\cvarg{src\_seq}{Pointer to the chain that can refer to other chains}
\cvarg{storage}{Storage location for the resulting polylines}
\cvarg{method}{Approximation method (see the description of the function \cvCPyCross{FindContours})}
\cvarg{parameter}{Method parameter (not used now)}
\cvarg{minimal\_perimeter}{Approximates only those contours whose perimeters are not less than \texttt{minimal\_perimeter}. Other chains are removed from the resulting structure}
\cvarg{recursive}{If not 0, the function approximates all chains that access can be obtained to from \texttt{src\_seq} by using the \texttt{h\_next} or \texttt{v\_next links}. If 0, the single chain is approximated}
\end{description}
This is a stand-alone approximation routine. The function \texttt{cvApproxChains} works exactly in the same way as \cvCPyCross{FindContours} with the corresponding approximation flag. The function returns pointer to the first resultant contour. Other approximated contours, if any, can be accessed via the \texttt{v\_next} or \texttt{h\_next} fields of the returned structure.
\cvCPyFunc{ApproxPoly}
Approximates polygonal curve(s) with the specified precision.
\cvdefC{
CvSeq* cvApproxPoly( \par const void* src\_seq,\par int header\_size,\par CvMemStorage* storage,\par int method,\par double parameter,\par int parameter2=0 );
\cvarg{header\_size}{Header size of the approximated curve[s]}
\fi
\cvarg{storage}{Container for the approximated contours. If it is NULL, the input sequences' storage is used}
\cvarg{method}{Approximation method; only \texttt{CV\_POLY\_APPROX\_DP} is supported, that corresponds to the Douglas-Peucker algorithm}
\cvarg{parameter}{Method-specific parameter; in the case of \texttt{CV\_POLY\_APPROX\_DP} it is a desired approximation accuracy}
\cvarg{parameter2}{If case if \texttt{src\_seq} is a sequence, the parameter determines whether the single sequence should be approximated or all sequences on the same level or below \texttt{src\_seq} (see \cvCPyCross{FindContours} for description of hierarchical contour structures). If \texttt{src\_seq} is an array CvMat* of points, the parameter specifies whether the curve is closed (\texttt{parameter2}!=0) or not (\texttt{parameter2} =0)}
\end{description}
The function approximates one or more curves and
returns the approximation result[s]. In the case of multiple curves,
the resultant tree will have the same structure as the input one (1:1
correspondence).
\cvCPyFunc{ArcLength}
Calculates the contour perimeter or the curve length.
\cvarg{curve}{Sequence or array of the curve points}
\cvarg{slice}{Starting and ending points of the curve, by default, the whole curve length is calculated}
\cvarg{isClosed}{Indicates whether the curve is closed or not. There are 3 cases:
\begin{itemize}
\item$\texttt{isClosed}=0$ the curve is assumed to be unclosed.
\item$\texttt{isClosed}>0$ the curve is assumed to be closed.
\item$\texttt{isClosed}<0$ if curve is sequence, the flag \texttt{CV\_SEQ\_FLAG\_CLOSED} of \texttt{((CvSeq*)curve)->flags} is checked to determine if the curve is closed or not, otherwise (curve is represented by array (CvMat*) of points) it is assumed to be unclosed.
\end{itemize}}
\end{description}
The function calculates the length or curve as the sum of lengths of segments between subsequent points
\cvCPyFunc{BoundingRect}
Calculates the up-right bounding rectangle of a point set.
\cvdefC{
CvRect cvBoundingRect( CvArr* points, int update=0 );
}\cvdefPy{BoundingRect(points,update=0)-> CvRect}
\begin{description}
\cvarg{points}{2D point set, either a sequence or vector (\texttt{CvMat}) of points}
\cvarg{update}{The update flag. See below.}
\end{description}
The function returns the up-right bounding rectangle for a 2d point set.
Here is the list of possible combination of the flag values and type of \texttt{points}:
\begin{tabular}{|c|c|p{3in}|}
\hline
update & points & action \\\hline
0 &\texttt{CvContour\*}& the bounding rectangle is not calculated, but it is taken from \texttt{rect} field of the contour header.\\\hline
1 &\texttt{CvContour\*}& the bounding rectangle is calculated and written to \texttt{rect} field of the contour header.\\\hline
0 &\texttt{CvSeq\*} or \texttt{CvMat\*}& the bounding rectangle is calculated and returned.\\\hline
1 &\texttt{CvSeq\*} or \texttt{CvMat\*}& runtime error is raised.\\\hline
\cvarg{signature1}{First signature, a $\texttt{size1}\times\texttt{dims}+1$ floating-point matrix. Each row stores the point weight followed by the point coordinates. The matrix is allowed to have a single column (weights only) if the user-defined cost matrix is used}
\cvarg{signature2}{Second signature of the same format as \texttt{signature1}, though the number of rows may be different. The total weights may be different, in this case an extra "dummy" point is added to either \texttt{signature1} or \texttt{signature2}}
\cvarg{distance\_type}{Metrics used; \texttt{CV\_DIST\_L1, CV\_DIST\_L2}, and \texttt{CV\_DIST\_C} stand for one of the standard metrics; \texttt{CV\_DIST\_USER} means that a user-defined function \texttt{distance\_func} or pre-calculated \texttt{cost\_matrix} is used}
\ifC
\cvarg{distance\_func}{The user-supplied distance function. It takes coordinates of two points and returns the distance between the points
\cvarg{distance\_func}{The user-supplied distance function. It takes coordinates of two points \texttt{pt0} and \texttt{pt1}, and returns the distance between the points, with sigature
\texttt{
func(pt0, pt1, userdata) -> float
}
}
\fi
\cvarg{cost\_matrix}{The user-defined $\texttt{size1}\times\texttt{size2}$ cost matrix. At least one of \texttt{cost\_matrix} and \texttt{distance\_func} must be NULL. Also, if a cost matrix is used, lower boundary (see below) can not be calculated, because it needs a metric function}
\cvarg{flow}{The resultant $\texttt{size1}\times\texttt{size2}$ flow matrix: $\texttt{flow}_{i,j}$ is a flow from $i$ th point of \texttt{signature1} to $j$ th point of \texttt{signature2}}
\cvarg{lower\_bound}{Optional input/output parameter: lower boundary of distance between the two signatures that is a distance between mass centers. The lower boundary may not be calculated if the user-defined cost matrix is used, the total weights of point configurations are not equal, or if the signatures consist of weights only (i.e. the signature matrices have a single column). The user \textbf{must} initialize \texttt{*lower\_bound}. If the calculated distance between mass centers is greater or equal to \texttt{*lower\_bound} (it means that the signatures are far enough) the function does not calculate EMD. In any case \texttt{*lower\_bound} is set to the calculated distance between mass centers on return. Thus, if user wants to calculate both distance between mass centers and EMD, \texttt{*lower\_bound} should be set to 0}
\cvarg{userdata}{Pointer to optional data that is passed into the user-defined distance function}
\end{description}
The function computes the earth mover distance and/or
a lower boundary of the distance between the two weighted point
configurations. One of the applications described in \cvCPyCross{RubnerSept98} is
multi-dimensional histogram comparison for image retrieval. EMD is a a
transportation problem that is solved using some modification of a simplex
algorithm, thus the complexity is exponential in the worst case, though, on average
it is much faster. In the case of a real metric the lower boundary
can be calculated even faster (using linear-time algorithm) and it can
be used to determine roughly whether the two signatures are far enough
so that they cannot relate to the same object.
\cvCPyFunc{CheckContourConvexity}
Tests contour convexity.
\cvdefC{
int cvCheckContourConvexity( const CvArr* contour );
}\cvdefPy{CheckContourConvexity(contour)-> int}
\begin{description}
\cvarg{contour}{Tested contour (sequence or array of points)}
\end{description}
The function tests whether the input contour is convex or not. The contour must be simple, without self-intersections.
Orientation of the contour affects the area sign, thus the function may return a \emph{negative} result. Use the \texttt{fabs()} function from C runtime to get the absolute value of the area.
\cvarg{storage}{Container for the reconstructed contour}
\cvarg{criteria}{Criteria, where to stop reconstruction}
\end{description}
The function restores the contour from its binary tree representation. The parameter \texttt{criteria} determines the accuracy and/or the number of tree levels used for reconstruction, so it is possible to build an approximated contour. The function returns the reconstructed contour.
\cvCPyFunc{ConvexHull2}
Finds the convex hull of a point set.
\cvdefC{
CvSeq* cvConvexHull2( \par const CvArr* input,\par void* storage=NULL,\par int orientation=CV\_CLOCKWISE,\par int return\_points=0 );
\cvarg{points}{Sequence or array of 2D points with 32-bit integer or floating-point coordinates}
\cvarg{storage}{The destination array (CvMat*) or memory storage (CvMemStorage*) that will store the convex hull. If it is an array, it should be 1d and have the same number of elements as the input array/sequence. On output the header is modified as to truncate the array down to the hull size. If \texttt{storage} is NULL then the convex hull will be stored in the same storage as the input sequence}
\cvarg{orientation}{Desired orientation of convex hull: \texttt{CV\_CLOCKWISE} or \texttt{CV\_COUNTER\_CLOCKWISE}}
\cvarg{return\_points}{If non-zero, the points themselves will be stored in the hull instead of indices if \texttt{storage} is an array, or pointers if \texttt{storage} is memory storage}
\end{description}
The function finds the convex hull of a 2D point set using Sklansky's algorithm. If \texttt{storage} is memory storage, the function creates a sequence containing the hull points or pointers to them, depending on \texttt{return\_points} value and returns the sequence on output. If \texttt{storage} is a CvMat, the function returns NULL.
\ifC
Example. Building convex hull for a sequence or array of points
\begin{lstlisting}
#include "cv.h"
#include "highgui.h"
#include <stdlib.h>
#define ARRAY 0 /* switch between array/sequence method by replacing 0<=>1 */
\cvarg{convexhull}{Convex hull obtained using \cvCPyCross{ConvexHull2} that should contain pointers or indices to the contour points, not the hull points themselves (the \texttt{return\_points} parameter in \cvCPyCross{ConvexHull2} should be 0)}
\cvarg{storage}{Container for the output sequence of convexity defects. If it is NULL, the contour or hull (in that order) storage is used}
\end{description}
The function finds all convexity defects of the input contour and returns a sequence of the CvConvexityDefect structures.
\cvCPyFunc{CreateContourTree}
Creates a hierarchical representation of a contour.
The function creates a binary tree representation for the input \texttt{contour} and returns the pointer to its root. If the parameter \texttt{threshold} is less than or equal to 0, the function creates a full binary tree representation. If the threshold is greater than 0, the function creates a representation with the precision \texttt{threshold}: if the vertices with the interceptive area of its base line are less than \texttt{threshold}, the tree should not be built any further. The function returns the created tree.
\cvarg{image}{The source, an 8-bit single channel image. Non-zero pixels are treated as 1's, zero pixels remain 0's - the image is treated as \texttt{binary}. To get such a binary image from grayscale, one may use \cvCPyCross{Threshold}, \cvCPyCross{AdaptiveThreshold} or \cvCPyCross{Canny}. The function modifies the source image's content}
\cvarg{storage}{Container of the retrieved contours}
\ifC
\cvarg{first\_contour}{Output parameter, will contain the pointer to the first outer contour}
\cvarg{header\_size}{Size of the sequence header, $\ge\texttt{sizeof(CvChain)}$ if $\texttt{method}=\texttt{CV\_CHAIN\_CODE}$,
and $\ge\texttt{sizeof(CvContour)}$ otherwise}
\fi
\cvarg{mode}{Retrieval mode
\begin{description}
\cvarg{CV\_RETR\_EXTERNAL}{retrives only the extreme outer contours}
\cvarg{CV\_RETR\_LIST}{retrieves all of the contours and puts them in the list}
\cvarg{CV\_RETR\_CCOMP}{retrieves all of the contours and organizes them into a two-level hierarchy: on the top level are the external boundaries of the components, on the second level are the boundaries of the holes}
\cvarg{CV\_RETR\_TREE}{retrieves all of the contours and reconstructs the full hierarchy of nested contours}
\end{description}}
\cvarg{method}{Approximation method (for all the modes, except \texttt{CV\_LINK\_RUNS}, which uses built-in approximation)
\begin{description}
\cvarg{CV\_CHAIN\_CODE}{outputs contours in the Freeman chain code. All other methods output polygons (sequences of vertices)}
\cvarg{CV\_CHAIN\_APPROX\_NONE}{translates all of the points from the chain code into points}
\cvarg{CV\_CHAIN\_APPROX\_SIMPLE}{compresses horizontal, vertical, and diagonal segments and leaves only their end points}
\cvarg{CV\_CHAIN\_APPROX\_TC89\_L1,CV\_CHAIN\_APPROX\_TC89\_KCOS}{applies one of the flavors of the Teh-Chin chain approximation algorithm.}
\cvarg{CV\_LINK\_RUNS}{uses a completely different contour retrieval algorithm by linking horizontal segments of 1's. Only the \texttt{CV\_RETR\_LIST} retrieval mode can be used with this method.}
\end{description}}
\cvarg{offset}{Offset, by which every contour point is shifted. This is useful if the contours are extracted from the image ROI and then they should be analyzed in the whole image context}
\end{description}
The function retrieves contours from the binary image using the algorithm
\cite{Suzuki85}. The contours are a useful tool for shape analysis and
object detection and recognition.
The function retrieves contours from the
binary image and returns the number of retrieved contours. The
pointer \texttt{first\_contour} is filled by the function. It will
contain a pointer to the first outermost contour or \texttt{NULL} if no
contours are detected (if the image is completely black). Other
contours may be reached from \texttt{first\_contour} using the
\texttt{h\_next} and \texttt{v\_next} links. The sample in the
\cvCPyCross{DrawContours} discussion shows how to use contours for
connected component detection. Contours can be also used for shape
analysis and object recognition - see
\ifC
\texttt{squares.c}
\else
\texttt{squares.py}
\fi
in the OpenCV sample directory.
\textbf{Note:} the source \texttt{image} is modified by this function.
\cvarg{points}{Sequence or array of 2D or 3D points with 32-bit integer or floating-point coordinates}
\cvarg{dist\_type}{The distance used for fitting (see the discussion)}
\cvarg{param}{Numerical parameter (\texttt{C}) for some types of distances, if 0 then some optimal value is chosen}
\cvarg{reps}{Sufficient accuracy for the radius (distance between the coordinate origin and the line). 0.01 is a good default value.}
\cvarg{aeps}{Sufficient accuracy for the angle. 0.01 is a good default value.}
\cvarg{line}{The output line parameters. In the case of a 2d fitting,
it is \cvC{an array}\cvPy{a tuple} of 4 floats \texttt{(vx, vy, x0, y0)} where \texttt{(vx, vy)} is a normalized vector collinear to the
line and \texttt{(x0, y0)} is some point on the line. in the case of a
3D fitting it is \cvC{an array}\cvPy{a tuple} of 6 floats \texttt{(vx, vy, vz, x0, y0, z0)}
where \texttt{(vx, vy, vz)} is a normalized vector collinear to the line
and \texttt{(x0, y0, z0)} is some point on the line}
\end{description}
The function fits a line to a 2D or 3D point set by minimizing $\sum_i \rho(r_i)$ where $r_i$ is the distance between the $i$ th point and the line and $\rho(r)$ is a distance function, one of:
\begin{description}
\item[dist\_type=CV\_DIST\_L2]
\[\rho(r)= r^2/2\quad\text{(the simplest and the fastest least-squares method)}\]
where $\eta_{ji}$ denote the normalized central moments.
These values are proved to be invariant to the image scale, rotation, and reflection except the seventh one, whose sign is changed by reflection. Of course, this invariance was proved with the assumption of infinite image resolution. In case of a raster images the computed Hu invariants for the original and transformed images will be a bit different.
\ifPy
\begin{lstlisting}
>>> import cv
>>> original = cv.LoadImageM("building.jpg", cv.CV_LOAD_IMAGE_GRAYSCALE)
\cvarg{method}{Similarity measure, only \texttt{CV\_CONTOUR\_TREES\_MATCH\_I1} is supported}
\cvarg{threshold}{Similarity threshold}
\end{description}
The function calculates the value of the matching measure for two contour trees. The similarity measure is calculated level by level from the binary tree roots. If at a certain level the difference between contours becomes less than \texttt{threshold}, the reconstruction process is interrupted and the current difference is returned.
\cvarg{object2}{Second contour or grayscale image}
\cvarg{method}{Comparison method;
\texttt{CV\_CONTOUR\_MATCH\_I1},
\texttt{CV\_CONTOURS\_MATCH\_I2}
or
\texttt{CV\_CONTOURS\_MATCH\_I3}}
\cvarg{parameter}{Method-specific parameter (is not used now)}
\end{description}
The function compares two shapes. The 3 implemented methods all use Hu moments (see \cvCPyCross{GetHuMoments}) ($A$ is \texttt{object1}, $B$ is \texttt{object2}):
The function finds a circumscribed rectangle of the minimal area for a 2D point set by building a convex hull for the set and applying the rotating calipers technique to the hull.
Picture. Minimal-area bounding rectangle for contour
\cvarg{arr}{Image (1-channel or 3-channel with COI set) or polygon (CvSeq of points or a vector of points)}
\cvarg{moments}{Pointer to returned moment's state structure}
\cvarg{binary}{(For images only) If the flag is non-zero, all of the zero pixel values are treated as zeroes, and all of the others are treated as 1's}
\end{description}
The function calculates spatial and central moments up to the third order and writes them to \texttt{moments}. The moments may then be used then to calculate the gravity center of the shape, its area, main axises and various shape characeteristics including 7 Hu invariants.
\cvarg{seq\_kind}{Type of the point sequence: point set (0), a curve (\texttt{CV\_SEQ\_KIND\_CURVE}), closed curve (\texttt{CV\_SEQ\_KIND\_CURVE+CV\_SEQ\_FLAG\_CLOSED}) etc.}
\cvarg{mat}{Input matrix. It should be a continuous, 1-dimensional vector of points, that is, it should have type \texttt{CV\_32SC2} or \texttt{CV\_32FC2}}
\cvarg{contour\_header}{Contour header, initialized by the function}
\cvarg{block}{Sequence block header, initialized by the function}
\end{description}
The function initializes a sequence
header to create a "virtual" sequence in which elements reside in
the specified matrix. No data is copied. The initialized sequence
header may be passed to any function that takes a point sequence
on input. No extra elements can be added to the sequence,
but some may be removed. The function is a specialized variant of
\cvCPyCross{MakeSeqHeaderForArray} and uses
the latter internally. It returns a pointer to the initialized contour
header. Note that the bounding rectangle (field \texttt{rect} of
\texttt{CvContour} strucuture) is not initialized by the function. If
\cvarg{image}{The 8-bit, single channel, binary source image}
\cvarg{storage}{Container of the retrieved contours}
\cvarg{header\_size}{Size of the sequence header, $>=sizeof(CvChain)$ if \texttt{method} =CV\_CHAIN\_CODE, and $>=sizeof(CvContour)$ otherwise}
\cvarg{mode}{Retrieval mode; see \cvCPyCross{FindContours}}
\cvarg{method}{Approximation method. It has the same meaning in \cvCPyCross{FindContours}, but \texttt{CV\_LINK\_RUNS} can not be used here}
\cvarg{offset}{ROI offset; see \cvCPyCross{FindContours}}
\end{description}
The function initializes and returns a pointer to the contour scanner. The scanner is used in \cvCPyCross{FindNextContour} to retrieve the rest of the contours.
Note that $\texttt{mu}_{00}=\texttt{m}_{00}$, $\texttt{nu}_{00}=1$$\texttt{nu}_{10}=\texttt{mu}_{10}=\texttt{mu}_{01}=\texttt{mu}_{10}=0$, hence the values are not stored.
The moments of a contour are defined in the same way, but computed using Green's formula
(see \url{http://en.wikipedia.org/wiki/Green_theorem}), therefore, because of a limited raster resolution, the moments computed for a contour will be slightly different from the moments computed for the same contour rasterized.
See also: \cvCppCross{contourArea}, \cvCppCross{arcLength}
where $\eta_{ji}$ stand for $\texttt{Moments::nu}_{ji}$.
These values are proved to be invariant to the image scale, rotation, and reflection except the seventh one, whose sign is changed by reflection. Of course, this invariance was proved with the assumption of infinite image resolution. In case of a raster images the computed Hu invariants for the original and transformed images will be a bit different.
\cvarg{image}{The source, an 8-bit single-channel image. Non-zero pixels are treated as 1's, zero pixels remain 0's - the image is treated as \texttt{binary}. You can use \cvCppCross{compare}, \cvCppCross{inRange}, \cvCppCross{threshold}, \cvCppCross{adaptiveThreshold}, \cvCppCross{Canny} etc. to create a binary image out of a grayscale or color one. The function modifies the \texttt{image} while extracting the contours}
\cvarg{contours}{The detected contours. Each contour is stored as a vector of points}
\cvarg{hiararchy}{The optional output vector that will contain information about the image topology. It will have as many elements as the number of contours. For each contour \texttt{contours[i]}, the elements \texttt{hierarchy[i][0]}, \texttt{hiearchy[i][1]}, \texttt{hiearchy[i][2]}, \texttt{hiearchy[i][3]} will be set to 0-based indices in \texttt{contours} of the next and previous contours at the same hierarchical level, the first child contour and the parent contour, respectively. If for some contour \texttt{i} there is no next, previous, parent or nested contours, the corresponding elements of \texttt{hierarchy[i]} will be negative}
\cvarg{mode}{The contour retrieval mode
\begin{description}
\cvarg{CV\_RETR\_EXTERNAL}{retrieves only the extreme outer contours; It will set \texttt{hierarchy[i][2]=hierarchy[i][3]=-1} for all the contours}
\cvarg{CV\_RETR\_LIST}{retrieves all of the contours without establishing any hierarchical relationships}
\cvarg{CV\_RETR\_CCOMP}{retrieves all of the contours and organizes them into a two-level hierarchy: on the top level are the external boundaries of the components, on the second level are the boundaries of the holes. If inside a hole of a connected component there is another contour, it will still be put on the top level}
\cvarg{CV\_RETR\_TREE}{retrieves all of the contours and reconstructs the full hierarchy of nested contours. This full hierarchy is built and shown in OpenCV \texttt{contours.c} demo}
\end{description}}
\cvarg{method}{The contour approximation method.
\begin{description}
\cvarg{CV\_CHAIN\_APPROX\_NONE}{stores absolutely all the contour points. That is, every 2 points of a contour stored with this method are 8-connected neighbors of each other}
\cvarg{CV\_CHAIN\_APPROX\_SIMPLE}{compresses horizontal, vertical, and diagonal segments and leaves only their end points. E.g. an up-right rectangular contour will be encoded with 4 points}
\cvarg{CV\_CHAIN\_APPROX\_TC89\_L1,CV\_CHAIN\_APPROX\_TC89\_KCOS}{applies one of the flavors of the Teh-Chin chain approximation algorithm; see \cite{TehChin89}}
\end{description}}
\cvarg{offset}{The optional offset, by which every contour point is shifted. This is useful if the contours are extracted from the image ROI and then they should be analyzed in the whole image context}
\end{description}
The function retrieves contours from the
binary image using the algorithm \cite{Suzuki85}. The contours are a useful tool for shape analysis and object detection and recognition. See \texttt{squares.c} in the OpenCV sample directory.
\textbf{Note:} the source \texttt{image} is modified by this function.
int contourIdx, const Scalar\& color, int thickness=1,\par
int lineType=8, const vector<Vec4i>\& hierarchy=vector<Vec4i>(),\par
int maxLevel=INT\_MAX, Point offset=Point() );}
\begin{description}
\cvarg{image}{The destination image}
\cvarg{contours}{All the input contours. Each contour is stored as a point vector}
\cvarg{contourIdx}{Indicates the contour to draw. If it is negative, all the contours are drawn}
\cvarg{color}{The contours' color}
\cvarg{thickness}{Thickness of lines the contours are drawn with.
If it is negative (e.g. \texttt{thickness=CV\_FILLED}), the contour interiors are
drawn.}
\cvarg{lineType}{The line connectivity; see \cvCppCross{line} description}
\cvarg{hierarchy}{The optional information about hierarchy. It is only needed if you want to draw only some of the contours (see \texttt{maxLevel})}
\cvarg{maxLevel}{Maximal level for drawn contours. If 0, only
the specified contour is drawn. If 1, the function draws the contour(s) and all the nested contours. If 2, the function draws the contours, all the nested contours and all the nested into nested contours etc. This parameter is only taken into account when there is \texttt{hierarchy} available.}
\cvarg{offset}{The optional contour shift parameter. Shift all the drawn contours by the specified $\texttt{offset}=(dx,dy)$}
\end{description}
The function draws contour outlines in the image if $\texttt{thickness}\ge0$ or fills the area bounded by the contours if $\texttt{thickness}<0$. Here is the example on how to retrieve connected components from the binary image and label them
\begin{lstlisting}
#include "cv.h"
#include "highgui.h"
using namespace cv;
int main( int argc, char** argv )
{
Mat src;
// the first command line parameter must be file name of binary
// (black-n-white) image
if( argc != 2 || !(src=imread(argv[1], 0)).data)
return -1;
Mat dst = Mat::zeros(src.rows, src.cols, CV_8UC3);
src = src > 1;
namedWindow( "Source", 1 );
imshow( "Source", src );
vector<vector<Point> > contours;
vector<Vec4i> hierarchy;
findContours( src, contours, hierarchy,
CV_RETR_CCOMP, CV_CHAIN_APPROX_SIMPLE );
// iterate through all the top-level contours,
// draw each connected component with its own random color
\cvarg{curve}{The polygon or curve to approximate. Must be $1\times N$ or $N \times1$ matrix of type \texttt{CV\_32SC2} or \texttt{CV\_32FC2}. You can also convert \texttt{vector<Point>} or \texttt{vector<Point2f} to the matrix by calling \texttt{Mat(const vector<T>\&)} constructor.}
\cvarg{approxCurve}{The result of the approximation; The type should match the type of the input curve}
\cvarg{epsilon}{Specifies the approximation accuracy. This is the maximum distance between the original curve and its approximation}
\cvarg{closed}{If true, the approximated curve is closed (i.e. its first and last vertices are connected), otherwise it's not}
\end{description}
The functions \texttt{approxPolyDP} approximate a curve or a polygon with another curve/polygon with less vertices, so that the distance between them is less or equal to the specified precision. It used Douglas-Peucker algorithm \url{http://en.wikipedia.org/wiki/Ramer-Douglas-Peucker_algorithm}
\cvarg{curve}{The input vector of 2D points, represented by \texttt{CV\_32SC2} or \texttt{CV\_32FC2} matrix, or by \texttt{vector<Point>} or \texttt{vector<Point2f>} converted to a matrix with \texttt{Mat(const vector<T>\&)} constructor}
\cvarg{closed}{Indicates, whether the curve is closed or not}
\end{description}
The function computes the curve length or the closed contour perimeter.
\cvCppFunc{boundingRect}
Calculates the up-right bounding rectangle of a point set.
\cvarg{points}{The input 2D point set, represented by \texttt{CV\_32SC2} or \texttt{CV\_32FC2} matrix, or by \texttt{vector<Point>} or \texttt{vector<Point2f>} converted to the matrix using \texttt{Mat(const vector<T>\&)} constructor.}
\end{description}
The function calculates and returns the minimal up-right bounding rectangle for the specified point set.
\cvCppFunc{estimateRigidTransform}
Computes optimal affine transformation between two 2D point sets
\cvarg{dst}{The second input 2D point set of the same size and the same type as \texttt{A}}
\cvarg{fullAffine}{If true, the function finds the optimal affine transformation with no any additional resrictions (i.e. there are 6 degrees of freedom); otherwise, the class of transformations to choose from is limited to combinations of translation, rotation and uniform scaling (i.e. there are 5 degrees of freedom)}
\end{description}
The function finds the optimal affine transform $[A|b]$ (a $2\times3$ floating-point matrix) that approximates best the transformation from $\texttt{srcpt}_i$ to $\texttt{dstpt}_i$:
\[[A^*|b^*]= arg \min_{[A|b]}\sum_i \|\texttt{dstpt}_i - A {\texttt{srcpt}_i}^T - b \|^2\]
where $[A|b]$ can be either arbitrary (when \texttt{fullAffine=true}) or have form
\[\begin{bmatrix}a_{11}& a_{12}& b_1\\-a_{12}& a_{11}& b_2\end{bmatrix}\] when \texttt{fullAffine=false}.
See also: \cvCppCross{getAffineTransform}, \cvCppCross{getPerspectiveTransform}, \cvCppCross{findHomography}
\cvCppFunc{estimateAffine3D}
Computes optimal affine transformation between two 3D point sets
\cvarg{contour}{The contour vertices, represented by \texttt{CV\_32SC2} or \texttt{CV\_32FC2} matrix, or by \texttt{vector<Point>} or \texttt{vector<Point2f>} converted to the matrix using \texttt{Mat(const vector<T>\&)} constructor.}
\end{description}
The function computes the contour area. Similarly to \cvCppCross{moments} the area is computed using the Green formula, thus the returned area and the number of non-zero pixels, if you draw the contour using \cvCppCross{drawContours} or \cvCppCross{fillPoly}, can be different.
\cvarg{points}{The input 2D point set, represented by \texttt{CV\_32SC2} or \texttt{CV\_32FC2} matrix, or by \texttt{vector<Point>} or \texttt{vector<Point2f>} converted to the matrix using \texttt{Mat(const vector<T>\&)} constructor.}
\cvarg{hull}{The output convex hull. It is either a vector of points that form the hull (must have the same type as the input points), or a vector of 0-based point indices of the hull points in the original array (since the set of convex hull points is a subset of the original point set).}
\cvarg{clockwise}{If true, the output convex hull will be oriented clockwise, otherwise it will be oriented counter-clockwise. Here, the usual screen coordinate system is assumed - the origin is at the top-left corner, x axis is oriented to the right, and y axis is oriented downwards.}
\end{description}
The functions find the convex hull of a 2D point set using Sklansky's algorithm \cite{Sklansky82} that has $O(N logN)$ or $O(N)$ complexity (where $N$ is the number of input points), depending on how the initial sorting is implemented (currently it is $O(N logN)$. See the OpenCV sample \texttt{convexhull.c} that demonstrates the use of the different function variants.
\cvarg{points}{The input 2D point set, represented by \texttt{CV\_32SC2} or \texttt{CV\_32FC2} matrix, or by \texttt{vector<Point>} or \texttt{vector<Point2f>} converted to the matrix using \texttt{Mat(const vector<T>\&)} constructor.}
\end{description}
The function calculates the ellipse that fits best
(in least-squares sense) a set of 2D points. It returns the rotated rectangle in which the ellipse is inscribed.
\cvCppFunc{fitLine}
Fits a line to a 2D or 3D point set.
\cvdefCpp{void fitLine( const Mat\& points, Vec4f\& line, int distType,\par
double param, double reps, double aeps );\newline
void fitLine( const Mat\& points, Vec6f\& line, int distType,\par
double param, double reps, double aeps );}
\begin{description}
\cvarg{points}{The input 2D point set, represented by \texttt{CV\_32SC2} or \texttt{CV\_32FC2} matrix, or by
\texttt{vector<Point>}, \texttt{vector<Point2f>}, \texttt{vector<Point3i>} or \texttt{vector<Point3f>} converted to the matrix by \texttt{Mat(const vector<T>\&)} constructor}
\cvarg{line}{The output line parameters. In the case of a 2d fitting,
it is a vector of 4 floats \texttt{(vx, vy,
x0, y0)} where \texttt{(vx, vy)} is a normalized vector collinear to the
line and \texttt{(x0, y0)} is some point on the line. in the case of a
3D fitting it is vector of 6 floats \texttt{(vx, vy, vz, x0, y0, z0)}
where \texttt{(vx, vy, vz)} is a normalized vector collinear to the line
and \texttt{(x0, y0, z0)} is some point on the line}
\cvarg{distType}{The distance used by the M-estimator (see the discussion)}
\cvarg{param}{Numerical parameter (\texttt{C}) for some types of distances, if 0 then some optimal value is chosen}
\cvarg{reps, aeps}{Sufficient accuracy for the radius (distance between the coordinate origin and the line) and angle, respectively; 0.01 would be a good default value for both.}
\end{description}
The functions \texttt{fitLine} fit a line to a 2D or 3D point set by minimizing $\sum_i \rho(r_i)$ where $r_i$ is the distance between the $i^{th}$ point and the line and $\rho(r)$ is a distance function, one of:
\begin{description}
\item[distType=CV\_DIST\_L2]
\[\rho(r)= r^2/2\quad\text{(the simplest and the fastest least-squares method)}\]
The algorithm is based on the M-estimator (\url{http://en.wikipedia.org/wiki/M-estimator}) technique, that iteratively fits the line using weighted least-squares algorithm and after each iteration the weights $w_i$ are adjusted to beinversely proportional to $\rho(r_i)$.
\cvarg{contour}{The tested contour, a matrix of type \texttt{CV\_32SC2} or \texttt{CV\_32FC2}, or \texttt{vector<Point>} or \texttt{vector<Point2f>} converted to the matrix using \texttt{Mat(const vector<T>\&)} constructor.}
\end{description}
The function tests whether the input contour is convex or not. The contour must be simple, i.e. without self-intersections, otherwise the function output is undefined.
\cvCppFunc{minAreaRect}
Finds the minimum area rotated rectangle enclosing a 2D point set.
\cvarg{points}{The input 2D point set, represented by \texttt{CV\_32SC2} or \texttt{CV\_32FC2} matrix, or by \texttt{vector<Point>} or \texttt{vector<Point2f>} converted to the matrix using \texttt{Mat(const vector<T>\&)} constructor.}
\end{description}
The function calculates and returns the minimum area bounding rectangle (possibly rotated) for the specified point set. See the OpenCV sample \texttt{minarea.c}
\cvCppFunc{minEnclosingCircle}
Finds the minimum area circle enclosing a 2D point set.
\cvarg{points}{The input 2D point set, represented by \texttt{CV\_32SC2} or \texttt{CV\_32FC2} matrix, or by \texttt{vector<Point>} or \texttt{vector<Point2f>} converted to the matrix using \texttt{Mat(const vector<T>\&)} constructor.}
\cvarg{center}{The output center of the circle}
\cvarg{radius}{The output radius of the circle}
\end{description}
The function finds the minimal enclosing circle of a 2D point set using iterative algorithm. See the OpenCV sample \texttt{minarea.c}
\cvarg{object1}{The first contour or grayscale image}
\cvarg{object2}{The second contour or grayscale image}
\cvarg{method}{Comparison method:
\texttt{CV\_CONTOUR\_MATCH\_I1},\\
\texttt{CV\_CONTOURS\_MATCH\_I2}\\
or
\texttt{CV\_CONTOURS\_MATCH\_I3} (see the discussion below)}
\cvarg{parameter}{Method-specific parameter (is not used now)}
\end{description}
The function compares two shapes. The 3 implemented methods all use Hu invariants (see \cvCppCross{HuMoments}) as following ($A$ denotes \texttt{object1}, $B$ denotes \texttt{object2}):
\cvarg{measureDist}{If true, the function estimates the signed distance from the point to the nearest contour edge; otherwise, the function only checks if the point is inside or not.}
\end{description}
The function determines whether the
point is inside a contour, outside, or lies on an edge (or coincides
with a vertex). It returns positive (inside), negative (outside) or zero (on an edge) value,
correspondingly. When \texttt{measureDist=false}, the return value
is +1, -1 and 0, respectively. Otherwise, the return value
it is a signed distance between the point and the nearest contour
edge.
Here is the sample output of the function, where each image pixel is tested against the contour.
