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Structural Analysis and Shape Descriptors
=========================================
.. index:: moments
moments
-----------
.. c:function:: Moments moments( const Mat& array, bool binaryImage=false )
Calculates all of the moments up to the third order of a polygon or rasterized shape.
where the class ``Moments`` is defined as: ::
class Moments
{
public:
Moments();
Moments(double m00, double m10, double m01, double m20, double m11,
double m02, double m30, double m21, double m12, double m03 );
Moments( const CvMoments& moments );
operator CvMoments() const;
// spatial moments
double m00, m10, m01, m20, m11, m02, m30, m21, m12, m03;
// central moments
double mu20, mu11, mu02, mu30, mu21, mu12, mu03;
// central normalized moments
double nu20, nu11, nu02, nu30, nu21, nu12, nu03;
};
:param array: A raster image (single-channel, 8-bit or floating-point 2D array) or an array
( :math:`1 \times N` or :math:`N \times 1` ) of 2D points ( ``Point`` or ``Point2f`` )
:param binaryImage: (For images only) If it is true, then all the non-zero image pixels are treated as 1's
The function computes moments, up to the 3rd order, of a vector shape or a rasterized shape.
In case of a raster image, the spatial moments
:math:`\texttt{Moments::m}_{ji}` are computed as:
.. math::
\texttt{m} _{ji}= \sum _{x,y} \left ( \texttt{array} (x,y) \cdot x^j \cdot y^i \right ),
the central moments
:math:`\texttt{Moments::mu}_{ji}` are computed as:
.. math::
\texttt{mu} _{ji}= \sum _{x,y} \left ( \texttt{array} (x,y) \cdot (x - \bar{x} )^j \cdot (y - \bar{y} )^i \right )
where
:math:`(\bar{x}, \bar{y})` is the mass center:
.. math::
\bar{x} = \frac{\texttt{m}_{10}}{\texttt{m}_{00}} , \; \bar{y} = \frac{\texttt{m}_{01}}{\texttt{m}_{00}}
and the normalized central moments
:math:`\texttt{Moments::nu}_{ij}` are computed as:
.. math::
\texttt{nu} _{ji}= \frac{\texttt{mu}_{ji}}{\texttt{m}_{00}^{(i+j)/2+1}} .
Note that
:math:`\texttt{mu}_{00}=\texttt{m}_{00}`,:math:`\texttt{nu}_{00}=1` :math:`\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
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:
:func:`contourArea`,:func:`arcLength`
.. index:: HuMoments
HuMoments
-------------
.. c:function:: void HuMoments( const Moments& moments, double h[7] )
Calculates the seven Hu invariants.
:param moments: The input moments, computed with :func:`moments`
:param h: The output Hu invariants
The function calculates the seven Hu invariants, see
http://en.wikipedia.org/wiki/Image_moment
, that are defined as:
.. math::
\begin{array}{l} h[0]= \eta _{20}+ \eta _{02} \\ h[1]=( \eta _{20}- \eta _{02})^{2}+4 \eta _{11}^{2} \\ h[2]=( \eta _{30}-3 \eta _{12})^{2}+ (3 \eta _{21}- \eta _{03})^{2} \\ h[3]=( \eta _{30}+ \eta _{12})^{2}+ ( \eta _{21}+ \eta _{03})^{2} \\ h[4]=( \eta _{30}-3 \eta _{12})( \eta _{30}+ \eta _{12})[( \eta _{30}+ \eta _{12})^{2}-3( \eta _{21}+ \eta _{03})^{2}]+(3 \eta _{21}- \eta _{03})( \eta _{21}+ \eta _{03})[3( \eta _{30}+ \eta _{12})^{2}-( \eta _{21}+ \eta _{03})^{2}] \\ h[5]=( \eta _{20}- \eta _{02})[( \eta _{30}+ \eta _{12})^{2}- ( \eta _{21}+ \eta _{03})^{2}]+4 \eta _{11}( \eta _{30}+ \eta _{12})( \eta _{21}+ \eta _{03}) \\ h[6]=(3 \eta _{21}- \eta _{03})( \eta _{21}+ \eta _{03})[3( \eta _{30}+ \eta _{12})^{2}-( \eta _{21}+ \eta _{03})^{2}]-( \eta _{30}-3 \eta _{12})( \eta _{21}+ \eta _{03})[3( \eta _{30}+ \eta _{12})^{2}-( \eta _{21}+ \eta _{03})^{2}] \\ \end{array}
where
:math:`\eta_{ji}` stand for
:math:`\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.
See also:
:func:`matchShapes`
.. index:: findContours
findContours
----------------
.. c:function:: void findContours( const Mat& image, vector<vector<Point> >& contours, vector<Vec4i>& hierarchy, int mode, int method, Point offset=Point())
.. c:function:: void findContours( const Mat& image, vector<vector<Point> >& contours, int mode, int method, Point offset=Point())
Finds the contours in a binary image.
:param 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 ``binary`` . You can use :func:`compare` , :func:`inRange` , :func:`threshold` , :func:`adaptiveThreshold` , :func:`Canny` etc. to create a binary image out of a grayscale or color one. The function modifies the ``image`` while extracting the contours
:param contours: The detected contours. Each contour is stored as a vector of points
:param 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 ``contours[i]`` , the elements ``hierarchy[i][0]`` , ``hiearchy[i][1]`` , ``hiearchy[i][2]`` , ``hiearchy[i][3]`` will be set to 0-based indices in ``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 ``i`` there is no next, previous, parent or nested contours, the corresponding elements of ``hierarchy[i]`` will be negative
:param mode: The contour retrieval mode
* **CV_RETR_EXTERNAL** retrieves only the extreme outer contours; It will set ``hierarchy[i][2]=hierarchy[i][3]=-1`` for all the contours
* **CV_RETR_LIST** retrieves all of the contours without establishing any hierarchical relationships
* **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
* **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 ``contours.c`` demo
:param method: The contour approximation method.
* **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
* **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
* **CV_CHAIN_APPROX_TC89_L1,CV_CHAIN_APPROX_TC89_KCOS** applies one of the flavors of the Teh-Chin chain approximation algorithm; see TehChin89
:param 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
The function retrieves contours from the
binary image using the algorithm
Suzuki85
. The contours are a useful tool for shape analysis and object detection and recognition. See ``squares.c`` in the OpenCV sample directory.
**Note:**
the source ``image`` is modified by this function.
.. index:: drawContours
drawContours
----------------
.. c:function:: void drawContours( Mat& image, const vector<vector<Point> >& contours, int contourIdx, const Scalar& color, int thickness=1, int lineType=8, const vector<Vec4i>& hierarchy=vector<Vec4i>(), int maxLevel=INT_MAX, Point offset=Point() )
Draws contours' outlines or filled contours.
:param image: The destination image
:param contours: All the input contours. Each contour is stored as a point vector
:param contourIdx: Indicates the contour to draw. If it is negative, all the contours are drawn
:param color: The contours' color
:param thickness: Thickness of lines the contours are drawn with.
If it is negative (e.g. ``thickness=CV_FILLED`` ), the contour interiors are
drawn.
:param lineType: The line connectivity; see :func:`line` description
:param hierarchy: The optional information about hierarchy. It is only needed if you want to draw only some of the contours (see ``maxLevel`` )
:param 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 ``hierarchy`` available.
:param offset: The optional contour shift parameter. Shift all the drawn contours by the specified :math:`\texttt{offset}=(dx,dy)`
The function draws contour outlines in the image if
:math:`\texttt{thickness} \ge 0` or fills the area bounded by the contours if
:math:`\texttt{thickness}<0` . Here is the example on how to retrieve connected components from the binary image and label them ::
#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
int idx = 0;
for( ; idx >= 0; idx = hierarchy[idx][0] )
{
Scalar color( rand()&255, rand()&255, rand()&255 );
drawContours( dst, contours, idx, color, CV_FILLED, 8, hierarchy );
}
namedWindow( "Components", 1 );
imshow( "Components", dst );
waitKey(0);
}
.. index:: approxPolyDP
approxPolyDP
----------------
.. c:function:: void approxPolyDP( const Mat& curve, vector<Point>& approxCurve, double epsilon, bool closed )
.. c:function:: void approxPolyDP( const Mat& curve, vector<Point2f>& approxCurve, double epsilon, bool closed )
Approximates polygonal curve(s) with the specified precision.
:param curve: The polygon or curve to approximate. Must be :math:`1 \times N` or :math:`N \times 1` matrix of type ``CV_32SC2`` or ``CV_32FC2`` . You can also convert ``vector<Point>`` or ``vector<Point2f`` to the matrix by calling ``Mat(const vector<T>&)`` constructor.
:param approxCurve: The result of the approximation; The type should match the type of the input curve
:param epsilon: Specifies the approximation accuracy. This is the maximum distance between the original curve and its approximation
:param closed: If true, the approximated curve is closed (i.e. its first and last vertices are connected), otherwise it's not
The functions ``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
http://en.wikipedia.org/wiki/Ramer-Douglas-Peucker_algorithm
.. index:: arcLength
arcLength
-------------
.. c:function:: double arcLength( const Mat& curve, bool closed )
Calculates a contour perimeter or a curve length.
:param curve: The input vector of 2D points, represented by ``CV_32SC2`` or ``CV_32FC2`` matrix, or by ``vector<Point>`` or ``vector<Point2f>`` converted to a matrix with ``Mat(const vector<T>&)`` constructor
:param closed: Indicates, whether the curve is closed or not
The function computes the curve length or the closed contour perimeter.
.. index:: boundingRect
boundingRect
----------------
.. c:function:: Rect boundingRect( const Mat& points )
Calculates the up-right bounding rectangle of a point set.
:param points: The input 2D point set, represented by ``CV_32SC2`` or ``CV_32FC2`` matrix, or by ``vector<Point>`` or ``vector<Point2f>`` converted to the matrix using ``Mat(const vector<T>&)`` constructor.
The function calculates and returns the minimal up-right bounding rectangle for the specified point set.
.. index:: estimateRigidTransform
estimateRigidTransform
--------------------------
.. c:function:: Mat estimateRigidTransform( const Mat& srcpt, const Mat& dstpt, bool fullAffine )
Computes optimal affine transformation between two 2D point sets
:param srcpt: The first input 2D point set
:param dst: The second input 2D point set of the same size and the same type as ``A``
:param 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)
The function finds the optimal affine transform
:math:`[A|b]` (a
:math:`2 \times 3` floating-point matrix) that approximates best the transformation from
:math:`\texttt{srcpt}_i` to
:math:`\texttt{dstpt}_i` :
.. math::
[A^*|b^*] = arg \min _{[A|b]} \sum _i \| \texttt{dstpt} _i - A { \texttt{srcpt} _i}^T - b \| ^2
where
:math:`[A|b]` can be either arbitrary (when ``fullAffine=true`` ) or have form
.. math::
\begin{bmatrix} a_{11} & a_{12} & b_1 \\ -a_{12} & a_{11} & b_2 \end{bmatrix}
when ``fullAffine=false`` .
See also:
:func:`getAffineTransform`,:func:`getPerspectiveTransform`,:func:`findHomography`
.. index:: estimateAffine3D
estimateAffine3D
--------------------
.. c:function:: int estimateAffine3D(const Mat& srcpt, const Mat& dstpt, Mat& out, vector<uchar>& outliers, double ransacThreshold = 3.0, double confidence = 0.99)
Computes optimal affine transformation between two 3D point sets
:param srcpt: The first input 3D point set
:param dstpt: The second input 3D point set
:param out: The output 3D affine transformation matrix :math:`3 \times 4`
:param outliers: The output vector indicating which points are outliers
:param ransacThreshold: The maximum reprojection error in RANSAC algorithm to consider a point an inlier
:param confidence: The confidence level, between 0 and 1, with which the matrix is estimated
The function estimates the optimal 3D affine transformation between two 3D point sets using RANSAC algorithm.
.. index:: contourArea
contourArea
---------------
.. c:function:: double contourArea( const Mat& contour )
Calculates the contour area
:param contour: The contour vertices, represented by ``CV_32SC2`` or ``CV_32FC2`` matrix, or by ``vector<Point>`` or ``vector<Point2f>`` converted to the matrix using ``Mat(const vector<T>&)`` constructor.
The function computes the contour area. Similarly to
:func:`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
:func:`drawContours` or
:func:`fillPoly` , can be different.
Here is a short example: ::
vector<Point> contour;
contour.push_back(Point2f(0, 0));
contour.push_back(Point2f(10, 0));
contour.push_back(Point2f(10, 10));
contour.push_back(Point2f(5, 4));
double area0 = contourArea(contour);
vector<Point> approx;
approxPolyDP(contour, approx, 5, true);
double area1 = contourArea(approx);
cout << "area0 =" << area0 << endl <<
"area1 =" << area1 << endl <<
"approx poly vertices" << approx.size() << endl;
.. index:: convexHull
convexHull
--------------
.. c:function:: void convexHull( const Mat& points, vector<int>& hull, bool clockwise=false )
.. c:function:: void convexHull( const Mat& points, vector<Point>& hull, bool clockwise=false )
.. c:function:: void convexHull( const Mat& points, vector<Point2f>& hull, bool clockwise=false )
Finds the convex hull of a point set.
:param points: The input 2D point set, represented by ``CV_32SC2`` or ``CV_32FC2`` matrix, or by ``vector<Point>`` or ``vector<Point2f>`` converted to the matrix using ``Mat(const vector<T>&)`` constructor.
:param 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).
:param 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.
The functions find the convex hull of a 2D point set using Sklansky's algorithm
Sklansky82
that has
:math:`O(N logN)` or
:math:`O(N)` complexity (where
:math:`N` is the number of input points), depending on how the initial sorting is implemented (currently it is
:math:`O(N logN)` . See the OpenCV sample ``convexhull.c`` that demonstrates the use of the different function variants.
.. index:: fitEllipse
fitEllipse
--------------
.. c:function:: RotatedRect fitEllipse( const Mat& points )
Fits an ellipse around a set of 2D points.
:param points: The input 2D point set, represented by ``CV_32SC2`` or ``CV_32FC2`` matrix, or by ``vector<Point>`` or ``vector<Point2f>`` converted to the matrix using ``Mat(const vector<T>&)`` constructor.
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.
.. index:: fitLine
fitLine
-----------
.. c:function:: void fitLine( const Mat& points, Vec4f& line, int distType, double param, double reps, double aeps )
.. c:function:: void fitLine( const Mat& points, Vec6f& line, int distType, double param, double reps, double aeps )
Fits a line to a 2D or 3D point set.
:param points: The input 2D point set, represented by ``CV_32SC2`` or ``CV_32FC2`` matrix, or by ``vector<Point>`` , ``vector<Point2f>`` , ``vector<Point3i>`` or ``vector<Point3f>`` converted to the matrix by ``Mat(const vector<T>&)`` constructor
:param line: The output line parameters. In the case of a 2d fitting,
it is a vector of 4 floats ``(vx, vy, x0, y0)`` where ``(vx, vy)`` is a normalized vector collinear to the
line and ``(x0, y0)`` is some point on the line. in the case of a
3D fitting it is vector of 6 floats ``(vx, vy, vz, x0, y0, z0)`` where ``(vx, vy, vz)`` is a normalized vector collinear to the line and ``(x0, y0, z0)`` is some point on the line
:param distType: The distance used by the M-estimator (see the discussion)
:param param: Numerical parameter ( ``C`` ) for some types of distances, if 0 then some optimal value is chosen
:param 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.
The functions ``fitLine`` fit a line to a 2D or 3D point set by minimizing
:math:`\sum_i \rho(r_i)` where
:math:`r_i` is the distance between the
:math:`i^{th}` point and the line and
:math:`\rho(r)` is a distance function, one of:
* distType=CV\_DIST\_L2
.. math::
\rho (r) = r^2/2 \quad \text{(the simplest and the fastest least-squares method)}
* distType=CV\_DIST\_L1
.. math::
\rho (r) = r
* distType=CV\_DIST\_L12
.. math::
\rho (r) = 2 \cdot ( \sqrt{1 + \frac{r^2}{2}} - 1)
* distType=CV\_DIST\_FAIR
.. math::
\rho \left (r \right ) = C^2 \cdot \left ( \frac{r}{C} - \log{\left(1 + \frac{r}{C}\right)} \right ) \quad \text{where} \quad C=1.3998
* distType=CV\_DIST\_WELSCH
.. math::
\rho \left (r \right ) = \frac{C^2}{2} \cdot \left ( 1 - \exp{\left(-\left(\frac{r}{C}\right)^2\right)} \right ) \quad \text{where} \quad C=2.9846
* distType=CV\_DIST\_HUBER
.. math::
\rho (r) = \fork{r^2/2}{if $r < C$}{C \cdot (r-C/2)}{otherwise} \quad \text{where} \quad C=1.345
The algorithm is based on the M-estimator (
http://en.wikipedia.org/wiki/M-estimator
) technique, that iteratively fits the line using weighted least-squares algorithm and after each iteration the weights
:math:`w_i` are adjusted to beinversely proportional to
:math:`\rho(r_i)` .
.. index:: isContourConvex
isContourConvex
-------------------
.. c:function:: bool isContourConvex( const Mat& contour )
Tests contour convexity.
:param contour: The tested contour, a matrix of type ``CV_32SC2`` or ``CV_32FC2`` , or ``vector<Point>`` or ``vector<Point2f>`` converted to the matrix using ``Mat(const vector<T>&)`` constructor.
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.
.. index:: minAreaRect
minAreaRect
---------------
.. c:function:: RotatedRect minAreaRect( const Mat& points )
Finds the minimum area rotated rectangle enclosing a 2D point set.
:param points: The input 2D point set, represented by ``CV_32SC2`` or ``CV_32FC2`` matrix, or by ``vector<Point>`` or ``vector<Point2f>`` converted to the matrix using ``Mat(const vector<T>&)`` constructor.
The function calculates and returns the minimum area bounding rectangle (possibly rotated) for the specified point set. See the OpenCV sample ``minarea.c``
.. index:: minEnclosingCircle
minEnclosingCircle
----------------------
.. c:function:: void minEnclosingCircle( const Mat& points, Point2f& center, float& radius )
Finds the minimum area circle enclosing a 2D point set.
:param points: The input 2D point set, represented by ``CV_32SC2`` or ``CV_32FC2`` matrix, or by ``vector<Point>`` or ``vector<Point2f>`` converted to the matrix using ``Mat(const vector<T>&)`` constructor.
:param center: The output center of the circle
:param radius: The output radius of the circle
The function finds the minimal enclosing circle of a 2D point set using iterative algorithm. See the OpenCV sample ``minarea.c``
.. index:: matchShapes
matchShapes
---------------
.. c:function:: double matchShapes( const Mat& object1, const Mat& object2, int method, double parameter=0 )
Compares two shapes.
:param object1: The first contour or grayscale image
:param object2: The second contour or grayscale image
:param method: Comparison method: ``CV_CONTOUR_MATCH_I1`` , \ ``CV_CONTOURS_MATCH_I2`` \
or ``CV_CONTOURS_MATCH_I3`` (see the discussion below)
:param parameter: Method-specific parameter (is not used now)
The function compares two shapes. The 3 implemented methods all use Hu invariants (see
:func:`HuMoments` ) as following (
:math:`A` denotes ``object1``,:math:`B` denotes ``object2`` ):
* method=CV\_CONTOUR\_MATCH\_I1
.. math::
I_1(A,B) = \sum _{i=1...7} \left | \frac{1}{m^A_i} - \frac{1}{m^B_i} \right |
* method=CV\_CONTOUR\_MATCH\_I2
.. math::
I_2(A,B) = \sum _{i=1...7} \left | m^A_i - m^B_i \right |
* method=CV\_CONTOUR\_MATCH\_I3
.. math::
I_3(A,B) = \sum _{i=1...7} \frac{ \left| m^A_i - m^B_i \right| }{ \left| m^A_i \right| }
where
.. math::
\begin{array}{l} m^A_i = \mathrm{sign} (h^A_i) \cdot \log{h^A_i} \\ m^B_i = \mathrm{sign} (h^B_i) \cdot \log{h^B_i} \end{array}
and
:math:`h^A_i, h^B_i` are the Hu moments of
:math:`A` and
:math:`B` respectively.
.. index:: pointPolygonTest
pointPolygonTest
--------------------
.. c:function:: double pointPolygonTest( const Mat& contour, Point2f pt, bool measureDist )
Performs point-in-contour test.
:param contour: The input contour
:param pt: The point tested against the contour
:param 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.
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 ``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.
.. image:: pics/pointpolygon.png
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