OpenCV
4.1.1
Open Source Computer Vision

Namespaces  
cv::traits  
Classes  
class  cv::GeneralizedHough 
finds arbitrary template in the grayscale image using Generalized Hough Transform More...  
class  cv::GeneralizedHoughBallard 
finds arbitrary template in the grayscale image using Generalized Hough Transform More...  
class  cv::GeneralizedHoughGuil 
finds arbitrary template in the grayscale image using Generalized Hough Transform More...  
class  cv::Moments 
struct returned by cv::moments More...  
Functions  
void  cv::approxPolyDP (InputArray curve, OutputArray approxCurve, double epsilon, bool closed) 
Approximates a polygonal curve(s) with the specified precision. More...  
double  cv::arcLength (InputArray curve, bool closed) 
Calculates a contour perimeter or a curve length. More...  
Rect  cv::boundingRect (InputArray array) 
Calculates the upright bounding rectangle of a point set or nonzero pixels of grayscale image. More...  
void  cv::boxPoints (RotatedRect box, OutputArray points) 
Finds the four vertices of a rotated rect. Useful to draw the rotated rectangle. More...  
int  cv::connectedComponents (InputArray image, OutputArray labels, int connectivity, int ltype, int ccltype) 
computes the connected components labeled image of boolean image More...  
int  cv::connectedComponents (InputArray image, OutputArray labels, int connectivity=8, int ltype=CV_32S) 
int  cv::connectedComponentsWithStats (InputArray image, OutputArray labels, OutputArray stats, OutputArray centroids, int connectivity, int ltype, int ccltype) 
computes the connected components labeled image of boolean image and also produces a statistics output for each label More...  
int  cv::connectedComponentsWithStats (InputArray image, OutputArray labels, OutputArray stats, OutputArray centroids, int connectivity=8, int ltype=CV_32S) 
double  cv::contourArea (InputArray contour, bool oriented=false) 
Calculates a contour area. More...  
void  cv::convexHull (InputArray points, OutputArray hull, bool clockwise=false, bool returnPoints=true) 
Finds the convex hull of a point set. More...  
void  cv::convexityDefects (InputArray contour, InputArray convexhull, OutputArray convexityDefects) 
Finds the convexity defects of a contour. More...  
Ptr< GeneralizedHoughBallard >  cv::createGeneralizedHoughBallard () 
Creates a smart pointer to a cv::GeneralizedHoughBallard class and initializes it. More...  
Ptr< GeneralizedHoughGuil >  cv::createGeneralizedHoughGuil () 
Creates a smart pointer to a cv::GeneralizedHoughGuil class and initializes it. More...  
void  cv::findContours (InputArray image, OutputArrayOfArrays contours, OutputArray hierarchy, int mode, int method, Point offset=Point()) 
Finds contours in a binary image. More...  
void  cv::findContours (InputArray image, OutputArrayOfArrays contours, int mode, int method, Point offset=Point()) 
RotatedRect  cv::fitEllipse (InputArray points) 
Fits an ellipse around a set of 2D points. More...  
RotatedRect  cv::fitEllipseAMS (InputArray points) 
Fits an ellipse around a set of 2D points. More...  
RotatedRect  cv::fitEllipseDirect (InputArray points) 
Fits an ellipse around a set of 2D points. More...  
void  cv::fitLine (InputArray points, OutputArray line, int distType, double param, double reps, double aeps) 
Fits a line to a 2D or 3D point set. More...  
void  cv::HuMoments (const Moments &moments, double hu[7]) 
Calculates seven Hu invariants. More...  
void  cv::HuMoments (const Moments &m, OutputArray hu) 
float  cv::intersectConvexConvex (InputArray _p1, InputArray _p2, OutputArray _p12, bool handleNested=true) 
finds intersection of two convex polygons More...  
bool  cv::isContourConvex (InputArray contour) 
Tests a contour convexity. More...  
double  cv::matchShapes (InputArray contour1, InputArray contour2, int method, double parameter) 
Compares two shapes. More...  
RotatedRect  cv::minAreaRect (InputArray points) 
Finds a rotated rectangle of the minimum area enclosing the input 2D point set. More...  
void  cv::minEnclosingCircle (InputArray points, Point2f ¢er, float &radius) 
Finds a circle of the minimum area enclosing a 2D point set. More...  
double  cv::minEnclosingTriangle (InputArray points, OutputArray triangle) 
Finds a triangle of minimum area enclosing a 2D point set and returns its area. More...  
Moments  cv::moments (InputArray array, bool binaryImage=false) 
Calculates all of the moments up to the third order of a polygon or rasterized shape. More...  
double  cv::pointPolygonTest (InputArray contour, Point2f pt, bool measureDist) 
Performs a pointincontour test. More...  
int  cv::rotatedRectangleIntersection (const RotatedRect &rect1, const RotatedRect &rect2, OutputArray intersectingRegion) 
Finds out if there is any intersection between two rotated rectangles. More...  
#include <opencv2/imgproc.hpp>
connected components algorithm
#include <opencv2/imgproc.hpp>
connected components algorithm output formats
#include <opencv2/imgproc.hpp>
the contour approximation algorithm
Enumerator  

CHAIN_APPROX_NONE Python: cv.CHAIN_APPROX_NONE  stores absolutely all the contour points. That is, any 2 subsequent points (x1,y1) and (x2,y2) of the contour will be either horizontal, vertical or diagonal neighbors, that is, max(abs(x1x2),abs(y2y1))==1. 
CHAIN_APPROX_SIMPLE Python: cv.CHAIN_APPROX_SIMPLE  compresses horizontal, vertical, and diagonal segments and leaves only their end points. For example, an upright rectangular contour is encoded with 4 points. 
CHAIN_APPROX_TC89_L1 Python: cv.CHAIN_APPROX_TC89_L1  applies one of the flavors of the TehChin chain approximation algorithm [212] 
CHAIN_APPROX_TC89_KCOS Python: cv.CHAIN_APPROX_TC89_KCOS  applies one of the flavors of the TehChin chain approximation algorithm [212] 
#include <opencv2/imgproc.hpp>
types of intersection between rectangles
enum cv::RetrievalModes 
#include <opencv2/imgproc.hpp>
mode of the contour retrieval algorithm
enum cv::ShapeMatchModes 
#include <opencv2/imgproc.hpp>
Shape matching methods.
\(A\) denotes object1, \(B\) denotes object2
\(\begin{array}{l} m^A_i = \mathrm{sign} (h^A_i) \cdot \log{h^A_i} \\ m^B_i = \mathrm{sign} (h^B_i) \cdot \log{h^B_i} \end{array}\)
and \(h^A_i, h^B_i\) are the Hu moments of \(A\) and \(B\) , respectively.
void cv::approxPolyDP  (  InputArray  curve, 
OutputArray  approxCurve,  
double  epsilon,  
bool  closed  
) 
Python:  

approxCurve  =  cv.approxPolyDP(  curve, epsilon, closed[, approxCurve]  ) 
#include <opencv2/imgproc.hpp>
Approximates a polygonal curve(s) with the specified precision.
The function cv::approxPolyDP approximates 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 uses the DouglasPeucker algorithm http://en.wikipedia.org/wiki/RamerDouglasPeucker_algorithm
curve  Input vector of a 2D point stored in std::vector or Mat 
approxCurve  Result of the approximation. The type should match the type of the input curve. 
epsilon  Parameter specifying the approximation accuracy. This is the maximum distance between the original curve and its approximation. 
closed  If true, the approximated curve is closed (its first and last vertices are connected). Otherwise, it is not closed. 
double cv::arcLength  (  InputArray  curve, 
bool  closed  
) 
Python:  

retval  =  cv.arcLength(  curve, closed  ) 
#include <opencv2/imgproc.hpp>
Calculates a contour perimeter or a curve length.
The function computes a curve length or a closed contour perimeter.
curve  Input vector of 2D points, stored in std::vector or Mat. 
closed  Flag indicating whether the curve is closed or not. 
Rect cv::boundingRect  (  InputArray  array  ) 
Python:  

retval  =  cv.boundingRect(  array  ) 
#include <opencv2/imgproc.hpp>
Calculates the upright bounding rectangle of a point set or nonzero pixels of grayscale image.
The function calculates and returns the minimal upright bounding rectangle for the specified point set or nonzero pixels of grayscale image.
array  Input grayscale image or 2D point set, stored in std::vector or Mat. 
void cv::boxPoints  (  RotatedRect  box, 
OutputArray  points  
) 
Python:  

points  =  cv.boxPoints(  box[, points]  ) 
#include <opencv2/imgproc.hpp>
Finds the four vertices of a rotated rect. Useful to draw the rotated rectangle.
The function finds the four vertices of a rotated rectangle. This function is useful to draw the rectangle. In C++, instead of using this function, you can directly use RotatedRect::points method. Please visit the tutorial on Creating Bounding rotated boxes and ellipses for contours for more information.
box  The input rotated rectangle. It may be the output of 
points  The output array of four vertices of rectangles. 
int cv::connectedComponents  (  InputArray  image, 
OutputArray  labels,  
int  connectivity,  
int  ltype,  
int  ccltype  
) 
Python:  

retval, labels  =  cv.connectedComponents(  image[, labels[, connectivity[, ltype]]]  )  
retval, labels  =  cv.connectedComponentsWithAlgorithm(  image, connectivity, ltype, ccltype[, labels]  ) 
#include <opencv2/imgproc.hpp>
computes the connected components labeled image of boolean image
image with 4 or 8 way connectivity  returns N, the total number of labels [0, N1] where 0 represents the background label. ltype specifies the output label image type, an important consideration based on the total number of labels or alternatively the total number of pixels in the source image. ccltype specifies the connected components labeling algorithm to use, currently Grana (BBDT) and Wu's (SAUF) algorithms are supported, see the ConnectedComponentsAlgorithmsTypes for details. Note that SAUF algorithm forces a row major ordering of labels while BBDT does not. This function uses parallel version of both Grana and Wu's algorithms if at least one allowed parallel framework is enabled and if the rows of the image are at least twice the number returned by getNumberOfCPUs.
image  the 8bit singlechannel image to be labeled 
labels  destination labeled image 
connectivity  8 or 4 for 8way or 4way connectivity respectively 
ltype  output image label type. Currently CV_32S and CV_16U are supported. 
ccltype  connected components algorithm type (see the ConnectedComponentsAlgorithmsTypes). 
int cv::connectedComponents  (  InputArray  image, 
OutputArray  labels,  
int  connectivity = 8 , 

int  ltype = CV_32S 

) 
Python:  

retval, labels  =  cv.connectedComponents(  image[, labels[, connectivity[, ltype]]]  )  
retval, labels  =  cv.connectedComponentsWithAlgorithm(  image, connectivity, ltype, ccltype[, labels]  ) 
#include <opencv2/imgproc.hpp>
This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.
image  the 8bit singlechannel image to be labeled 
labels  destination labeled image 
connectivity  8 or 4 for 8way or 4way connectivity respectively 
ltype  output image label type. Currently CV_32S and CV_16U are supported. 
int cv::connectedComponentsWithStats  (  InputArray  image, 
OutputArray  labels,  
OutputArray  stats,  
OutputArray  centroids,  
int  connectivity,  
int  ltype,  
int  ccltype  
) 
Python:  

retval, labels, stats, centroids  =  cv.connectedComponentsWithStats(  image[, labels[, stats[, centroids[, connectivity[, ltype]]]]]  )  
retval, labels, stats, centroids  =  cv.connectedComponentsWithStatsWithAlgorithm(  image, connectivity, ltype, ccltype[, labels[, stats[, centroids]]]  ) 
#include <opencv2/imgproc.hpp>
computes the connected components labeled image of boolean image and also produces a statistics output for each label
image with 4 or 8 way connectivity  returns N, the total number of labels [0, N1] where 0 represents the background label. ltype specifies the output label image type, an important consideration based on the total number of labels or alternatively the total number of pixels in the source image. ccltype specifies the connected components labeling algorithm to use, currently Grana's (BBDT) and Wu's (SAUF) algorithms are supported, see the ConnectedComponentsAlgorithmsTypes for details. Note that SAUF algorithm forces a row major ordering of labels while BBDT does not. This function uses parallel version of both Grana and Wu's algorithms (statistics included) if at least one allowed parallel framework is enabled and if the rows of the image are at least twice the number returned by getNumberOfCPUs.
image  the 8bit singlechannel image to be labeled 
labels  destination labeled image 
stats  statistics output for each label, including the background label, see below for available statistics. Statistics are accessed via stats(label, COLUMN) where COLUMN is one of ConnectedComponentsTypes. The data type is CV_32S. 
centroids  centroid output for each label, including the background label. Centroids are accessed via centroids(label, 0) for x and centroids(label, 1) for y. The data type CV_64F. 
connectivity  8 or 4 for 8way or 4way connectivity respectively 
ltype  output image label type. Currently CV_32S and CV_16U are supported. 
ccltype  connected components algorithm type (see ConnectedComponentsAlgorithmsTypes). 
int cv::connectedComponentsWithStats  (  InputArray  image, 
OutputArray  labels,  
OutputArray  stats,  
OutputArray  centroids,  
int  connectivity = 8 , 

int  ltype = CV_32S 

) 
Python:  

retval, labels, stats, centroids  =  cv.connectedComponentsWithStats(  image[, labels[, stats[, centroids[, connectivity[, ltype]]]]]  )  
retval, labels, stats, centroids  =  cv.connectedComponentsWithStatsWithAlgorithm(  image, connectivity, ltype, ccltype[, labels[, stats[, centroids]]]  ) 
#include <opencv2/imgproc.hpp>
This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.
image  the 8bit singlechannel image to be labeled 
labels  destination labeled image 
stats  statistics output for each label, including the background label, see below for available statistics. Statistics are accessed via stats(label, COLUMN) where COLUMN is one of ConnectedComponentsTypes. The data type is CV_32S. 
centroids  centroid output for each label, including the background label. Centroids are accessed via centroids(label, 0) for x and centroids(label, 1) for y. The data type CV_64F. 
connectivity  8 or 4 for 8way or 4way connectivity respectively 
ltype  output image label type. Currently CV_32S and CV_16U are supported. 
double cv::contourArea  (  InputArray  contour, 
bool  oriented = false 

) 
Python:  

retval  =  cv.contourArea(  contour[, oriented]  ) 
#include <opencv2/imgproc.hpp>
Calculates a contour area.
The function computes a contour area. Similarly to moments , the area is computed using the Green formula. Thus, the returned area and the number of nonzero pixels, if you draw the contour using drawContours or fillPoly , can be different. Also, the function will most certainly give a wrong results for contours with selfintersections.
Example:
contour  Input vector of 2D points (contour vertices), stored in std::vector or Mat. 
oriented  Oriented area flag. If it is true, the function returns a signed area value, depending on the contour orientation (clockwise or counterclockwise). Using this feature you can determine orientation of a contour by taking the sign of an area. By default, the parameter is false, which means that the absolute value is returned. 
void cv::convexHull  (  InputArray  points, 
OutputArray  hull,  
bool  clockwise = false , 

bool  returnPoints = true 

) 
Python:  

hull  =  cv.convexHull(  points[, hull[, clockwise[, returnPoints]]]  ) 
#include <opencv2/imgproc.hpp>
Finds the convex hull of a point set.
The function cv::convexHull finds the convex hull of a 2D point set using the Sklansky's algorithm [198] that has O(N logN) complexity in the current implementation.
points  Input 2D point set, stored in std::vector or Mat. 
hull  Output convex hull. It is either an integer vector of indices or vector of points. In the first case, the hull elements are 0based indices of the convex hull points in the original array (since the set of convex hull points is a subset of the original point set). In the second case, hull elements are the convex hull points themselves. 
clockwise  Orientation flag. If it is true, the output convex hull is oriented clockwise. Otherwise, it is oriented counterclockwise. The assumed coordinate system has its X axis pointing to the right, and its Y axis pointing upwards. 
returnPoints  Operation flag. In case of a matrix, when the flag is true, the function returns convex hull points. Otherwise, it returns indices of the convex hull points. When the output array is std::vector, the flag is ignored, and the output depends on the type of the vector: std::vector<int> implies returnPoints=false, std::vector<Point> implies returnPoints=true. 
points
and hull
should be different arrays, inplace processing isn't supported.Check the corresponding tutorial for more details.
useful links:
https://www.learnopencv.com/convexhullusingopencvinpythonandc/
void cv::convexityDefects  (  InputArray  contour, 
InputArray  convexhull,  
OutputArray  convexityDefects  
) 
Python:  

convexityDefects  =  cv.convexityDefects(  contour, convexhull[, convexityDefects]  ) 
#include <opencv2/imgproc.hpp>
Finds the convexity defects of a contour.
The figure below displays convexity defects of a hand contour:
contour  Input contour. 
convexhull  Convex hull obtained using convexHull that should contain indices of the contour points that make the hull. 
convexityDefects  The output vector of convexity defects. In C++ and the new Python/Java interface each convexity defect is represented as 4element integer vector (a.k.a. Vec4i): (start_index, end_index, farthest_pt_index, fixpt_depth), where indices are 0based indices in the original contour of the convexity defect beginning, end and the farthest point, and fixpt_depth is fixedpoint approximation (with 8 fractional bits) of the distance between the farthest contour point and the hull. That is, to get the floatingpoint value of the depth will be fixpt_depth/256.0. 
Ptr<GeneralizedHoughBallard> cv::createGeneralizedHoughBallard  (  ) 
Python:  

retval  =  cv.createGeneralizedHoughBallard(  ) 
#include <opencv2/imgproc.hpp>
Creates a smart pointer to a cv::GeneralizedHoughBallard class and initializes it.
Ptr<GeneralizedHoughGuil> cv::createGeneralizedHoughGuil  (  ) 
Python:  

retval  =  cv.createGeneralizedHoughGuil(  ) 
#include <opencv2/imgproc.hpp>
Creates a smart pointer to a cv::GeneralizedHoughGuil class and initializes it.
void cv::findContours  (  InputArray  image, 
OutputArrayOfArrays  contours,  
OutputArray  hierarchy,  
int  mode,  
int  method,  
Point  offset = Point() 

) 
Python:  

contours, hierarchy  =  cv.findContours(  image, mode, method[, contours[, hierarchy[, offset]]]  ) 
#include <opencv2/imgproc.hpp>
Finds contours in a binary image.
The function retrieves contours from the binary image using the algorithm [204] . The contours are a useful tool for shape analysis and object detection and recognition. See squares.cpp in the OpenCV sample directory.
image  Source, an 8bit singlechannel image. Nonzero pixels are treated as 1's. Zero pixels remain 0's, so the image is treated as binary . You can use compare, inRange, threshold , adaptiveThreshold, Canny, and others to create a binary image out of a grayscale or color one. If mode equals to RETR_CCOMP or RETR_FLOODFILL, the input can also be a 32bit integer image of labels (CV_32SC1). 
contours  Detected contours. Each contour is stored as a vector of points (e.g. std::vector<std::vector<cv::Point> >). 
hierarchy  Optional output vector (e.g. std::vector<cv::Vec4i>), containing information about the image topology. It has as many elements as the number of contours. For each ith contour contours[i], the elements hierarchy[i][0] , hierarchy[i][1] , hierarchy[i][2] , and hierarchy[i][3] are set to 0based 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 the contour i there are no next, previous, parent, or nested contours, the corresponding elements of hierarchy[i] will be negative. 
mode  Contour retrieval mode, see RetrievalModes 
method  Contour approximation method, see ContourApproximationModes 
offset  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. 
void cv::findContours  (  InputArray  image, 
OutputArrayOfArrays  contours,  
int  mode,  
int  method,  
Point  offset = Point() 

) 
Python:  

contours, hierarchy  =  cv.findContours(  image, mode, method[, contours[, hierarchy[, offset]]]  ) 
#include <opencv2/imgproc.hpp>
This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.
RotatedRect cv::fitEllipse  (  InputArray  points  ) 
Python:  

retval  =  cv.fitEllipse(  points  ) 
#include <opencv2/imgproc.hpp>
Fits an ellipse around a set of 2D points.
The function calculates the ellipse that fits (in a leastsquares sense) a set of 2D points best of all. It returns the rotated rectangle in which the ellipse is inscribed. The first algorithm described by [69] is used. Developer should keep in mind that it is possible that the returned ellipse/rotatedRect data contains negative indices, due to the data points being close to the border of the containing Mat element.
points  Input 2D point set, stored in std::vector<> or Mat 
RotatedRect cv::fitEllipseAMS  (  InputArray  points  ) 
Python:  

retval  =  cv.fitEllipseAMS(  points  ) 
#include <opencv2/imgproc.hpp>
Fits an ellipse around a set of 2D points.
The function calculates the ellipse that fits a set of 2D points. It returns the rotated rectangle in which the ellipse is inscribed. The Approximate Mean Square (AMS) proposed by [211] is used.
For an ellipse, this basis set is \( \chi= \left(x^2, x y, y^2, x, y, 1\right) \), which is a set of six free coefficients \( A^T=\left\{A_{\text{xx}},A_{\text{xy}},A_{\text{yy}},A_x,A_y,A_0\right\} \). However, to specify an ellipse, all that is needed is five numbers; the major and minor axes lengths \( (a,b) \), the position \( (x_0,y_0) \), and the orientation \( \theta \). This is because the basis set includes lines, quadratics, parabolic and hyperbolic functions as well as elliptical functions as possible fits. If the fit is found to be a parabolic or hyperbolic function then the standard fitEllipse method is used. The AMS method restricts the fit to parabolic, hyperbolic and elliptical curves by imposing the condition that \( A^T ( D_x^T D_x + D_y^T D_y) A = 1 \) where the matrices \( Dx \) and \( Dy \) are the partial derivatives of the design matrix \( D \) with respect to x and y. The matrices are formed row by row applying the following to each of the points in the set:
\begin{align*} D(i,:)&=\left\{x_i^2, x_i y_i, y_i^2, x_i, y_i, 1\right\} & D_x(i,:)&=\left\{2 x_i,y_i,0,1,0,0\right\} & D_y(i,:)&=\left\{0,x_i,2 y_i,0,1,0\right\} \end{align*}
The AMS method minimizes the cost function
\begin{equation*} \epsilon ^2=\frac{ A^T D^T D A }{ A^T (D_x^T D_x + D_y^T D_y) A^T } \end{equation*}
The minimum cost is found by solving the generalized eigenvalue problem.
\begin{equation*} D^T D A = \lambda \left( D_x^T D_x + D_y^T D_y\right) A \end{equation*}
points  Input 2D point set, stored in std::vector<> or Mat 
RotatedRect cv::fitEllipseDirect  (  InputArray  points  ) 
Python:  

retval  =  cv.fitEllipseDirect(  points  ) 
#include <opencv2/imgproc.hpp>
Fits an ellipse around a set of 2D points.
The function calculates the ellipse that fits a set of 2D points. It returns the rotated rectangle in which the ellipse is inscribed. The Direct least square (Direct) method by [70] is used.
For an ellipse, this basis set is \( \chi= \left(x^2, x y, y^2, x, y, 1\right) \), which is a set of six free coefficients \( A^T=\left\{A_{\text{xx}},A_{\text{xy}},A_{\text{yy}},A_x,A_y,A_0\right\} \). However, to specify an ellipse, all that is needed is five numbers; the major and minor axes lengths \( (a,b) \), the position \( (x_0,y_0) \), and the orientation \( \theta \). This is because the basis set includes lines, quadratics, parabolic and hyperbolic functions as well as elliptical functions as possible fits. The Direct method confines the fit to ellipses by ensuring that \( 4 A_{xx} A_{yy} A_{xy}^2 > 0 \). The condition imposed is that \( 4 A_{xx} A_{yy} A_{xy}^2=1 \) which satisfies the inequality and as the coefficients can be arbitrarily scaled is not overly restrictive.
\begin{equation*} \epsilon ^2= A^T D^T D A \quad \text{with} \quad A^T C A =1 \quad \text{and} \quad C=\left(\begin{matrix} 0 & 0 & 2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 2 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix} \right) \end{equation*}
The minimum cost is found by solving the generalized eigenvalue problem.
\begin{equation*} D^T D A = \lambda \left( C\right) A \end{equation*}
The system produces only one positive eigenvalue \( \lambda\) which is chosen as the solution with its eigenvector \(\mathbf{u}\). These are used to find the coefficients
\begin{equation*} A = \sqrt{\frac{1}{\mathbf{u}^T C \mathbf{u}}} \mathbf{u} \end{equation*}
The scaling factor guarantees that \(A^T C A =1\).
points  Input 2D point set, stored in std::vector<> or Mat 
void cv::fitLine  (  InputArray  points, 
OutputArray  line,  
int  distType,  
double  param,  
double  reps,  
double  aeps  
) 
Python:  

line  =  cv.fitLine(  points, distType, param, reps, aeps[, line]  ) 
#include <opencv2/imgproc.hpp>
Fits a line to a 2D or 3D point set.
The function fitLine fits a line to a 2D or 3D point set by minimizing \(\sum_i \rho(r_i)\) where \(r_i\) is a distance between the \(i^{th}\) point, the line and \(\rho(r)\) is a distance function, one of the following:
\[\rho (r) = r^2/2 \quad \text{(the simplest and the fastest leastsquares method)}\]
\[\rho (r) = r\]
\[\rho (r) = 2 \cdot ( \sqrt{1 + \frac{r^2}{2}}  1)\]
\[\rho \left (r \right ) = C^2 \cdot \left ( \frac{r}{C}  \log{\left(1 + \frac{r}{C}\right)} \right ) \quad \text{where} \quad C=1.3998\]
\[\rho \left (r \right ) = \frac{C^2}{2} \cdot \left ( 1  \exp{\left(\left(\frac{r}{C}\right)^2\right)} \right ) \quad \text{where} \quad C=2.9846\]
\[\rho (r) = \fork{r^2/2}{if \(r < C\)}{C \cdot (rC/2)}{otherwise} \quad \text{where} \quad C=1.345\]
The algorithm is based on the Mestimator ( http://en.wikipedia.org/wiki/Mestimator ) technique that iteratively fits the line using the weighted leastsquares algorithm. After each iteration the weights \(w_i\) are adjusted to be inversely proportional to \(\rho(r_i)\) .
points  Input vector of 2D or 3D points, stored in std::vector<> or Mat. 
line  Output line parameters. In case of 2D fitting, it should be a vector of 4 elements (like Vec4f)  (vx, vy, x0, y0), where (vx, vy) is a normalized vector collinear to the line and (x0, y0) is a point on the line. In case of 3D fitting, it should be a vector of 6 elements (like Vec6f)  (vx, vy, vz, x0, y0, z0), where (vx, vy, vz) is a normalized vector collinear to the line and (x0, y0, z0) is a point on the line. 
distType  Distance used by the Mestimator, see DistanceTypes 
param  Numerical parameter ( C ) for some types of distances. If it is 0, an optimal value is chosen. 
reps  Sufficient accuracy for the radius (distance between the coordinate origin and the line). 
aeps  Sufficient accuracy for the angle. 0.01 would be a good default value for reps and aeps. 
void cv::HuMoments  (  const Moments &  moments, 
double  hu[7]  
) 
Python:  

hu  =  cv.HuMoments(  m[, hu]  ) 
#include <opencv2/imgproc.hpp>
Calculates seven Hu invariants.
The function calculates seven Hu invariants (introduced in [100]; see also http://en.wikipedia.org/wiki/Image_moment) defined as:
\[\begin{array}{l} hu[0]= \eta _{20}+ \eta _{02} \\ hu[1]=( \eta _{20} \eta _{02})^{2}+4 \eta _{11}^{2} \\ hu[2]=( \eta _{30}3 \eta _{12})^{2}+ (3 \eta _{21} \eta _{03})^{2} \\ hu[3]=( \eta _{30}+ \eta _{12})^{2}+ ( \eta _{21}+ \eta _{03})^{2} \\ hu[4]=( \eta _{30}3 \eta _{12})( \eta _{30}+ \eta _{12})[( \eta _{30}+ \eta _{12})^{2}3( \eta _{21}+ \eta _{03})^{2}]+(3 \eta _{21} \eta _{03})( \eta _{21}+ \eta _{03})[3( \eta _{30}+ \eta _{12})^{2}( \eta _{21}+ \eta _{03})^{2}] \\ hu[5]=( \eta _{20} \eta _{02})[( \eta _{30}+ \eta _{12})^{2} ( \eta _{21}+ \eta _{03})^{2}]+4 \eta _{11}( \eta _{30}+ \eta _{12})( \eta _{21}+ \eta _{03}) \\ hu[6]=(3 \eta _{21} \eta _{03})( \eta _{21}+ \eta _{03})[3( \eta _{30}+ \eta _{12})^{2}( \eta _{21}+ \eta _{03})^{2}]( \eta _{30}3 \eta _{12})( \eta _{21}+ \eta _{03})[3( \eta _{30}+ \eta _{12})^{2}( \eta _{21}+ \eta _{03})^{2}] \\ \end{array}\]
where \(\eta_{ji}\) stands for \(\texttt{Moments::nu}_{ji}\) .
These values are proved to be invariants to the image scale, rotation, and reflection except the seventh one, whose sign is changed by reflection. This invariance is proved with the assumption of infinite image resolution. In case of raster images, the computed Hu invariants for the original and transformed images are a bit different.
moments  Input moments computed with moments . 
hu  Output Hu invariants. 
void cv::HuMoments  (  const Moments &  m, 
OutputArray  hu  
) 
Python:  

hu  =  cv.HuMoments(  m[, hu]  ) 
#include <opencv2/imgproc.hpp>
This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.
float cv::intersectConvexConvex  (  InputArray  _p1, 
InputArray  _p2,  
OutputArray  _p12,  
bool  handleNested = true 

) 
Python:  

retval, _p12  =  cv.intersectConvexConvex(  _p1, _p2[, _p12[, handleNested]]  ) 
#include <opencv2/imgproc.hpp>
finds intersection of two convex polygons
bool cv::isContourConvex  (  InputArray  contour  ) 
Python:  

retval  =  cv.isContourConvex(  contour  ) 
#include <opencv2/imgproc.hpp>
Tests a contour convexity.
The function tests whether the input contour is convex or not. The contour must be simple, that is, without selfintersections. Otherwise, the function output is undefined.
contour  Input vector of 2D points, stored in std::vector<> or Mat 
double cv::matchShapes  (  InputArray  contour1, 
InputArray  contour2,  
int  method,  
double  parameter  
) 
Python:  

retval  =  cv.matchShapes(  contour1, contour2, method, parameter  ) 
#include <opencv2/imgproc.hpp>
Compares two shapes.
The function compares two shapes. All three implemented methods use the Hu invariants (see HuMoments)
contour1  First contour or grayscale image. 
contour2  Second contour or grayscale image. 
method  Comparison method, see ShapeMatchModes 
parameter  Methodspecific parameter (not supported now). 
RotatedRect cv::minAreaRect  (  InputArray  points  ) 
Python:  

retval  =  cv.minAreaRect(  points  ) 
#include <opencv2/imgproc.hpp>
Finds a rotated rectangle of the minimum area enclosing the input 2D point set.
The function calculates and returns the minimumarea bounding rectangle (possibly rotated) for a specified point set. Developer should keep in mind that the returned RotatedRect can contain negative indices when data is close to the containing Mat element boundary.
points  Input vector of 2D points, stored in std::vector<> or Mat 
void cv::minEnclosingCircle  (  InputArray  points, 
Point2f &  center,  
float &  radius  
) 
Python:  

center, radius  =  cv.minEnclosingCircle(  points  ) 
#include <opencv2/imgproc.hpp>
Finds a circle of the minimum area enclosing a 2D point set.
The function finds the minimal enclosing circle of a 2D point set using an iterative algorithm.
points  Input vector of 2D points, stored in std::vector<> or Mat 
center  Output center of the circle. 
radius  Output radius of the circle. 
double cv::minEnclosingTriangle  (  InputArray  points, 
OutputArray  triangle  
) 
Python:  

retval, triangle  =  cv.minEnclosingTriangle(  points[, triangle]  ) 
#include <opencv2/imgproc.hpp>
Finds a triangle of minimum area enclosing a 2D point set and returns its area.
The function finds a triangle of minimum area enclosing the given set of 2D points and returns its area. The output for a given 2D point set is shown in the image below. 2D points are depicted in red* and the enclosing triangle in yellow.
The implementation of the algorithm is based on O'Rourke's [167] and Klee and Laskowski's [113] papers. O'Rourke provides a \(\theta(n)\) algorithm for finding the minimal enclosing triangle of a 2D convex polygon with n vertices. Since the minEnclosingTriangle function takes a 2D point set as input an additional preprocessing step of computing the convex hull of the 2D point set is required. The complexity of the convexHull function is \(O(n log(n))\) which is higher than \(\theta(n)\). Thus the overall complexity of the function is \(O(n log(n))\).
points  Input vector of 2D points with depth CV_32S or CV_32F, stored in std::vector<> or Mat 
triangle  Output vector of three 2D points defining the vertices of the triangle. The depth of the OutputArray must be CV_32F. 
Moments cv::moments  (  InputArray  array, 
bool  binaryImage = false 

) 
Python:  

retval  =  cv.moments(  array[, binaryImage]  ) 
#include <opencv2/imgproc.hpp>
Calculates all of the moments up to the third order of a polygon or rasterized shape.
The function computes moments, up to the 3rd order, of a vector shape or a rasterized shape. The results are returned in the structure cv::Moments.
array  Raster image (singlechannel, 8bit or floatingpoint 2D array) or an array ( \(1 \times N\) or \(N \times 1\) ) of 2D points (Point or Point2f ). 
binaryImage  If it is true, all nonzero image pixels are treated as 1's. The parameter is used for images only. 
double cv::pointPolygonTest  (  InputArray  contour, 
Point2f  pt,  
bool  measureDist  
) 
Python:  

retval  =  cv.pointPolygonTest(  contour, pt, measureDist  ) 
#include <opencv2/imgproc.hpp>
Performs a pointincontour test.
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 is a signed distance between the point and the nearest contour edge.
See below a sample output of the function where each image pixel is tested against the contour:
contour  Input contour. 
pt  Point tested against the contour. 
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 a contour or not. 
int cv::rotatedRectangleIntersection  (  const RotatedRect &  rect1, 
const RotatedRect &  rect2,  
OutputArray  intersectingRegion  
) 
Python:  

retval, intersectingRegion  =  cv.rotatedRectangleIntersection(  rect1, rect2[, intersectingRegion]  ) 
#include <opencv2/imgproc.hpp>
Finds out if there is any intersection between two rotated rectangles.
If there is then the vertices of the intersecting region are returned as well.
Below are some examples of intersection configurations. The hatched pattern indicates the intersecting region and the red vertices are returned by the function.
rect1  First rectangle 
rect2  Second rectangle 
intersectingRegion  The output array of the vertices of the intersecting region. It returns at most 8 vertices. Stored as std::vector<cv::Point2f> or cv::Mat as Mx1 of type CV_32FC2. 