OpenCV: Structural Analysis and Shape Descriptors

| OpenCV 3.4.2 Open Source Computer Vision

Namespaces | Classes | Enumerations | Functions
Structural Analysis and Shape Descriptors
Image processing

## Namespaces
| cv::traits

## Classes
class | cv::Moments
| struct returned by cv::moments More...

## Enumerations
enum | cv::ConnectedComponentsAlgorithmsTypes { cv::CCL_WU = 0, cv::CCL_DEFAULT = -1, cv::CCL_GRANA = 1 }
| connected components algorithm More...
enum | cv::ConnectedComponentsTypes { cv::CC_STAT_LEFT = 0, cv::CC_STAT_TOP = 1, cv::CC_STAT_WIDTH = 2, cv::CC_STAT_HEIGHT = 3, cv::CC_STAT_AREA = 4, cv::CC_STAT_MAX = 5 }
| connected components algorithm output formats More...
enum | cv::ContourApproximationModes { cv::CHAIN_APPROX_NONE = 1, cv::CHAIN_APPROX_SIMPLE = 2, cv::CHAIN_APPROX_TC89_L1 = 3, cv::CHAIN_APPROX_TC89_KCOS = 4 }
| the contour approximation algorithm More...
enum | cv::RectanglesIntersectTypes { cv::INTERSECT_NONE = 0, cv::INTERSECT_PARTIAL = 1, cv::INTERSECT_FULL = 2 }
enum | cv::RetrievalModes { cv::RETR_EXTERNAL = 0, cv::RETR_LIST = 1, cv::RETR_CCOMP = 2, cv::RETR_TREE = 3, cv::RETR_FLOODFILL = 4 }
| mode of the contour retrieval algorithm More...
enum | cv::ShapeMatchModes { cv::CONTOURS_MATCH_I1 =1, cv::CONTOURS_MATCH_I2 =2, cv::CONTOURS_MATCH_I3 =3 }
| Shape matching methods. 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 points)
| Calculates the up-right bounding rectangle of a point set. 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...
void | cv::findContours (InputOutputArray image, OutputArrayOfArrays contours, OutputArray hierarchy, int mode, int method, Point offset=Point())
| Finds contours in a binary image. More...
void | cv::findContours (InputOutputArray 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 &center, 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 point-in-contour 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...

## Detailed Description

## Enumeration Type Documentation

## ◆ ConnectedComponentsAlgorithmsTypes

enum cv::ConnectedComponentsAlgorithmsTypes

connected components algorithm

Enumerator
CCL_WU | SAUF algorithm for 8-way connectivity, SAUF algorithm for 4-way connectivity.
CCL_DEFAULT | BBDT algorithm for 8-way connectivity, SAUF algorithm for 4-way connectivity.
CCL_GRANA | BBDT algorithm for 8-way connectivity, SAUF algorithm for 4-way connectivity.

## ◆ ConnectedComponentsTypes

enum cv::ConnectedComponentsTypes

connected components algorithm output formats

Enumerator
CC_STAT_LEFT | The leftmost (x) coordinate which is the inclusive start of the bounding box in the horizontal direction.
CC_STAT_TOP | The topmost (y) coordinate which is the inclusive start of the bounding box in the vertical direction.
CC_STAT_WIDTH | The horizontal size of the bounding box.
CC_STAT_HEIGHT | The vertical size of the bounding box.
CC_STAT_AREA | The total area (in pixels) of the connected component.
CC_STAT_MAX |

## ◆ ContourApproximationModes

enum cv::ContourApproximationModes

the contour approximation algorithm

Enumerator
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(x1-x2),abs(y2-y1))==1.
CHAIN_APPROX_SIMPLE | compresses horizontal, vertical, and diagonal segments and leaves only their end points. For example, an up-right rectangular contour is encoded with 4 points.
CHAIN_APPROX_TC89_L1 | applies one of the flavors of the Teh-Chin chain approximation algorithm [TehChin89]
CHAIN_APPROX_TC89_KCOS | applies one of the flavors of the Teh-Chin chain approximation algorithm [TehChin89]

## ◆ RectanglesIntersectTypes

enum cv::RectanglesIntersectTypes

types of intersection between rectangles

Enumerator
INTERSECT_NONE | No intersection.
INTERSECT_PARTIAL | There is a partial intersection.
INTERSECT_FULL | One of the rectangle is fully enclosed in the other.

## ◆ RetrievalModes

enum cv::RetrievalModes

mode of the contour retrieval algorithm

Enumerator
RETR_EXTERNAL | retrieves only the extreme outer contours. It sets `hierarchy[i][2]=hierarchy[i][3]=-1` for all the contours.
RETR_LIST | retrieves all of the contours without establishing any hierarchical relationships.
RETR_CCOMP | retrieves all of the contours and organizes them into a two-level hierarchy. At the top level, there are external boundaries of the components. At the second level, there are boundaries of the holes. If there is another contour inside a hole of a connected component, it is still put at the top level.
RETR_TREE | retrieves all of the contours and reconstructs a full hierarchy of nested contours.
RETR_FLOODFILL |

## ◆ ShapeMatchModes

enum cv::ShapeMatchModes

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.

Enumerator
CONTOURS_MATCH_I1 | \[I_1(A,B) = \sum _{i=1...7} \left | \frac{1}{m^A_i} - \frac{1}{m^B_i} \right |\]
CONTOURS_MATCH_I2 | \[I_2(A,B) = \sum _{i=1...7} \left | m^A_i - m^B_i \right |\]
CONTOURS_MATCH_I3 | \[I_3(A,B) = \max _{i=1...7} \frac{ \left| m^A_i - m^B_i \right| }{ \left| m^A_i \right| }\]

## Function Documentation

## ◆ approxPolyDP()

void cv::approxPolyDP | ( | InputArray | curve,
|  | OutputArray | approxCurve,
|  | double | epsilon,
|  | bool | closed
| ) |  |

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 Douglas-Peucker algorithm http://en.wikipedia.org/wiki/Ramer-Douglas-Peucker_algorithm

Parameters

  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.

Examples:
  contours2.cpp.

## ◆ arcLength()

double cv::arcLength | ( | InputArray | curve,
|  | bool | closed
| ) |  |

Calculates a contour perimeter or a curve length.

The function computes a curve length or a closed contour perimeter.

Parameters

  curve | Input vector of 2D points, stored in std::vector or Mat.
  closed | Flag indicating whether the curve is closed or not.

## ◆ boundingRect()

Rect cv::boundingRect | ( | InputArray | points | ) |

Calculates the up-right bounding rectangle of a point set.

The function calculates and returns the minimal up-right bounding rectangle for the specified point set.

Parameters

  points | Input 2D point set, stored in std::vector or Mat.

## ◆ boxPoints()

void cv::boxPoints | ( | RotatedRect | box,
|  | OutputArray | points
| ) |  |

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.

Parameters

  box | The input rotated rectangle. It may be the output of
  points | The output array of four vertices of rectangles.

## ◆ connectedComponents() [1/2]

int cv::connectedComponents | ( | InputArray | image,
|  | OutputArray | labels,
|  | int | connectivity,
|  | int | ltype,
|  | int | ccltype
| ) |  |

computes the connected components labeled image of boolean image

image with 4 or 8 way connectivity - returns N, the total number of labels [0, N-1] 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.

Parameters

  image | the 8-bit single-channel image to be labeled
  labels | destination labeled image
  connectivity | 8 or 4 for 8-way or 4-way connectivity respectively
  ltype | output image label type. Currently CV_32S and CV_16U are supported.
  ccltype | connected components algorithm type (see the ConnectedComponentsAlgorithmsTypes).

## ◆ connectedComponents() [2/2]

int cv::connectedComponents | ( | InputArray | image,
|  | OutputArray | labels,
|  | int | connectivity = `8`,
|  | int | ltype = `CV_32S`
| ) |  |

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters

  image | the 8-bit single-channel image to be labeled
  labels | destination labeled image
  connectivity | 8 or 4 for 8-way or 4-way connectivity respectively
  ltype | output image label type. Currently CV_32S and CV_16U are supported.

## ◆ connectedComponentsWithStats() [1/2]

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

image with 4 or 8 way connectivity - returns N, the total number of labels [0, N-1] 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.

Parameters

  image | the 8-bit single-channel 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 8-way or 4-way connectivity respectively
  ltype | output image label type. Currently CV_32S and CV_16U are supported.
  ccltype | connected components algorithm type (see ConnectedComponentsAlgorithmsTypes).

## ◆ connectedComponentsWithStats() [2/2]

int cv::connectedComponentsWithStats | ( | InputArray | image,
|  | OutputArray | labels,
|  | OutputArray | stats,
|  | OutputArray | centroids,
|  | int | connectivity = `8`,
|  | int | ltype = `CV_32S`
| ) |  |

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters

  image | the 8-bit single-channel 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 8-way or 4-way connectivity respectively
  ltype | output image label type. Currently CV_32S and CV_16U are supported.

## ◆ contourArea()

double cv::contourArea | ( | InputArray | contour,
|  | bool | oriented = `false`
| ) |  |

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 non-zero 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 self-intersections.

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;

Parameters

  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 counter-clockwise). 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.

Examples:
  segment_objects.cpp.

## ◆ convexHull()

void cv::convexHull | ( | InputArray | points,
|  | OutputArray | hull,
|  | bool | clockwise = `false`,
|  | bool | returnPoints = `true`
| ) |  |

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 [Sklansky82] that has O(N logN) complexity in the current implementation.

Parameters

  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 0-based 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 counter-clockwise. 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.

Note
  `points` and `hull` should be different arrays, inplace processing isn't supported.

Examples:
  convexhull.cpp.

## ◆ convexityDefects()

void cv::convexityDefects | ( | InputArray | contour,
|  | InputArray | convexhull,
|  | OutputArray | convexityDefects
| ) |  |

Finds the convexity defects of a contour.

The figure below displays convexity defects of a hand contour:

image

Parameters

  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 4-element integer vector (a.k.a. Vec4i): (start_index, end_index, farthest_pt_index, fixpt_depth), where indices are 0-based indices in the original contour of the convexity defect beginning, end and the farthest point, and fixpt_depth is fixed-point approximation (with 8 fractional bits) of the distance between the farthest contour point and the hull. That is, to get the floating-point value of the depth will be fixpt_depth/256.0.

## ◆ findContours() [1/2]

void cv::findContours | ( | InputOutputArray | image,
|  | OutputArrayOfArrays | contours,
|  | OutputArray | hierarchy,
|  | int | mode,
|  | int | method,
|  | Point | offset = `Point()`
| ) |  |

Finds contours in a binary image.

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.cpp in the OpenCV sample directory.

Note
  Since opencv 3.2 source image is not modified by this function.

Parameters

  image | Source, an 8-bit single-channel image. Non-zero 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 32-bit 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 i-th contour contours[i], the elements hierarchy[i][0] , hierarchy[i][1] , hierarchy[i][2] , and hierarchy[i][3] are 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 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.

Examples:
  contours2.cpp, fitellipse.cpp, segment_objects.cpp, shape_example.cpp, and watershed.cpp.

## ◆ findContours() [2/2]

void cv::findContours | ( | InputOutputArray | image,
|  | OutputArrayOfArrays | contours,
|  | int | mode,
|  | int | method,
|  | Point | offset = `Point()`
| ) |  |

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ fitEllipse()

RotatedRect cv::fitEllipse | ( | InputArray | points | ) |

Fits an ellipse around a set of 2D points.

The function calculates the ellipse that fits (in a least-squares 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 [Fitzgibbon95] 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.

Parameters

  points | Input 2D point set, stored in std::vector<> or Mat

Examples:
  fitellipse.cpp.

## ◆ fitEllipseAMS()

RotatedRect cv::fitEllipseAMS | ( | InputArray | points | ) |

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 [Taubin1991] 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*}

Parameters

  points | Input 2D point set, stored in std::vector<> or Mat

Examples:
  fitellipse.cpp.

## ◆ fitEllipseDirect()

RotatedRect cv::fitEllipseDirect | ( | InputArray | points | ) |

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 [Fitzgibbon1999] 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 \(A^T C A =1\).

Parameters

  points | Input 2D point set, stored in std::vector<> or Mat

Examples:
  fitellipse.cpp.

## ◆ fitLine()

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.

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:

- DIST_L2

  \[\rho (r) = r^2/2 \quad \text{(the simplest and the fastest least-squares method)}\]

- DIST_L1

  \[\rho (r) = r\]

- DIST_L12

  \[\rho (r) = 2 \cdot ( \sqrt{1 + \frac{r^2}{2}} - 1)\]

- DIST_FAIR

  \[\rho \left (r \right ) = C^2 \cdot \left ( \frac{r}{C} - \log{\left(1 + \frac{r}{C}\right)} \right ) \quad \text{where} \quad C=1.3998\]

- DIST_WELSCH

  \[\rho \left (r \right ) = \frac{C^2}{2} \cdot \left ( 1 - \exp{\left(-\left(\frac{r}{C}\right)^2\right)} \right ) \quad \text{where} \quad C=2.9846\]

- DIST_HUBER

  \[\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 the weighted least-squares algorithm. After each iteration the weights \(w_i\) are adjusted to be inversely proportional to \(\rho(r_i)\) .

Parameters

  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 M-estimator, 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.

## ◆ HuMoments() [1/2]

void cv::HuMoments | ( | const Moments & | moments,
|  | double | hu[7]
| ) |  |

Calculates seven Hu invariants.

The function calculates seven Hu invariants (introduced in [Hu62]; 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.

Parameters

  moments | Input moments computed with moments .
  hu | Output Hu invariants.

See also
  matchShapes

## ◆ HuMoments() [2/2]

void cv::HuMoments | ( | const Moments & | m,
|  | OutputArray | hu
| ) |  |

This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

## ◆ intersectConvexConvex()

float cv::intersectConvexConvex | ( | InputArray | _p1,
|  | InputArray | _p2,
|  | OutputArray | _p12,
|  | bool | handleNested = `true`
| ) |  |

finds intersection of two convex polygons

## ◆ isContourConvex()

bool cv::isContourConvex | ( | InputArray | contour | ) |

Tests a contour convexity.

The function tests whether the input contour is convex or not. The contour must be simple, that is, without self-intersections. Otherwise, the function output is undefined.

Parameters

  contour | Input vector of 2D points, stored in std::vector<> or Mat

## ◆ matchShapes()

double cv::matchShapes | ( | InputArray | contour1,
|  | InputArray | contour2,
|  | int | method,
|  | double | parameter
| ) |  |

Compares two shapes.

The function compares two shapes. All three implemented methods use the Hu invariants (see HuMoments)

Parameters

  contour1 | First contour or grayscale image.
  contour2 | Second contour or grayscale image.
  method | Comparison method, see ShapeMatchModes
  parameter | Method-specific parameter (not supported now).

## ◆ minAreaRect()

RotatedRect cv::minAreaRect | ( | InputArray | points | ) |

Finds a rotated rectangle of the minimum area enclosing the input 2D point set.

The function calculates and returns the minimum-area 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.

Parameters

  points | Input vector of 2D points, stored in std::vector<> or Mat

Examples:
  minarea.cpp.

## ◆ minEnclosingCircle()

void cv::minEnclosingCircle | ( | InputArray | points,
|  | Point2f & | center,
|  | float & | radius
| ) |  |

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.

Parameters

  points | Input vector of 2D points, stored in std::vector<> or Mat
  center | Output center of the circle.
  radius | Output radius of the circle.

Examples:
  minarea.cpp.

## ◆ minEnclosingTriangle()

double cv::minEnclosingTriangle | ( | InputArray | points,
|  | OutputArray | triangle
| ) |  |

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.

Sample output of the minimum enclosing triangle function

The implementation of the algorithm is based on O'Rourke's [ORourke86] and Klee and Laskowski's [KleeLaskowski85] 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 \(O(n log(n))\).

Parameters

  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.

Examples:
  minarea.cpp.

## ◆ moments()

Moments cv::moments | ( | InputArray | array,
|  | bool | binaryImage = `false`
| ) |  |

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.

Parameters

  array | Raster image (single-channel, 8-bit or floating-point 2D array) or an array ( \(1 \times N\) or \(N \times 1\) ) of 2D points (Point or Point2f ).
  binaryImage | If it is true, all non-zero image pixels are treated as 1's. The parameter is used for images only.

Returns
  moments.

Note
  Only applicable to contour moments calculations from Python bindings: Note that the numpy type for the input array should be either np.int32 or np.float32.

See also
  contourArea, arcLength

## ◆ pointPolygonTest()

double cv::pointPolygonTest | ( | InputArray | contour,
|  | Point2f | pt,
|  | bool | measureDist
| ) |  |

Performs a point-in-contour 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:

sample output

Parameters

  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.

## ◆ rotatedRectangleIntersection()

int cv::rotatedRectangleIntersection | ( | const RotatedRect & | rect1,
|  | const RotatedRect & | rect2,
|  | OutputArray | intersectingRegion
| ) |  |

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.

intersection examples

Parameters

  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.

Returns
  One of RectanglesIntersectTypes