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opencv/modules/imgproc/src/min_enclosing_triangle.cpp

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/*M///////////////////////////////////////////////////////////////////////////////////////
//
// IMPORTANT: READ BEFORE DOWNLOADING, COPYING, INSTALLING OR USING.
//
// By downloading, copying, installing or using the software you agree to this license.
// If you do not agree to this license, do not download, install,
// copy or use the software.
//
// INFORMATION REGARDING THE CONTRIBUTION:
//
// Author: Ovidiu Parvu
// Affiliation: Brunel University
// Created: 11.09.2013
// E-mail: <ovidiu.parvu[AT]gmail.com>
// Web: http://people.brunel.ac.uk/~cspgoop
//
// These functions were implemented during Ovidiu Parvu's first year as a PhD student at
// Brunel University, London, UK. The PhD project is supervised by prof. David Gilbert (principal)
// and prof. Nigel Saunders (second).
//
// THE IMPLEMENTATION OF THE MODULES IS BASED ON THE FOLLOWING PAPERS:
//
// [1] V. Klee and M. C. Laskowski, "Finding the smallest triangles containing a given convex
// polygon", Journal of Algorithms, vol. 6, no. 3, pp. 359-375, Sep. 1985.
// [2] J. O'Rourke, A. Aggarwal, S. Maddila, and M. Baldwin, "An optimal algorithm for finding
// minimal enclosing triangles", Journal of Algorithms, vol. 7, no. 2, pp. 258-269, Jun. 1986.
//
// The overall complexity of the algorithm is theta(n) where "n" represents the number
// of vertices in the convex polygon.
//
//
//
// License Agreement
// For Open Source Computer Vision Library
//
// Copyright (C) 2000, Intel Corporation, all rights reserved.
// Copyright (C) 2013, OpenCV Foundation, all rights reserved.
// Copyright (C) 2013, Ovidiu Parvu, all rights reserved.
// Third party copyrights are property of their respective owners.
//
// Redistribution and use in source and binary forms, with or without modification,
// are permitted provided that the following conditions are met:
//
// * Redistribution's of source code must retain the above copyright notice,
// this list of conditions and the following disclaimer.
//
// * Redistribution's in binary form must reproduce the above copyright notice,
// this list of conditions and the following disclaimer in the documentation
// and/or other materials provided with the distribution.
//
// * The name of the copyright holders may not be used to endorse or promote products
// derived from this software without specific prior written permission.
//
// This software is provided by the copyright holders and contributors "as is" and
// any express or implied warranties, including, but not limited to, the implied
// warranties of merchantability and fitness for a particular purpose are disclaimed.
// In no event shall the Intel Corporation or contributors be liable for any direct,
// indirect, incidental, special, exemplary, or consequential damages
// (including, but not limited to, procurement of substitute goods or services;
// loss of use, data, or profits; or business interruption) however caused
// and on any theory of liability, whether in contract, strict liability,
// or tort (including negligence or otherwise) arising in any way out of
// the use of this software, even if advised of the possibility of such damage.
//
//M*/
#include "precomp.hpp"
#include <algorithm>
#include <cmath>
#include <limits>
#include <vector>
///////////////////////////////// Constants definitions //////////////////////////////////
// Intersection of line and polygon
#define INTERSECTS_BELOW 1
#define INTERSECTS_ABOVE 2
#define INTERSECTS_CRITICAL 3
// Error messages
#define ERR_SIDE_B_GAMMA "The position of side B could not be determined, because gamma(b) could not be computed."
#define ERR_VERTEX_C_ON_SIDE_B "The position of the vertex C on side B could not be determined, because the considered lines do not intersect."
// Possible values for validation flag
#define VALIDATION_SIDE_A_TANGENT 0
#define VALIDATION_SIDE_B_TANGENT 1
#define VALIDATION_SIDES_FLUSH 2
// Threshold value for comparisons
#define EPSILON 1E-5
/////////////////////////////////// Global variables /////////////////////////////////////
namespace minEnclosingTriangle {
static unsigned int G_validationFlag;
static cv::Point2f G_vertexA;
static cv::Point2f G_vertexB;
static cv::Point2f G_vertexC;
static cv::Point2f G_sideAStartVertex;
static cv::Point2f G_sideAEndVertex;
static cv::Point2f G_sideBStartVertex;
static cv::Point2f G_sideBEndVertex;
static cv::Point2f G_sideCStartVertex;
static cv::Point2f G_sideCEndVertex;
static double G_triangleArea;
static unsigned int G_a;
static unsigned int G_b;
static unsigned int G_c;
static unsigned int G_nrOfPoints;
static std::vector<cv::Point2f> G_polygon;
};
////////////////////////////// Helper functions declarations /////////////////////////////
namespace minEnclosingTriangle {
static void advance(unsigned int &index);
static void advanceBToRightChain();
static bool almostEqual(double number1, double number2);
static double angleOfLineWrtOxAxis(const cv::Point2f &a, const cv::Point2f &b);
static bool areEqualPoints(const cv::Point2f &point1, const cv::Point2f &point2);
static bool areIdenticalLines(const std::vector<double> &side1Params,
const std::vector<double> &side2Params, double sideCExtraParam);
static bool areIdenticalLines(double a1, double b1, double c1, double a2, double b2, double c2);
static bool areIntersectingLines(const std::vector<double> &side1Params,
const std::vector<double> &side2Params,
double sideCExtraParam, cv::Point2f &intersectionPoint1,
cv::Point2f &intersectionPoint2);
static bool areOnTheSameSideOfLine(const cv::Point2f &p1, const cv::Point2f &p2,
const cv::Point2f &a, const cv::Point2f &b);
static double areaOfTriangle(const cv::Point2f &a, const cv::Point2f &b, const cv::Point2f &c);
static void copyResultingTriangle(const std::vector<cv::Point2f> &resultingTriangle, cv::OutputArray triangle);
static void createConvexHull(cv::InputArray points);
static double distanceBtwPoints(const cv::Point2f &a, const cv::Point2f &b);
static double distanceFromPointToLine(const cv::Point2f &a, const cv::Point2f &linePointB,
const cv::Point2f &linePointC);
static bool findGammaIntersectionPoints(unsigned int polygonPointIndex, const cv::Point2f &side1StartVertex,
const cv::Point2f &side1EndVertex, const cv::Point2f &side2StartVertex,
const cv::Point2f &side2EndVertex, cv::Point2f &intersectionPoint1,
cv::Point2f &intersectionPoint2);
static void findMinEnclosingTriangle(cv::InputArray points,
CV_OUT cv::OutputArray triangle, CV_OUT double &area);
static void findMinEnclosingTriangle(std::vector<cv::Point2f> &triangle, double &area);
static void findMinimumAreaEnclosingTriangle(std::vector<cv::Point2f> &triangle, double &area);
static cv::Point2f findVertexCOnSideB();
static bool gamma(unsigned int polygonPointIndex, cv::Point2f &gammaPoint);
static bool greaterOrEqual(double number1, double number2);
static double height(const cv::Point2f &polygonPoint);
static double height(unsigned int polygonPointIndex);
static void initialise(std::vector<cv::Point2f> &triangle, double &area);
static unsigned int intersects(double angleGammaAndPoint, unsigned int polygonPointIndex);
static bool intersectsAbove(const cv::Point2f &gammaPoint, unsigned int polygonPointIndex);
static unsigned int intersectsAboveOrBelow(unsigned int succPredIndex, unsigned int pointIndex);
static bool intersectsBelow(const cv::Point2f &gammaPoint, unsigned int polygonPointIndex);
static bool isAngleBetween(double angle1, double angle2, double angle3);
static bool isAngleBetweenNonReflex(double angle1, double angle2, double angle3);
static bool isFlushAngleBtwPredAndSucc(double &angleFlushEdge, double anglePred, double angleSucc);
static bool isGammaAngleBtw(double &gammaAngle, double angle1, double angle2);
static bool isGammaAngleEqualTo(double &gammaAngle, double angle);
static bool isLocalMinimalTriangle();
static bool isNotBTangency();
static bool isOppositeAngleBetweenNonReflex(double angle1, double angle2, double angle3);
static bool isPointOnLineSegment(const cv::Point2f &point, const cv::Point2f &lineSegmentStart,
const cv::Point2f &lineSegmentEnd);
static bool isValidMinimalTriangle();
static bool lessOrEqual(double number1, double number2);
static void lineEquationDeterminedByPoints(const cv::Point2f &p, const cv::Point2f &q,
double &a, double &b, double &c);
static std::vector<double> lineEquationParameters(const cv::Point2f& p, const cv::Point2f &q);
static bool lineIntersection(const cv::Point2f &a1, const cv::Point2f &b1, const cv::Point2f &a2,
const cv::Point2f &b2, cv::Point2f &intersection);
static bool lineIntersection(double a1, double b1, double c1, double a2, double b2, double c2,
cv::Point2f &intersection);
static double maximum(double number1, double number2, double number3);
static cv::Point2f middlePoint(const cv::Point2f &a, const cv::Point2f &b);
static bool middlePointOfSideB(cv::Point2f& middlePointOfSideB);
static void moveAIfLowAndBIfHigh();
static double oppositeAngle(double angle);
static unsigned int predecessor(unsigned int index);
static void returnMinimumAreaEnclosingTriangle(std::vector<cv::Point2f> &triangle, double &area);
static void searchForBTangency();
static int sign(double number);
static unsigned int successor(unsigned int index);
static void updateMinimumAreaEnclosingTriangle(std::vector<cv::Point2f> &triangle, double &area);
static void updateSideB();
static void updateSidesBA();
static void updateSidesCA();
};
///////////////////////////////////// Main functions /////////////////////////////////////
//! Find the minimum enclosing triangle for the given set of points and return its area
/*!
* @param points Set of points
* @param triangle Minimum area triangle enclosing the given set of points
*/
double cv::minEnclosingTriangle(cv::InputArray points, CV_OUT cv::OutputArray triangle) {
double area;
minEnclosingTriangle::findMinEnclosingTriangle(points, triangle, area);
return area;
}
/////////////////////////////// Helper functions definition //////////////////////////////
namespace minEnclosingTriangle {
//! Find the minimum enclosing triangle and its area
/*!
* @param points Set of points
* @param triangle Minimum area triangle enclosing the given set of points
* @param area Area of the minimum area enclosing triangle
*/
static void findMinEnclosingTriangle(cv::InputArray points,
CV_OUT cv::OutputArray triangle, CV_OUT double &area) {
std::vector<cv::Point2f> resultingTriangle;
createConvexHull(points);
findMinEnclosingTriangle(resultingTriangle, area);
copyResultingTriangle(resultingTriangle, triangle);
}
//! Create the convex hull of the given set of points
/*!
* @param points The provided set of points
*/
static void createConvexHull(cv::InputArray points) {
cv::Mat pointsMat = points.getMat();
std::vector<cv::Point2f> pointsVector;
CV_Assert((pointsMat.checkVector(2) > 0) &&
((pointsMat.depth() == CV_32F) || (pointsMat.depth() == CV_32S)));
pointsMat.convertTo(pointsVector, CV_32F);
convexHull(pointsVector, G_polygon, true, true);
}
//! Find the minimum enclosing triangle and its area
/*!
* The overall complexity of the algorithm is theta(n) where "n" represents the number
* of vertices in the convex polygon
*
* @param triangle Minimum area triangle enclosing the given polygon
* @param area Area of the minimum area enclosing triangle
*/
static void findMinEnclosingTriangle(std::vector<cv::Point2f> &triangle, double &area) {
initialise(triangle, area);
if (G_polygon.size() > 3) {
findMinimumAreaEnclosingTriangle(triangle, area);
} else {
returnMinimumAreaEnclosingTriangle(triangle, area);
}
}
//! Copy resultingTriangle to the OutputArray triangle
/*!
* @param resultingTriangle Minimum area triangle enclosing the given polygon found by the algorithm
* @param triangle Minimum area triangle enclosing the given polygon returned to the user
*/
static void copyResultingTriangle(const std::vector<cv::Point2f> &resultingTriangle,
cv::OutputArray triangle) {
cv::Mat(resultingTriangle).copyTo(triangle);
}
//! Initialisation function
/*!
* @param triangle Minimum area triangle enclosing the given polygon
* @param area Area of the minimum area enclosing triangle
*/
static void initialise(std::vector<cv::Point2f> &triangle, double &area) {
G_nrOfPoints = static_cast<unsigned int>(G_polygon.size());
area = std::numeric_limits<double>::max();
// Clear all points previously stored in the vector
triangle.clear();
// Initialise the values of the indices for the algorithm
G_a = 1;
G_b = 2;
G_c = 0;
}
//! Find the minimum area enclosing triangle for the given polygon
/*!
* @param triangle Minimum area triangle enclosing the given polygon
* @param area Area of the minimum area enclosing triangle
*/
static void findMinimumAreaEnclosingTriangle(std::vector<cv::Point2f> &triangle, double &area) {
for (G_c = 0; G_c < G_nrOfPoints; G_c++) {
advanceBToRightChain();
moveAIfLowAndBIfHigh();
searchForBTangency();
updateSidesCA();
if (isNotBTangency()) {
updateSidesBA();
} else {
updateSideB();
}
if (isLocalMinimalTriangle()) {
updateMinimumAreaEnclosingTriangle(triangle, area);
}
}
}
//! Return the minimum area enclosing (pseudo-)triangle in case the convex polygon has at most three points
/*!
* @param triangle Minimum area triangle enclosing the given polygon
* @param area Area of the minimum area enclosing triangle
*/
static void returnMinimumAreaEnclosingTriangle(std::vector<cv::Point2f> &triangle, double &area) {
for (int i = 0; i < 3; i++) {
triangle.push_back(G_polygon[i % G_nrOfPoints]);
}
area = areaOfTriangle(triangle[0], triangle[1], triangle[2]);
}
//! Advance b to the right chain
/*!
* See paper [2] for more details
*/
static void advanceBToRightChain() {
while (greaterOrEqual(height(successor(G_b)), height(G_b))) {
advance(G_b);
}
}
//! Move "a" if it is low and "b" if it is high
/*!
* See paper [2] for more details
*/
static void moveAIfLowAndBIfHigh() {
cv::Point2f gammaOfA;
while(height(G_b) > height(G_a)) {
if ((gamma(G_a, gammaOfA)) && (intersectsBelow(gammaOfA, G_b))) {
advance(G_b);
} else {
advance(G_a);
}
}
}
//! Search for the tangency of side B
/*!
* See paper [2] for more details
*/
static void searchForBTangency() {
cv::Point2f gammaOfB;
while (((gamma(G_b, gammaOfB)) && (intersectsBelow(gammaOfB, G_b))) &&
(greaterOrEqual(height(G_b), height(predecessor(G_a))))) {
advance(G_b);
}
}
//! Check if tangency for side B was not obtained
/*!
* See paper [2] for more details
*/
static bool isNotBTangency() {
cv::Point2f gammaOfB;
if (((gamma(G_b, gammaOfB)) && (intersectsAbove(gammaOfB, G_b))) ||
(height(G_b) < height(predecessor(G_a)))) {
return true;
}
return false;
}
//! Update sides A and C
/*!
* Side C will have as start and end vertices the polygon points "c" and "c-1"
* Side A will have as start and end vertices the polygon points "a" and "a-1"
*/
static void updateSidesCA() {
G_sideCStartVertex = G_polygon[predecessor(G_c)];
G_sideCEndVertex = G_polygon[G_c];
G_sideAStartVertex = G_polygon[predecessor(G_a)];
G_sideAEndVertex = G_polygon[G_a];
}
//! Update sides B and possibly A if tangency for side B was not obtained
/*!
* See paper [2] for more details
*/
static void updateSidesBA() {
// Side B is flush with edge [b, b-1]
G_sideBStartVertex = G_polygon[predecessor(G_b)];
G_sideBEndVertex = G_polygon[G_b];
// Find middle point of side B
cv::Point2f sideBMiddlePoint;
if ((middlePointOfSideB(sideBMiddlePoint)) &&
(height(sideBMiddlePoint) < height(predecessor(G_a)))) {
G_sideAStartVertex = G_polygon[predecessor(G_a)];
G_sideAEndVertex = findVertexCOnSideB();
G_validationFlag = VALIDATION_SIDE_A_TANGENT;
} else {
G_validationFlag = VALIDATION_SIDES_FLUSH;
}
}
//! Set side B if tangency for side B was obtained
/*!
* See paper [2] for more details
*/
static void updateSideB() {
if (!gamma(G_b, G_sideBStartVertex)) {
CV_Error(cv::Error::StsInternal, ERR_SIDE_B_GAMMA);
}
G_sideBEndVertex = G_polygon[G_b];
G_validationFlag = VALIDATION_SIDE_B_TANGENT;
}
//! Update the triangle vertices after all sides were set and check if a local minimal triangle was found or not
/*!
* See paper [2] for more details
*/
static bool isLocalMinimalTriangle() {
if ((!lineIntersection(G_sideAStartVertex, G_sideAEndVertex,
G_sideBStartVertex, G_sideBEndVertex, G_vertexC)) ||
(!lineIntersection(G_sideAStartVertex, G_sideAEndVertex,
G_sideCStartVertex, G_sideCEndVertex, G_vertexB)) ||
(!lineIntersection(G_sideBStartVertex, G_sideBEndVertex,
G_sideCStartVertex, G_sideCEndVertex, G_vertexA))) {
return false;
}
return isValidMinimalTriangle();
}
//! Check if the found minimal triangle is valid
/*!
* This means that all midpoints of the triangle should touch the polygon
*
* See paper [2] for more details
*/
static bool isValidMinimalTriangle() {
cv::Point2f midpointSideA = middlePoint(G_vertexB, G_vertexC);
cv::Point2f midpointSideB = middlePoint(G_vertexA, G_vertexC);
cv::Point2f midpointSideC = middlePoint(G_vertexA, G_vertexB);
bool sideAValid = (G_validationFlag == VALIDATION_SIDE_A_TANGENT)
? (areEqualPoints(midpointSideA, G_polygon[predecessor(G_a)]))
: (isPointOnLineSegment(midpointSideA, G_sideAStartVertex, G_sideAEndVertex));
bool sideBValid = (G_validationFlag == VALIDATION_SIDE_B_TANGENT)
? (areEqualPoints(midpointSideB, G_polygon[G_b]))
: (isPointOnLineSegment(midpointSideB, G_sideBStartVertex, G_sideBEndVertex));
bool sideCValid = isPointOnLineSegment(midpointSideC, G_sideCStartVertex, G_sideCEndVertex);
return (sideAValid && sideBValid && sideCValid);
}
//! Update the current minimum area enclosing triangle if the newly obtained one has a smaller area
/*!
* @param triangle Minimum area triangle enclosing the given polygon
* @param area Area of the minimum area triangle enclosing the given polygon
*/
static void updateMinimumAreaEnclosingTriangle(std::vector<cv::Point2f> &triangle, double &area) {
G_triangleArea = areaOfTriangle(G_vertexA, G_vertexB, G_vertexC);
if (G_triangleArea < area) {
triangle.clear();
triangle.push_back(G_vertexA);
triangle.push_back(G_vertexB);
triangle.push_back(G_vertexC);
area = G_triangleArea;
}
}
//! Return the middle point of side B
static bool middlePointOfSideB(cv::Point2f& middlePointOfSideB) {
cv::Point2f vertexA, vertexC;
if ((!lineIntersection(G_sideBStartVertex, G_sideBEndVertex, G_sideCStartVertex, G_sideCEndVertex, vertexA)) ||
(!lineIntersection(G_sideBStartVertex, G_sideBEndVertex, G_sideAStartVertex, G_sideAEndVertex, vertexC))) {
return false;
}
middlePointOfSideB = middlePoint(vertexA, vertexC);
return true;
}
//! Check if the line intersects below
/*!
* Check if the line determined by gammaPoint and polygon[polygonPointIndex] intersects
* the polygon below the point polygon[polygonPointIndex]
*
* @param gammaPoint Gamma(p)
* @param polygonPointIndex Index of the polygon point which is considered when determining the line
*/
static bool intersectsBelow(const cv::Point2f &gammaPoint, unsigned int polygonPointIndex) {
double angleOfGammaAndPoint = angleOfLineWrtOxAxis(G_polygon[polygonPointIndex], gammaPoint);
return (intersects(angleOfGammaAndPoint, polygonPointIndex) == INTERSECTS_BELOW);
}
//! Check if the line intersects above
/*!
* Check if the line determined by gammaPoint and polygon[polygonPointIndex] intersects
* the polygon above the point polygon[polygonPointIndex]
*
* @param gammaPoint Gamma(p)
* @param polygonPointIndex Index of the polygon point which is considered when determining the line
*/
static bool intersectsAbove(const cv::Point2f &gammaPoint, unsigned int polygonPointIndex) {
double angleOfGammaAndPoint = angleOfLineWrtOxAxis(gammaPoint, G_polygon[polygonPointIndex]);
return (intersects(angleOfGammaAndPoint, polygonPointIndex) == INTERSECTS_ABOVE);
}
//! Check if/where the line determined by gammaPoint and polygon[polygonPointIndex] intersects the polygon
/*!
* @param angleGammaAndPoint Angle determined by gammaPoint and polygon[polygonPointIndex] wrt Ox axis
* @param polygonPointIndex Index of the polygon point which is considered when determining the line
*/
static unsigned int intersects(double angleGammaAndPoint, unsigned int polygonPointIndex) {
double anglePointPredecessor = angleOfLineWrtOxAxis(G_polygon[predecessor(polygonPointIndex)],
G_polygon[polygonPointIndex]);
double anglePointSuccessor = angleOfLineWrtOxAxis(G_polygon[successor(polygonPointIndex)],
G_polygon[polygonPointIndex]);
double angleFlushEdge = angleOfLineWrtOxAxis(G_polygon[predecessor(G_c)],
G_polygon[G_c]);
if (isFlushAngleBtwPredAndSucc(angleFlushEdge, anglePointPredecessor, anglePointSuccessor)) {
if ((isGammaAngleBtw(angleGammaAndPoint, anglePointPredecessor, angleFlushEdge)) ||
(almostEqual(angleGammaAndPoint, anglePointPredecessor))) {
return intersectsAboveOrBelow(predecessor(polygonPointIndex), polygonPointIndex);
} else if ((isGammaAngleBtw(angleGammaAndPoint, anglePointSuccessor, angleFlushEdge)) ||
(almostEqual(angleGammaAndPoint, anglePointSuccessor))) {
return intersectsAboveOrBelow(successor(polygonPointIndex), polygonPointIndex);
}
} else {
if (
(isGammaAngleBtw(angleGammaAndPoint, anglePointPredecessor, anglePointSuccessor)) ||
(
(isGammaAngleEqualTo(angleGammaAndPoint, anglePointPredecessor)) &&
(!isGammaAngleEqualTo(angleGammaAndPoint, angleFlushEdge))
) ||
(
(isGammaAngleEqualTo(angleGammaAndPoint, anglePointSuccessor)) &&
(!isGammaAngleEqualTo(angleGammaAndPoint, angleFlushEdge))
)
) {
return INTERSECTS_BELOW;
}
}
return INTERSECTS_CRITICAL;
}
//! If (gamma(x) x) intersects P between successorOrPredecessorIndex and pointIntex is it above/below?
/*!
* @param succPredIndex Index of the successor or predecessor
* @param pointIndex Index of the point x in the polygon
*/
static unsigned int intersectsAboveOrBelow(unsigned int succPredIndex, unsigned int pointIndex) {
if (height(succPredIndex) > height(pointIndex)) {
return INTERSECTS_ABOVE;
} else {
return INTERSECTS_BELOW;
}
}
//! Find gamma for a given point "p" specified by its index
/*!
* The function returns true if gamma exists i.e. if lines (a a-1) and (x y) intersect
* and false otherwise. In case the two lines intersect in point intersectionPoint, gamma is computed.
*
* Considering that line (x y) is a line parallel to (c c-1) and that the distance between the lines is equal
* to 2 * height(p), we can have two possible (x y) lines.
*
* Therefore, we will compute two intersection points between the lines (x y) and (a a-1) and take the
* point which is on the same side of line (c c-1) as the polygon.
*
* See paper [2] and formula for distance from point to a line for more details
*
* @param polygonPointIndex Index of the polygon point
* @param gammaPoint Point gamma(polygon[polygonPointIndex])
*/
static bool gamma(unsigned int polygonPointIndex, cv::Point2f &gammaPoint) {
cv::Point2f intersectionPoint1, intersectionPoint2;
// Get intersection points if they exist
if (!findGammaIntersectionPoints(polygonPointIndex, G_polygon[G_a], G_polygon[predecessor(G_a)],
G_polygon[G_c], G_polygon[predecessor(G_c)],
intersectionPoint1, intersectionPoint2)) {
return false;
}
// Select the point which is on the same side of line C as the polygon
if (areOnTheSameSideOfLine(intersectionPoint1, G_polygon[successor(G_c)],
G_polygon[G_c], G_polygon[predecessor(G_c)])) {
gammaPoint = intersectionPoint1;
} else {
gammaPoint = intersectionPoint2;
}
return true;
}
//! Find the intersection points to compute gamma(point)
/*!
* @param polygonPointIndex Index of the polygon point for which the distance is known
* @param side1StartVertex Start vertex for side 1
* @param side1EndVertex End vertex for side 1
* @param side2StartVertex Start vertex for side 2
* @param side2EndVertex End vertex for side 2
* @param intersectionPoint1 First intersection point between one pair of lines
* @param intersectionPoint2 Second intersection point between other pair of lines
*/
static bool findGammaIntersectionPoints(unsigned int polygonPointIndex, const cv::Point2f &side1StartVertex,
const cv::Point2f &side1EndVertex, const cv::Point2f &side2StartVertex,
const cv::Point2f &side2EndVertex, cv::Point2f &intersectionPoint1,
cv::Point2f &intersectionPoint2) {
std::vector<double> side1Params = lineEquationParameters(side1StartVertex, side1EndVertex);
std::vector<double> side2Params = lineEquationParameters(side2StartVertex, side2EndVertex);
// Compute side C extra parameter using the formula for distance from a point to a line
double polygonPointHeight = height(polygonPointIndex);
double distFormulaDenom = sqrt((side2Params[0] * side2Params[0]) + (side2Params[1] * side2Params[1]));
double sideCExtraParam = 2 * polygonPointHeight * distFormulaDenom;
// Get intersection points if they exist or if lines are identical
if (!areIntersectingLines(side1Params, side2Params, sideCExtraParam, intersectionPoint1, intersectionPoint2)) {
return false;
} else if (areIdenticalLines(side1Params, side2Params, sideCExtraParam)) {
intersectionPoint1 = side1StartVertex;
intersectionPoint2 = side1EndVertex;
}
return true;
}
//! Check if the given lines are identical or not
/*!
* The lines are specified as:
* ax + by + c = 0
* OR
* ax + by + c (+/-) sideCExtraParam = 0
*
* @param side1Params Vector containing the values of a, b and c for side 1
* @param side2Params Vector containing the values of a, b and c for side 2
* @param sideCExtraParam Extra parameter for the flush edge C
*/
static bool areIdenticalLines(const std::vector<double> &side1Params,
const std::vector<double> &side2Params, double sideCExtraParam) {
return (
(areIdenticalLines(side1Params[0], side1Params[1], -(side1Params[2]),
side2Params[0], side2Params[1], -(side2Params[2]) - sideCExtraParam)) ||
(areIdenticalLines(side1Params[0], side1Params[1], -(side1Params[2]),
side2Params[0], side2Params[1], -(side2Params[2]) + sideCExtraParam))
);
}
//! Check if the given lines intersect or not. If the lines intersect find their intersection points.
/*!
* The lines are specified as:
* ax + by + c = 0
* OR
* ax + by + c (+/-) sideCExtraParam = 0
*
* @param side1Params Vector containing the values of a, b and c for side 1
* @param side2Params Vector containing the values of a, b and c for side 2
* @param sideCExtraParam Extra parameter for the flush edge C
* @param intersectionPoint1 The first intersection point, if it exists
* @param intersectionPoint2 The second intersection point, if it exists
*/
static bool areIntersectingLines(const std::vector<double> &side1Params,
const std::vector<double> &side2Params,
double sideCExtraParam, cv::Point2f &intersectionPoint1,
cv::Point2f &intersectionPoint2) {
return (
(lineIntersection(side1Params[0], side1Params[1], -(side1Params[2]),
side2Params[0], side2Params[1], -(side2Params[2]) - sideCExtraParam,
intersectionPoint1)) &&
(lineIntersection(side1Params[0], side1Params[1], -(side1Params[2]),
side2Params[0], side2Params[1], -(side2Params[2]) + sideCExtraParam,
intersectionPoint2))
);
}
//! Get the line equation parameters "a", "b" and "c" for the line determined by points "p" and "q"
/*!
* The equation of the line is considered in the general form:
* ax + by + c = 0
*
* @param p One point for defining the equation of the line
* @param q Second point for defining the equation of the line
*/
static std::vector<double> lineEquationParameters(const cv::Point2f& p, const cv::Point2f &q) {
std::vector<double> lineEquationParameters;
double a, b, c;
lineEquationDeterminedByPoints(p, q, a, b, c);
lineEquationParameters.push_back(a);
lineEquationParameters.push_back(b);
lineEquationParameters.push_back(c);
return lineEquationParameters;
}
//! Find vertex C which lies on side B at a distance = 2 * height(a-1) from side C
/*!
* Considering that line (x y) is a line parallel to (c c-1) and that the distance between the lines is equal
* to 2 * height(a-1), we can have two possible (x y) lines.
*
* Therefore, we will compute two intersection points between the lines (x y) and (b b-1) and take the
* point which is on the same side of line (c c-1) as the polygon.
*
* See paper [2] and formula for distance from point to a line for more details
*/
static cv::Point2f findVertexCOnSideB() {
cv::Point2f intersectionPoint1, intersectionPoint2;
// Get intersection points if they exist
if (!findGammaIntersectionPoints(predecessor(G_a), G_sideBStartVertex, G_sideBEndVertex,
G_sideCStartVertex, G_sideCEndVertex,
intersectionPoint1, intersectionPoint2)) {
CV_Error(cv::Error::StsInternal, ERR_VERTEX_C_ON_SIDE_B);
}
// Select the point which is on the same side of line C as the polygon
if (areOnTheSameSideOfLine(intersectionPoint1, G_polygon[successor(G_c)],
G_polygon[G_c], G_polygon[predecessor(G_c)])) {
return intersectionPoint1;
} else {
return intersectionPoint2;
}
}
//! Compute the height of the point
/*!
* See paper [2] for more details
*
* @param polygonPoint Polygon point
*/
static double height(const cv::Point2f &polygonPoint) {
cv::Point2f pointC = G_polygon[G_c];
cv::Point2f pointCPredecessor = G_polygon[predecessor(G_c)];
return distanceFromPointToLine(polygonPoint, pointC, pointCPredecessor);
}
//! Compute the height of the point specified by the given index
/*!
* See paper [2] for more details
*
* @param polygonPointIndex Index of the polygon point
*/
static double height(unsigned int polygonPointIndex) {
cv::Point2f pointC = G_polygon[G_c];
cv::Point2f pointCPredecessor = G_polygon[predecessor(G_c)];
cv::Point2f polygonPoint = G_polygon[polygonPointIndex];
return distanceFromPointToLine(polygonPoint, pointC, pointCPredecessor);
}
//! Advance the given index with one position
/*!
* @param index Index of the point
*/
static void advance(unsigned int &index) {
index = successor(index);
}
//! Return the succesor of the provided point index
/*!
* The succesor of the last polygon point is the first polygon point
* (circular referencing)
*
* @param index Index of the point
*/
static unsigned int successor(unsigned int index) {
return ((index + 1) % G_nrOfPoints);
}
//! Return the predecessor of the provided point index
/*!
* The predecessor of the first polygon point is the last polygon point
* (circular referencing)
*
* @param index Index of the point
*/
static unsigned int predecessor(unsigned int index) {
return (index == 0) ? (G_nrOfPoints - 1)
: (index - 1);
}
//! Check if the flush edge angle/opposite angle lie between the predecessor and successor angle
/*!
* Check if the angle of the flush edge or its opposite angle lie between the angle of
* the predecessor and successor
*
* @param angleFlushEdge Angle of the flush edge
* @param anglePred Angle of the predecessor
* @param angleSucc Angle of the successor
*/
static bool isFlushAngleBtwPredAndSucc(double &angleFlushEdge, double anglePred, double angleSucc) {
if (isAngleBetweenNonReflex(angleFlushEdge, anglePred, angleSucc)) {
return true;
} else if (isOppositeAngleBetweenNonReflex(angleFlushEdge, anglePred, angleSucc)) {
angleFlushEdge = oppositeAngle(angleFlushEdge);
return true;
}
return false;
}
//! Check if the angle of the line (gamma(p) p) or its opposite angle is equal to the given angle
/*!
* @param gammaAngle Angle of the line (gamma(p) p)
* @param angle Angle to compare against
*/
static bool isGammaAngleEqualTo(double &gammaAngle, double angle) {
return (almostEqual(gammaAngle, angle));
}
//! Check if the angle of the line (gamma(p) p) or its opposite angle lie between angle1 and angle2
/*!
* @param gammaAngle Angle of the line (gamma(p) p)
* @param angle1 One of the boundary angles
* @param angle2 Another boundary angle
*/
static bool isGammaAngleBtw(double &gammaAngle, double angle1, double angle2) {
return (isAngleBetweenNonReflex(gammaAngle, angle1, angle2));
}
//! Get the angle of the line measured from the Ox axis in counterclockwise direction
/*!
* The line is specified by points "a" and "b". The value of the angle is expressed in degrees.
*
* @param a Point a
* @param b Point b
*/
static double angleOfLineWrtOxAxis(const cv::Point2f &a, const cv::Point2f &b) {
double y = b.y - a.y;
double x = b.x - a.x;
double angle = (std::atan2(y, x) * 180 / CV_PI);
return (angle < 0) ? (angle + 360)
: angle;
}
//! Check if angle1 lies between non reflex angle determined by angles 2 and 3
/*!
* @param angle1 The angle which lies between angle2 and angle3 or not
* @param angle2 One of the boundary angles
* @param angle3 The other boundary angle
*/
static bool isAngleBetweenNonReflex(double angle1, double angle2, double angle3) {
if (std::abs(angle2 - angle3) > 180) {
if (angle2 > angle3) {
return (((angle2 < angle1) && (lessOrEqual(angle1, 360))) ||
((lessOrEqual(0, angle1)) && (angle1 < angle3)));
} else {
return (((angle3 < angle1) && (lessOrEqual(angle1, 360))) ||
((lessOrEqual(0, angle1)) && (angle1 < angle2)));
}
} else {
return isAngleBetween(angle1, angle2, angle3);
}
}
//! Check if the opposite of angle1, ((angle1 + 180) % 360), lies between non reflex angle determined by angles 2 and 3
/*!
* @param angle1 The angle which lies between angle2 and angle3 or not
* @param angle2 One of the boundary angles
* @param angle3 The other boundary angle
*/
static bool isOppositeAngleBetweenNonReflex(double angle1, double angle2, double angle3) {
double angle1Opposite = oppositeAngle(angle1);
return (isAngleBetweenNonReflex(angle1Opposite, angle2, angle3));
}
//! Check if angle1 lies between angles 2 and 3
/*!
* @param angle1 The angle which lies between angle2 and angle3 or not
* @param angle2 One of the boundary angles
* @param angle3 The other boundary angle
*/
static bool isAngleBetween(double angle1, double angle2, double angle3) {
if ((((int)(angle2 - angle3)) % 180) > 0) {
return ((angle3 < angle1) && (angle1 < angle2));
} else {
return ((angle2 < angle1) && (angle1 < angle3));
}
}
//! Return the angle opposite to the given angle
/*!
* if (angle < 180) then
* return (angle + 180);
* else
* return (angle - 180);
* endif
*
* @param angle Angle
*/
static double oppositeAngle(double angle) {
return (angle > 180) ? (angle - 180)
: (angle + 180);
}
//! Compute the distance from a point "a" to a line specified by two points "B" and "C"
/*!
* Formula used:
*
* |(x_c - x_b) * (y_b - y_a) - (x_b - x_a) * (y_c - y_b)|
* d = -------------------------------------------------------
* sqrt(((x_c - x_b)^2) + ((y_c - y_b)^2))
*
* Reference: http://mathworld.wolfram.com/Point-LineDistance2-Dimensional.html
* (Last access: 15.09.2013)
*
* @param a Point from which the distance is measures
* @param linePointB One of the points determining the line
* @param linePointC One of the points determining the line
*/
static double distanceFromPointToLine(const cv::Point2f &a, const cv::Point2f &linePointB,
const cv::Point2f &linePointC) {
double term1 = linePointC.x - linePointB.x;
double term2 = linePointB.y - a.y;
double term3 = linePointB.x - a.x;
double term4 = linePointC.y - linePointB.y;
double nominator = std::abs((term1 * term2) - (term3 * term4));
double denominator = std::sqrt((term1 * term1) + (term4 * term4));
return (denominator != 0) ? (nominator / denominator)
: 0;
}
//! Compute the distance between two points
/*! Compute the Euclidean distance between two points
*
* @param a Point a
* @param b Point b
*/
static double distanceBtwPoints(const cv::Point2f &a, const cv::Point2f &b) {
double xDiff = a.x - b.x;
double yDiff = a.y - b.y;
return std::sqrt((xDiff * xDiff) + (yDiff * yDiff));
}
//! Compute the area of a triangle defined by three points
/*!
* The area is computed using the determinant method.
* An example is depicted at http://demonstrations.wolfram.com/TheAreaOfATriangleUsingADeterminant/
* (Last access: 15.09.2013)
*
* @param a Point a
* @param b Point b
* @param c Point c
*/
static double areaOfTriangle(const cv::Point2f &a, const cv::Point2f &b, const cv::Point2f &c) {
double posTerm = (a.x * b.y) + (a.y * c.x) + (b.x * c.y);
double negTerm = (b.y * c.x) + (a.x * c.y) + (a.y * b.x);
double determinant = posTerm - negTerm;
return std::abs(determinant) / 2;
}
//! Get the point in the middle of the segment determined by points "a" and "b"
/*!
* @param a Point a
* @param b Point b
*/
static cv::Point2f middlePoint(const cv::Point2f &a, const cv::Point2f &b) {
double middleX = static_cast<double>((a.x + b.x) / 2);
double middleY = static_cast<double>((a.y + b.y) / 2);
return cv::Point2f(static_cast<float>(middleX), static_cast<float>(middleY));
}
//! Determine the intersection point of two lines, if this point exists
/*! Two lines intersect if they are not parallel (Parallel lines intersect at
* +/- infinity, but we do not consider this case here).
*
* The lines are specified in the following form:
* A1x + B1x = C1
* A2x + B2x = C2
*
* If det (= A1*B2 - A2*B1) == 0, then lines are parallel
* else they intersect
*
* If they intersect, then let us denote the intersection point with P(x, y) where:
* x = (C1*B2 - C2*B1) / (det)
* y = (C2*A1 - C1*A2) / (det)
*
* @param a1 A1
* @param b1 B1
* @param c1 C1
* @param a2 A2
* @param b2 B2
* @param c2 C2
* @param intersection The intersection point, if this point exists
*/
static bool lineIntersection(double a1, double b1, double c1, double a2, double b2, double c2,
cv::Point2f &intersection) {
double det = (a1 * b2) - (a2 * b1);
if (!(almostEqual(det, 0))) {
intersection.x = static_cast<float>(((c1 * b2) - (c2 * b1)) / (det));
intersection.y = static_cast<float>(((c2 * a1) - (c1 * a2)) / (det));
return true;
}
return false;
}
//! Determine the intersection point of two lines, if this point exists
/*! Two lines intersect if they are not parallel (Parallel lines intersect at
* +/- infinity, but we do not consider this case here).
*
* The lines are specified by a pair of points each. If they intersect, then
* the function returns true, else it returns false.
*
* Lines can be specified in the following form:
* A1x + B1x = C1
* A2x + B2x = C2
*
* If det (= A1*B2 - A2*B1) == 0, then lines are parallel
* else they intersect
*
* If they intersect, then let us denote the intersection point with P(x, y) where:
* x = (C1*B2 - C2*B1) / (det)
* y = (C2*A1 - C1*A2) / (det)
*
* @param a1 First point for determining the first line
* @param b1 Second point for determining the first line
* @param a2 First point for determining the second line
* @param b2 Second point for determining the second line
* @param intersection The intersection point, if this point exists
*/
static bool lineIntersection(const cv::Point2f &a1, const cv::Point2f &b1, const cv::Point2f &a2,
const cv::Point2f &b2, cv::Point2f &intersection) {
double A1 = b1.y - a1.y;
double B1 = a1.x - b1.x;
double C1 = (a1.x * A1) + (a1.y * B1);
double A2 = b2.y - a2.y;
double B2 = a2.x - b2.x;
double C2 = (a2.x * A2) + (a2.y * B2);
double det = (A1 * B2) - (A2 * B1);
if (!almostEqual(det, 0)) {
intersection.x = static_cast<float>(((C1 * B2) - (C2 * B1)) / (det));
intersection.y = static_cast<float>(((C2 * A1) - (C1 * A2)) / (det));
return true;
}
return false;
}
//! Get the values of "a", "b" and "c" of the line equation ax + by + c = 0 knowing that point "p" and "q" are on the line
/*!
* a = q.y - p.y
* b = p.x - q.x
* c = - (p.x * a) - (p.y * b)
*
* @param p Point p
* @param q Point q
* @param a Parameter "a" from the line equation
* @param b Parameter "b" from the line equation
* @param c Parameter "c" from the line equation
*/
static void lineEquationDeterminedByPoints(const cv::Point2f &p, const cv::Point2f &q,
double &a, double &b, double &c) {
CV_Assert(areEqualPoints(p, q) == false);
a = q.y - p.y;
b = p.x - q.x;
c = ((-p.y) * b) - (p.x * a);
}
//! Check if p1 and p2 are on the same side of the line determined by points a and b
/*!
* @param p1 Point p1
* @param p2 Point p2
* @param a First point for determining line
* @param b Second point for determining line
*/
static bool areOnTheSameSideOfLine(const cv::Point2f &p1, const cv::Point2f &p2,
const cv::Point2f &a, const cv::Point2f &b) {
double a1, b1, c1;
lineEquationDeterminedByPoints(a, b, a1, b1, c1);
double p1OnLine = (a1 * p1.x) + (b1 * p1.y) + c1;
double p2OnLine = (a1 * p2.x) + (b1 * p2.y) + c1;
return (sign(p1OnLine) == sign(p2OnLine));
}
//! Check if one point lies between two other points
/*!
* @param point Point lying possibly outside the line segment
* @param lineSegmentStart First point determining the line segment
* @param lineSegmentEnd Second point determining the line segment
*/
static bool isPointOnLineSegment(const cv::Point2f &point, const cv::Point2f &lineSegmentStart,
const cv::Point2f &lineSegmentEnd) {
double d1 = distanceBtwPoints(point, lineSegmentStart);
double d2 = distanceBtwPoints(point, lineSegmentEnd);
double lineSegmentLength = distanceBtwPoints(lineSegmentStart, lineSegmentEnd);
return (almostEqual(d1 + d2, lineSegmentLength));
}
//! Check if two lines are identical
/*!
* Lines are be specified in the following form:
* A1x + B1x = C1
* A2x + B2x = C2
*
* If (A1/A2) == (B1/B2) == (C1/C2), then the lines are identical
* else they are not
*
* @param a1 A1
* @param b1 B1
* @param c1 C1
* @param a2 A2
* @param b2 B2
* @param c2 C2
*/
static bool areIdenticalLines(double a1, double b1, double c1, double a2, double b2, double c2) {
double a1B2 = a1 * b2;
double a2B1 = a2 * b1;
double a1C2 = a1 * c2;
double a2C1 = a2 * c1;
double b1C2 = b1 * c2;
double b2C1 = b2 * c1;
return ((almostEqual(a1B2, a2B1)) && (almostEqual(b1C2, b2C1)) && (almostEqual(a1C2, a2C1)));
}
//! Check if points point1 and point2 are equal or not
/*!
* @param point1 One point
* @param point2 The other point
*/
static bool areEqualPoints(const cv::Point2f &point1, const cv::Point2f &point2) {
return (almostEqual(point1.x, point2.x) && almostEqual(point1.y, point2.y));
}
//! Return the sign of the number
/*!
* The sign function returns:
* -1, if number < 0
* +1, if number > 0
* 0, otherwise
*/
static int sign(double number) {
return (number > 0) ? 1 : ((number < 0) ? -1 : 0);
}
//! Return the maximum of the provided numbers
static double maximum(double number1, double number2, double number3) {
return std::max(std::max(number1, number2), number3);
}
//! Check if the two numbers are equal (almost)
/*!
* The expression for determining if two real numbers are equal is:
* if (Abs(x - y) <= EPSILON * Max(1.0f, Abs(x), Abs(y))).
*
* @param number1 First number
* @param number2 Second number
*/
static bool almostEqual(double number1, double number2) {
return (std::abs(number1 - number2) <= (EPSILON * maximum(1.0, std::abs(number1), std::abs(number2))));
}
//! Check if the first number is greater than or equal to the second number
/*!
* @param number1 First number
* @param number2 Second number
*/
static bool greaterOrEqual(double number1, double number2) {
return ((number1 > number2) || (almostEqual(number1, number2)));
}
//! Check if the first number is less than or equal to the second number
/*!
* @param number1 First number
* @param number2 Second number
*/
static bool lessOrEqual(double number1, double number2) {
return ((number1 < number2) || (almostEqual(number1, number2)));
}
};
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