如何在2D中绘制透视正确的网格

我有一个应用程序,在图像/照片的顶部定义一个真实世界的矩形,当然在2D它可能不是一个矩形,因为你从一个角度看它.

问题是,例如矩形需要在其上绘制网格线,例如,如果它是3x5,那么我需要从侧面1到侧面3绘制2条线,从侧面2到侧面4绘制4条线.

到目前为止,我将每条线分成等距的部分,以获得所有网格线的起点和终点.然而,矩形越大,角度越大,这些线越"不正确",因为离你越远的水平线应该越靠近.

有谁知道我应该搜索的算法的名称？

是的我知道你可以用3D做到这一点,但是对于这个特殊的应用程序,我仅限于2D.

这是解决方案:http: //freespace.virgin.net/hugo.elias/graphics/x_persp.htm

基本思路是通过对角连接角可以找到矩形的透视正确"中心".两条结果线的交点是您的透视正确中心.从那里你将你的矩形细分为四个较小的矩形,然后重复这个过程.次数取决于您想要的准确度.您可以细分到像素大小的下方,以获得有效完美的视角.

然后在您的子矩形中,您只需应用标准的未经修正的"纹理"三角形,或矩形等等.

您可以执行此算法,而无需解决构建"真实"3D世界的复杂问题.如果你确实有一个真正的3D世界建模,但你的短信不是在硬件中进行透视校正,或者你需要一种高效的方法来获得透视正确的平面,而不需要每个像素渲染技巧.

图像:双线性和透视变换的示例(注意:顶部和底部水平网格线的高度实际上是其余线高度的一半,在两个图纸上)

========================================

我知道这是一个老问题,但我有一个通用的解决方案,所以我决定发布它,它将对未来的读者有用.下面的代码可以绘制任意透视网格而无需重复计算.

我实际上开始遇到类似的问题:绘制2D透视网格,然后转换下划线图像以恢复透视.

我开始在这里阅读:http: //www.imagemagick.org/Usage/distorts/#bilinear_forward

然后在这里(Leptonica图书馆):http: //www.leptonica.com/affine.html

我找到了这个:

当您从有限距离的某个任意方向观察平面中的物体时,图像中会出现额外的"梯形失真"失真.这是一个投影变换,它保持直线直线,但不保持线之间的角度.这种扭曲不能通过线性仿射变换来描述,并且实际上不同于分母中的x和y依赖项.

转型不是线性的,正如许多人已经在这个帖子中指出的那样.它涉及求解8个方程的线性系统(一次)来计算8个所需系数,然后您可以使用它们来转换任意数量的点.

为了避免在我的项目中包含所有Leptonica库,我从中获取了一些代码,我删除了所有特殊的Leptonica数据类型和宏,我修复了一些内存泄漏并将其转换为C++类(主要是出于封装原因)只做一件事:它将(Qt)QPointF float(x,y)坐标映射到相应的Perspective Coordinate.

如果要将代码调整到另一个C++库,则重新定义/替换的唯一方法是QPointF坐标类.

我希望未来的读者会发现它很有用.下面的代码分为3部分:

A.关于如何使用genImageProjective C++类绘制2D透视网格的示例

B. genImageProjective.h文件

C. genImageProjective.cpp文件

//============================================================
// C++ Code Example on how to use the 
//     genImageProjective class to draw a perspective 2D Grid
//============================================================

#include "genImageProjective.h"

// Input: 4 Perspective-Tranformed points:
//        perspPoints[0] = top-left
//        perspPoints[1] = top-right
//        perspPoints[2] = bottom-right
//        perspPoints[3] = bottom-left
void drawGrid(QPointF *perspPoints)
{
(...)
        // Setup a non-transformed area rectangle
        // I use a simple square rectangle here because in this case we are not interested in the source-rectangle,
        //  (we want to just draw a grid on the perspPoints[] area)
        //   but you can use any arbitrary rectangle to perform a real mapping to the perspPoints[] area
        QPointF topLeft = QPointF(0,0);
        QPointF topRight = QPointF(1000,0);
        QPointF bottomRight = QPointF(1000,1000);
        QPointF bottomLeft = QPointF(0,1000);
        float width = topRight.x() - topLeft.x();
        float height = bottomLeft.y() - topLeft.y();

        // Setup Projective trasform object
        genImageProjective imageProjective;
        imageProjective.sourceArea[0] = topLeft;
        imageProjective.sourceArea[1] = topRight;
        imageProjective.sourceArea[2] = bottomRight;
        imageProjective.sourceArea[3] = bottomLeft;
        imageProjective.destArea[0] = perspPoints[0];
        imageProjective.destArea[1] = perspPoints[1];
        imageProjective.destArea[2] = perspPoints[2];
        imageProjective.destArea[3] = perspPoints[3];
        // Compute projective transform coefficients
        if (imageProjective.computeCoeefficients() != 0)
            return; // This can actually fail if any 3 points of Source or Dest are colinear

        // Initialize Grid parameters (without transform)
        float gridFirstLine = 0.1f; // The normalized position of first Grid Line (0.0 to 1.0)
        float gridStep = 0.1f;      // The normalized Grd size (=distance between grid lines: 0.0 to 1.0)

        // Draw Horizonal Grid lines
        QPointF lineStart, lineEnd, tempPnt;
        for (float pos = gridFirstLine; pos <= 1.0f; pos += gridStep)
        {
            // Compute Grid Line Start
            tempPnt = QPointF(topLeft.x(), topLeft.y() + pos*width);
            imageProjective.mapSourceToDestPoint(tempPnt, lineStart);
            // Compute Grid Line End
            tempPnt = QPointF(topRight.x(), topLeft.y() + pos*width);
            imageProjective.mapSourceToDestPoint(tempPnt, lineEnd);

            // Draw Horizontal Line (use your prefered method to draw the line)
            (...)
        }
        // Draw Vertical Grid lines
        for (float pos = gridFirstLine; pos <= 1.0f; pos += gridStep)
        {
            // Compute Grid Line Start
            tempPnt = QPointF(topLeft.x() + pos*height, topLeft.y());
            imageProjective.mapSourceToDestPoint(tempPnt, lineStart);
            // Compute Grid Line End
            tempPnt = QPointF(topLeft.x() + pos*height, bottomLeft.y());
            imageProjective.mapSourceToDestPoint(tempPnt, lineEnd);

            // Draw Vertical Line (use your prefered method to draw the line)
            (...)
        }
(...)
}

==========================================



//========================================
//C++ Header File: genImageProjective.h
//========================================

#ifndef GENIMAGE_H
#define GENIMAGE_H

#include 

// Class to transform an Image Point using Perspective transformation
class genImageProjective
{
public:
    genImageProjective();

    int computeCoeefficients(void);
    int mapSourceToDestPoint(QPointF& sourcePoint, QPointF& destPoint);

public:
    QPointF sourceArea[4]; // Source Image area limits (Rectangular)
    QPointF destArea[4];   // Destination Image area limits (Perspectivelly Transformed)

private:
    static int gaussjordan(float  **a, float  *b, int n);

    bool coefficientsComputed;
    float vc[8];           // Vector of Transform Coefficients
};

#endif // GENIMAGE_H
//========================================


//========================================
//C++ CPP File: genImageProjective.cpp
//========================================

#include 
#include "genImageProjective.h"

// ----------------------------------------------------
// class genImageProjective
// ----------------------------------------------------
genImageProjective::genImageProjective()
{
    sourceArea[0] = sourceArea[1] = sourceArea[2] = sourceArea[3] = QPointF(0,0);
    destArea[0] = destArea[1] = destArea[2] = destArea[3] = QPointF(0,0);
    coefficientsComputed = false;
}


// --------------------------------------------------------------
// Compute projective transform coeeeficients
// RetValue: 0: Success, !=0: Error
/*-------------------------------------------------------------*
 *                Projective coordinate transformation         *
 *-------------------------------------------------------------*/
/*!
 *  computeCoeefficients()
 *
 *      Input:  this->sourceArea[4]: (source 4 points; unprimed)
 *              this->destArea[4]:   (transformed 4 points; primed)
 *              this->vc  (computed vector of transform coefficients)
 *      Return: 0 if OK; <0 on error
 *
 *  We have a set of 8 equations, describing the projective
 *  transformation that takes 4 points (sourceArea) into 4 other
 *  points (destArea).  These equations are:
 *
 *          x1' = (c[0]*x1 + c[1]*y1 + c[2]) / (c[6]*x1 + c[7]*y1 + 1)
 *          y1' = (c[3]*x1 + c[4]*y1 + c[5]) / (c[6]*x1 + c[7]*y1 + 1)
 *          x2' = (c[0]*x2 + c[1]*y2 + c[2]) / (c[6]*x2 + c[7]*y2 + 1)
 *          y2' = (c[3]*x2 + c[4]*y2 + c[5]) / (c[6]*x2 + c[7]*y2 + 1)
 *          x3' = (c[0]*x3 + c[1]*y3 + c[2]) / (c[6]*x3 + c[7]*y3 + 1)
 *          y3' = (c[3]*x3 + c[4]*y3 + c[5]) / (c[6]*x3 + c[7]*y3 + 1)
 *          x4' = (c[0]*x4 + c[1]*y4 + c[2]) / (c[6]*x4 + c[7]*y4 + 1)
 *          y4' = (c[3]*x4 + c[4]*y4 + c[5]) / (c[6]*x4 + c[7]*y4 + 1)
 *
 *  Multiplying both sides of each eqn by the denominator, we get
 *
 *           AC = B
 *
 *  where B and C are column vectors
 *
 *         B = [ x1' y1' x2' y2' x3' y3' x4' y4' ]
 *         C = [ c[0] c[1] c[2] c[3] c[4] c[5] c[6] c[7] ]
 *
 *  and A is the 8x8 matrix
 *
 *             x1   y1     1     0   0    0   -x1*x1'  -y1*x1'
 *              0    0     0    x1   y1   1   -x1*y1'  -y1*y1'
 *             x2   y2     1     0   0    0   -x2*x2'  -y2*x2'
 *              0    0     0    x2   y2   1   -x2*y2'  -y2*y2'
 *             x3   y3     1     0   0    0   -x3*x3'  -y3*x3'
 *              0    0     0    x3   y3   1   -x3*y3'  -y3*y3'
 *             x4   y4     1     0   0    0   -x4*x4'  -y4*x4'
 *              0    0     0    x4   y4   1   -x4*y4'  -y4*y4'
 *
 *  These eight equations are solved here for the coefficients C.
 *
 *  These eight coefficients can then be used to find the mapping
 *  (x,y) --> (x',y'):
 *
 *           x' = (c[0]x + c[1]y + c[2]) / (c[6]x + c[7]y + 1)
 *           y' = (c[3]x + c[4]y + c[5]) / (c[6]x + c[7]y + 1)
 *
 */
int genImageProjective::computeCoeefficients(void)
{
    int retValue = 0;
    int     i;
    float  *a[8];  /* 8x8 matrix A  */
    float  *b = this->vc; /* rhs vector of primed coords X'; coeffs returned in vc[] */

    b[0] = destArea[0].x();
    b[1] = destArea[0].y();
    b[2] = destArea[1].x();
    b[3] = destArea[1].y();
    b[4] = destArea[2].x();
    b[5] = destArea[2].y();
    b[6] = destArea[3].x();
    b[7] = destArea[3].y();

    for (i = 0; i < 8; i++)
        a[i] = NULL;
    for (i = 0; i < 8; i++)
    {
        if ((a[i] = (float *)calloc(8, sizeof(float))) == NULL)
        {
            retValue = -100; // ERROR_INT("a[i] not made", procName, 1);
            goto Terminate;
        }
    }

    a[0][0] = sourceArea[0].x();
    a[0][1] = sourceArea[0].y();
    a[0][2] = 1.;
    a[0][6] = -sourceArea[0].x() * b[0];
    a[0][7] = -sourceArea[0].y() * b[0];
    a[1][3] = sourceArea[0].x();
    a[1][4] = sourceArea[0].y();
    a[1][5] = 1;
    a[1][6] = -sourceArea[0].x() * b[1];
    a[1][7] = -sourceArea[0].y() * b[1];
    a[2][0] = sourceArea[1].x();
    a[2][1] = sourceArea[1].y();
    a[2][2] = 1.;
    a[2][6] = -sourceArea[1].x() * b[2];
    a[2][7] = -sourceArea[1].y() * b[2];
    a[3][3] = sourceArea[1].x();
    a[3][4] = sourceArea[1].y();
    a[3][5] = 1;
    a[3][6] = -sourceArea[1].x() * b[3];
    a[3][7] = -sourceArea[1].y() * b[3];
    a[4][0] = sourceArea[2].x();
    a[4][1] = sourceArea[2].y();
    a[4][2] = 1.;
    a[4][6] = -sourceArea[2].x() * b[4];
    a[4][7] = -sourceArea[2].y() * b[4];
    a[5][3] = sourceArea[2].x();
    a[5][4] = sourceArea[2].y();
    a[5][5] = 1;
    a[5][6] = -sourceArea[2].x() * b[5];
    a[5][7] = -sourceArea[2].y() * b[5];
    a[6][0] = sourceArea[3].x();
    a[6][1] = sourceArea[3].y();
    a[6][2] = 1.;
    a[6][6] = -sourceArea[3].x() * b[6];
    a[6][7] = -sourceArea[3].y() * b[6];
    a[7][3] = sourceArea[3].x();
    a[7][4] = sourceArea[3].y();
    a[7][5] = 1;
    a[7][6] = -sourceArea[3].x() * b[7];
    a[7][7] = -sourceArea[3].y() * b[7];

    retValue = gaussjordan(a, b, 8);

Terminate:
    // Clean up
    for (i = 0; i < 8; i++)
    {
        if (a[i])
            free(a[i]);
    }

    this->coefficientsComputed = (retValue == 0);
    return retValue;
}


/*-------------------------------------------------------------*
 *               Gauss-jordan linear equation solver           *
 *-------------------------------------------------------------*/
/*
 *  gaussjordan()
 *
 *      Input:   a  (n x n matrix)
 *               b  (rhs column vector)
 *               n  (dimension)
 *      Return:  0 if ok, 1 on error
 *
 *      Note side effects:
 *            (1) the matrix a is transformed to its inverse
 *            (2) the vector b is transformed to the solution X to the
 *                linear equation AX = B
 *
 *      Adapted from "Numerical Recipes in C, Second Edition", 1992
 *      pp. 36-41 (gauss-jordan elimination)
 */
#define  SWAP(a,b)   {temp = (a); (a) = (b); (b) = temp;}
int genImageProjective::gaussjordan(float  **a, float  *b, int n)
{
    int retValue = 0;
    int i, icol=0, irow=0, j, k, l, ll;
    int *indexc = NULL, *indexr = NULL, *ipiv = NULL;
    float  big, dum, pivinv, temp;

    if (!a)
    {
        retValue = -1; // ERROR_INT("a not defined", procName, 1);
        goto Terminate;
    }
    if (!b)
    {
        retValue = -2; // ERROR_INT("b not defined", procName, 1);
        goto Terminate;
    }

    if ((indexc = (int *)calloc(n, sizeof(int))) == NULL)
    {
        retValue = -3; // ERROR_INT("indexc not made", procName, 1);
        goto Terminate;
    }
    if ((indexr = (int *)calloc(n, sizeof(int))) == NULL)
    {
        retValue = -4; // ERROR_INT("indexr not made", procName, 1);
        goto Terminate;
    }
    if ((ipiv = (int *)calloc(n, sizeof(int))) == NULL)
    {
        retValue = -5; // ERROR_INT("ipiv not made", procName, 1);
        goto Terminate;
    }

    for (i = 0; i < n; i++)
    {
        big = 0.0;
        for (j = 0; j < n; j++)
        {
            if (ipiv[j] != 1)
            {
                for (k = 0; k < n; k++)
                {
                    if (ipiv[k] == 0)
                    {
                        if (fabs(a[j][k]) >= big)
                        {
                            big = fabs(a[j][k]);
                            irow = j;
                            icol = k;
                        }
                    }
                    else if (ipiv[k] > 1)
                    {
                        retValue = -6; // ERROR_INT("singular matrix", procName, 1);
                        goto Terminate;
                    }
                }
            }
        }
        ++(ipiv[icol]);

        if (irow != icol)
        {
            for (l = 0; l < n; l++)
                SWAP(a[irow][l], a[icol][l]);
            SWAP(b[irow], b[icol]);
        }

        indexr[i] = irow;
        indexc[i] = icol;
        if (a[icol][icol] == 0.0)
        {
            retValue = -7; // ERROR_INT("singular matrix", procName, 1);
            goto Terminate;
        }
        pivinv = 1.0 / a[icol][icol];
        a[icol][icol] = 1.0;
        for (l = 0; l < n; l++)
            a[icol][l] *= pivinv;
        b[icol] *= pivinv;

        for (ll = 0; ll < n; ll++)
        {
            if (ll != icol)
            {
                dum = a[ll][icol];
                a[ll][icol] = 0.0;
                for (l = 0; l < n; l++)
                    a[ll][l] -= a[icol][l] * dum;
                b[ll] -= b[icol] * dum;
            }
        }
    }

    for (l = n - 1; l >= 0; l--)
    {
        if (indexr[l] != indexc[l])
        {
            for (k = 0; k < n; k++)
                SWAP(a[k][indexr[l]], a[k][indexc[l]]);
        }
    }

Terminate:
    if (indexr)
        free(indexr);
    if (indexc)
        free(indexc);
    if (ipiv)
        free(ipiv);
    return retValue;
}


// --------------------------------------------------------------
// Map a source point to destination using projective transform
// --------------------------------------------------------------
// Params:
//  sourcePoint: initial point
//  destPoint:   transformed point
// RetValue: 0: Success, !=0: Error
// --------------------------------------------------------------
//  Notes:
//   1. You must call once computeCoeefficients() to compute
//      the this->vc[] vector of 8 coefficients, before you call
//      mapSourceToDestPoint().
//   2. If there was an error or the 8 coefficients were not computed,
//      a -1 is returned and destPoint is just set to sourcePoint value.
// --------------------------------------------------------------
int genImageProjective::mapSourceToDestPoint(QPointF& sourcePoint, QPointF& destPoint)
{
    if (coefficientsComputed)
    {
        float factor = 1.0f / (vc[6] * sourcePoint.x() + vc[7] * sourcePoint.y() + 1.);
        destPoint.setX( factor * (vc[0] * sourcePoint.x() + vc[1] * sourcePoint.y() + vc[2]) );
        destPoint.setY( factor * (vc[3] * sourcePoint.x() + vc[4] * sourcePoint.y() + vc[5]) );
        return 0;
    }
    else // There was an error while computing coefficients
    {
        destPoint = sourcePoint; // just copy the source to destination...
        return -1;               // ...and return an error
    }
}
//========================================