homography.c

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/* Copyright (C) 2013-2016, The Regents of The University of Michigan.
All rights reserved.

This software was developed in the APRIL Robotics Lab under the
direction of Edwin Olson, ebolson@umich.edu. This software may be
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*/

#include <math.h>
#include <stdio.h>

#include "common/matd.h"
#include "common/zarray.h"
#include "common/homography.h"
#include "common/math_util.h"

// correspondences is a list of float[4]s, consisting of the points x
// and y concatenated. We will compute a homography such that y = Hx
matd_t *homography_compute(zarray_t *correspondences, int flags)
{
    // compute centroids of both sets of points (yields a better
    // conditioned information matrix)
    double x_cx = 0, x_cy = 0;
    double y_cx = 0, y_cy = 0;

    for (int i = 0; i < zarray_size(correspondences); i++) {
        float *c;
        zarray_get_volatile(correspondences, i, &c);

        x_cx += c[0];
        x_cy += c[1];
        y_cx += c[2];
        y_cy += c[3];
    }

    int sz = zarray_size(correspondences);
    x_cx /= sz;
    x_cy /= sz;
    y_cx /= sz;
    y_cy /= sz;

    // NB We don't normalize scale; it seems implausible that it could
    // possibly make any difference given the dynamic range of IEEE
    // doubles.

    matd_t *A = matd_create(9,9);
    for (int i = 0; i < zarray_size(correspondences); i++) {
        float *c;
        zarray_get_volatile(correspondences, i, &c);

        // (below world is "x", and image is "y")
        double worldx = c[0] - x_cx;
        double worldy = c[1] - x_cy;
        double imagex = c[2] - y_cx;
        double imagey = c[3] - y_cy;

        double a03 = -worldx;
        double a04 = -worldy;
        double a05 = -1;
        double a06 = worldx*imagey;
        double a07 = worldy*imagey;
        double a08 = imagey;

        MATD_EL(A, 3, 3) += a03*a03;
        MATD_EL(A, 3, 4) += a03*a04;
        MATD_EL(A, 3, 5) += a03*a05;
        MATD_EL(A, 3, 6) += a03*a06;
        MATD_EL(A, 3, 7) += a03*a07;
        MATD_EL(A, 3, 8) += a03*a08;
        MATD_EL(A, 4, 4) += a04*a04;
        MATD_EL(A, 4, 5) += a04*a05;
        MATD_EL(A, 4, 6) += a04*a06;
        MATD_EL(A, 4, 7) += a04*a07;
        MATD_EL(A, 4, 8) += a04*a08;
        MATD_EL(A, 5, 5) += a05*a05;
        MATD_EL(A, 5, 6) += a05*a06;
        MATD_EL(A, 5, 7) += a05*a07;
        MATD_EL(A, 5, 8) += a05*a08;
        MATD_EL(A, 6, 6) += a06*a06;
        MATD_EL(A, 6, 7) += a06*a07;
        MATD_EL(A, 6, 8) += a06*a08;
        MATD_EL(A, 7, 7) += a07*a07;
        MATD_EL(A, 7, 8) += a07*a08;
        MATD_EL(A, 8, 8) += a08*a08;

        double a10 = worldx;
        double a11 = worldy;
        double a12 = 1;
        double a16 = -worldx*imagex;
        double a17 = -worldy*imagex;
        double a18 = -imagex;

        MATD_EL(A, 0, 0) += a10*a10;
        MATD_EL(A, 0, 1) += a10*a11;
        MATD_EL(A, 0, 2) += a10*a12;
        MATD_EL(A, 0, 6) += a10*a16;
        MATD_EL(A, 0, 7) += a10*a17;
        MATD_EL(A, 0, 8) += a10*a18;
        MATD_EL(A, 1, 1) += a11*a11;
        MATD_EL(A, 1, 2) += a11*a12;
        MATD_EL(A, 1, 6) += a11*a16;
        MATD_EL(A, 1, 7) += a11*a17;
        MATD_EL(A, 1, 8) += a11*a18;
        MATD_EL(A, 2, 2) += a12*a12;
        MATD_EL(A, 2, 6) += a12*a16;
        MATD_EL(A, 2, 7) += a12*a17;
        MATD_EL(A, 2, 8) += a12*a18;
        MATD_EL(A, 6, 6) += a16*a16;
        MATD_EL(A, 6, 7) += a16*a17;
        MATD_EL(A, 6, 8) += a16*a18;
        MATD_EL(A, 7, 7) += a17*a17;
        MATD_EL(A, 7, 8) += a17*a18;
        MATD_EL(A, 8, 8) += a18*a18;

        double a20 = -worldx*imagey;
        double a21 = -worldy*imagey;
        double a22 = -imagey;
        double a23 = worldx*imagex;
        double a24 = worldy*imagex;
        double a25 = imagex;

        MATD_EL(A, 0, 0) += a20*a20;
        MATD_EL(A, 0, 1) += a20*a21;
        MATD_EL(A, 0, 2) += a20*a22;
        MATD_EL(A, 0, 3) += a20*a23;
        MATD_EL(A, 0, 4) += a20*a24;
        MATD_EL(A, 0, 5) += a20*a25;
        MATD_EL(A, 1, 1) += a21*a21;
        MATD_EL(A, 1, 2) += a21*a22;
        MATD_EL(A, 1, 3) += a21*a23;
        MATD_EL(A, 1, 4) += a21*a24;
        MATD_EL(A, 1, 5) += a21*a25;
        MATD_EL(A, 2, 2) += a22*a22;
        MATD_EL(A, 2, 3) += a22*a23;
        MATD_EL(A, 2, 4) += a22*a24;
        MATD_EL(A, 2, 5) += a22*a25;
        MATD_EL(A, 3, 3) += a23*a23;
        MATD_EL(A, 3, 4) += a23*a24;
        MATD_EL(A, 3, 5) += a23*a25;
        MATD_EL(A, 4, 4) += a24*a24;
        MATD_EL(A, 4, 5) += a24*a25;
        MATD_EL(A, 5, 5) += a25*a25;
    }

    // make symmetric
    for (int i = 0; i < 9; i++)
        for (int j = i+1; j < 9; j++)
            MATD_EL(A, j, i) = MATD_EL(A, i, j);

    matd_t *H = matd_create(3,3);

    if (flags & HOMOGRAPHY_COMPUTE_FLAG_INVERSE) {
        // compute singular vector by (carefully) inverting the rank-deficient matrix.

        if (1) {
            matd_t *Ainv = matd_inverse(A);
            double scale = 0;

            for (int i = 0; i < 9; i++)
                scale += sq(MATD_EL(Ainv, i, 0));
            scale = sqrt(scale);

            for (int i = 0; i < 3; i++)
                for (int j = 0; j < 3; j++)
                    MATD_EL(H, i, j) = MATD_EL(Ainv, 3*i+j, 0) / scale;

            matd_destroy(Ainv);
        } else {

            matd_t *b = matd_create_data(9, 1, (double[]) { 1, 0, 0, 0, 0, 0, 0, 0, 0 });
            matd_t *Ainv = NULL;

            if (0) {
                matd_plu_t *lu = matd_plu(A);
                Ainv = matd_plu_solve(lu, b);
                matd_plu_destroy(lu);
            } else {
                matd_chol_t *chol = matd_chol(A);
                Ainv = matd_chol_solve(chol, b);
                matd_chol_destroy(chol);
            }

            double scale = 0;

            for (int i = 0; i < 9; i++)
                scale += sq(MATD_EL(Ainv, i, 0));
            scale = sqrt(scale);

            for (int i = 0; i < 3; i++)
                for (int j = 0; j < 3; j++)
                    MATD_EL(H, i, j) = MATD_EL(Ainv, 3*i+j, 0) / scale;

            matd_destroy(b);
            matd_destroy(Ainv);
        }

    } else {
        // compute singular vector using SVD. A bit slower, but more accurate.
        matd_svd_t svd = matd_svd_flags(A, MATD_SVD_NO_WARNINGS);

        for (int i = 0; i < 3; i++)
            for (int j = 0; j < 3; j++)
                MATD_EL(H, i, j) = MATD_EL(svd.U, 3*i+j, 8);

        matd_destroy(svd.U);
        matd_destroy(svd.S);
        matd_destroy(svd.V);

    }

    matd_t *Tx = matd_identity(3);
    MATD_EL(Tx,0,2) = -x_cx;
    MATD_EL(Tx,1,2) = -x_cy;

    matd_t *Ty = matd_identity(3);
    MATD_EL(Ty,0,2) = y_cx;
    MATD_EL(Ty,1,2) = y_cy;

    matd_t *H2 = matd_op("M*M*M", Ty, H, Tx);

    matd_destroy(A);
    matd_destroy(Tx);
    matd_destroy(Ty);
    matd_destroy(H);

    return H2;
}


// assuming that the projection matrix is:
// [ fx 0  cx 0 ]
// [  0 fy cy 0 ]
// [  0  0  1 0 ]
//
// And that the homography is equal to the projection matrix times the
// model matrix, recover the model matrix (which is returned). Note
// that the third column of the model matrix is missing in the
// expresison below, reflecting the fact that the homography assumes
// all points are at z=0 (i.e., planar) and that the element of z is
// thus omitted.  (3x1 instead of 4x1).
//
// [ fx 0  cx 0 ] [ R00  R01  TX ]    [ H00 H01 H02 ]
// [  0 fy cy 0 ] [ R10  R11  TY ] =  [ H10 H11 H12 ]
// [  0  0  1 0 ] [ R20  R21  TZ ] =  [ H20 H21 H22 ]
//                [  0    0    1 ]
//
// fx*R00 + cx*R20 = H00   (note, H only known up to scale; some additional adjustments required; see code.)
// fx*R01 + cx*R21 = H01
// fx*TX  + cx*TZ  = H02
// fy*R10 + cy*R20 = H10
// fy*R11 + cy*R21 = H11
// fy*TY  + cy*TZ  = H12
// R20 = H20
// R21 = H21
// TZ  = H22

matd_t *homography_to_pose(const matd_t *H, double fx, double fy, double cx, double cy)
{
    // Note that every variable that we compute is proportional to the scale factor of H.
    double R20 = MATD_EL(H, 2, 0);
    double R21 = MATD_EL(H, 2, 1);
    double TZ  = MATD_EL(H, 2, 2);
    double R00 = (MATD_EL(H, 0, 0) - cx*R20) / fx;
    double R01 = (MATD_EL(H, 0, 1) - cx*R21) / fx;
    double TX  = (MATD_EL(H, 0, 2) - cx*TZ)  / fx;
    double R10 = (MATD_EL(H, 1, 0) - cy*R20) / fy;
    double R11 = (MATD_EL(H, 1, 1) - cy*R21) / fy;
    double TY  = (MATD_EL(H, 1, 2) - cy*TZ)  / fy;

    // compute the scale by requiring that the rotation columns are unit length
    // (Use geometric average of the two length vectors we have)
    double length1 = sqrtf(R00*R00 + R10*R10 + R20*R20);
    double length2 = sqrtf(R01*R01 + R11*R11 + R21*R21);
    double s = 1.0 / sqrtf(length1 * length2);

    // get sign of S by requiring the tag to be in front the camera;
    // we assume camera looks in the -Z direction.
    if (TZ > 0)
        s *= -1;

    R20 *= s;
    R21 *= s;
    TZ  *= s;
    R00 *= s;
    R01 *= s;
    TX  *= s;
    R10 *= s;
    R11 *= s;
    TY  *= s;

    // now recover [R02 R12 R22] by noting that it is the cross product of the other two columns.
    double R02 = R10*R21 - R20*R11;
    double R12 = R20*R01 - R00*R21;
    double R22 = R00*R11 - R10*R01;

    // Improve rotation matrix by applying polar decomposition.
    if (1) {
        // do polar decomposition. This makes the rotation matrix
        // "proper", but probably increases the reprojection error. An
        // iterative alignment step would be superior.

        matd_t *R = matd_create_data(3, 3, (double[]) { R00, R01, R02,
                                                       R10, R11, R12,
                                                       R20, R21, R22 });

        matd_svd_t svd = matd_svd(R);
        matd_destroy(R);

        R = matd_op("M*M'", svd.U, svd.V);

        matd_destroy(svd.U);
        matd_destroy(svd.S);
        matd_destroy(svd.V);

        R00 = MATD_EL(R, 0, 0);
        R01 = MATD_EL(R, 0, 1);
        R02 = MATD_EL(R, 0, 2);
        R10 = MATD_EL(R, 1, 0);
        R11 = MATD_EL(R, 1, 1);
        R12 = MATD_EL(R, 1, 2);
        R20 = MATD_EL(R, 2, 0);
        R21 = MATD_EL(R, 2, 1);
        R22 = MATD_EL(R, 2, 2);

        matd_destroy(R);
    }

    return matd_create_data(4, 4, (double[]) { R00, R01, R02, TX,
                                               R10, R11, R12, TY,
                                               R20, R21, R22, TZ,
                                                0, 0, 0, 1 });
}

// Similar to above
// Recover the model view matrix assuming that the projection matrix is:
//
// [ F  0  A  0 ]     (see glFrustrum)
// [ 0  G  B  0 ]
// [ 0  0  C  D ]
// [ 0  0 -1  0 ]

matd_t *homography_to_model_view(const matd_t *H, double F, double G, double A, double B, double C, double D)
{
    // Note that every variable that we compute is proportional to the scale factor of H.
    double R20 = -MATD_EL(H, 2, 0);
    double R21 = -MATD_EL(H, 2, 1);
    double TZ  = -MATD_EL(H, 2, 2);
    double R00 = (MATD_EL(H, 0, 0) - A*R20) / F;
    double R01 = (MATD_EL(H, 0, 1) - A*R21) / F;
    double TX  = (MATD_EL(H, 0, 2) - A*TZ)  / F;
    double R10 = (MATD_EL(H, 1, 0) - B*R20) / G;
    double R11 = (MATD_EL(H, 1, 1) - B*R21) / G;
    double TY  = (MATD_EL(H, 1, 2) - B*TZ)  / G;

    // compute the scale by requiring that the rotation columns are unit length
    // (Use geometric average of the two length vectors we have)
    double length1 = sqrtf(R00*R00 + R10*R10 + R20*R20);
    double length2 = sqrtf(R01*R01 + R11*R11 + R21*R21);
    double s = 1.0 / sqrtf(length1 * length2);

    // get sign of S by requiring the tag to be in front of the camera
    // (which is Z < 0) for our conventions.
    if (TZ > 0)
        s *= -1;

    R20 *= s;
    R21 *= s;
    TZ  *= s;
    R00 *= s;
    R01 *= s;
    TX  *= s;
    R10 *= s;
    R11 *= s;
    TY  *= s;

    // now recover [R02 R12 R22] by noting that it is the cross product of the other two columns.
    double R02 = R10*R21 - R20*R11;
    double R12 = R20*R01 - R00*R21;
    double R22 = R00*R11 - R10*R01;

    // TODO XXX: Improve rotation matrix by applying polar decomposition.

    return matd_create_data(4, 4, (double[]) { R00, R01, R02, TX,
        R10, R11, R12, TY,
        R20, R21, R22, TZ,
        0, 0, 0, 1 });
}

// Only uses the upper 3x3 matrix.
/*
static void matrix_to_quat(const matd_t *R, double q[4])
{
    // see: "from quaternion to matrix and back"

    // trace: get the same result if R is 4x4 or 3x3:
    double T = MATD_EL(R, 0, 0) + MATD_EL(R, 1, 1) + MATD_EL(R, 2, 2) + 1;
    double S = 0;

    double m0  = MATD_EL(R, 0, 0);
    double m1  = MATD_EL(R, 1, 0);
    double m2  = MATD_EL(R, 2, 0);
    double m4  = MATD_EL(R, 0, 1);
    double m5  = MATD_EL(R, 1, 1);
    double m6  = MATD_EL(R, 2, 1);
    double m8  = MATD_EL(R, 0, 2);
    double m9  = MATD_EL(R, 1, 2);
    double m10 = MATD_EL(R, 2, 2);

    if (T > 0.0000001) {
        S = sqrtf(T) * 2;
        q[1] = -( m9 - m6 ) / S;
        q[2] = -( m2 - m8 ) / S;
        q[3] = -( m4 - m1 ) / S;
        q[0] = 0.25 * S;
    } else if ( m0 > m5 && m0 > m10 )  {        // Column 0:
        S  = sqrtf( 1.0 + m0 - m5 - m10 ) * 2;
        q[1] = -0.25 * S;
        q[2] = -(m4 + m1 ) / S;
        q[3] = -(m2 + m8 ) / S;
        q[0] = (m9 - m6 ) / S;
    } else if ( m5 > m10 ) {                    // Column 1:
        S  = sqrtf( 1.0 + m5 - m0 - m10 ) * 2;
        q[1] = -(m4 + m1 ) / S;
        q[2] = -0.25 * S;
        q[3] = -(m9 + m6 ) / S;
        q[0] = (m2 - m8 ) / S;
    } else {
        // Column 2:
        S  = sqrtf( 1.0 + m10 - m0 - m5 ) * 2;
        q[1] = -(m2 + m8 ) / S;
        q[2] = -(m9 + m6 ) / S;
        q[3] = -0.25 * S;
        q[0] = (m4 - m1 ) / S;
    }

    double mag2 = 0;
    for (int i = 0; i < 4; i++)
        mag2 += q[i]*q[i];
    double norm = 1.0 / sqrtf(mag2);
    for (int i = 0; i < 4; i++)
        q[i] *= norm;
}
*/

// overwrites upper 3x3 area of matrix M. Doesn't touch any other elements of M.
void quat_to_matrix(const double q[4], matd_t *M)
{
    double w = q[0], x = q[1], y = q[2], z = q[3];

    MATD_EL(M, 0, 0) = w*w + x*x - y*y - z*z;
    MATD_EL(M, 0, 1) = 2*x*y - 2*w*z;
    MATD_EL(M, 0, 2) = 2*x*z + 2*w*y;

    MATD_EL(M, 1, 0) = 2*x*y + 2*w*z;
    MATD_EL(M, 1, 1) = w*w - x*x + y*y - z*z;
    MATD_EL(M, 1, 2) = 2*y*z - 2*w*x;

    MATD_EL(M, 2, 0) = 2*x*z - 2*w*y;
    MATD_EL(M, 2, 1) = 2*y*z + 2*w*x;
    MATD_EL(M, 2, 2) = w*w - x*x - y*y + z*z;
}