1 | function [H,Hnorm,inv_Hnorm] = compute_homography(m,M); |
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2 | |
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3 | %compute_homography |
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4 | % |
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5 | %[H,Hnorm,inv_Hnorm] = compute_homography(m,M) |
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6 | % |
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7 | %Computes the planar homography between the point coordinates on the plane (M) and the image |
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8 | %point coordinates (m). |
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9 | % |
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10 | %INPUT: m: homogeneous coordinates in the image plane (3xN matrix) |
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11 | % M: homogeneous coordinates in the plane in 3D (3xN matrix) |
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12 | % |
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13 | %OUTPUT: H: Homography matrix (3x3 homogeneous matrix) |
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14 | % Hnorm: Normalization matrix used on the points before homography computation |
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15 | % (useful for numerical stability is points in pixel coordinates) |
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16 | % inv_Hnorm: The inverse of Hnorm |
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17 | % |
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18 | %Definition: m ~ H*M where "~" means equal up to a non zero scalar factor. |
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19 | % |
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20 | %Method: First computes an initial guess for the homography through quasi-linear method. |
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21 | % Then, if the total number of points is larger than 4, optimize the solution by minimizing |
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22 | % the reprojection error (in the least squares sense). |
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23 | % |
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24 | % |
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25 | %Important functions called within that program: |
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26 | % |
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27 | %comp_distortion_oulu: Undistorts pixel coordinates. |
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28 | % |
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29 | %compute_homography.m: Computes the planar homography between points on the grid in 3D, and the image plane. |
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30 | % |
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31 | %project_points.m: Computes the 2D image projections of a set of 3D points, and also returns te Jacobian |
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32 | % matrix (derivative with respect to the intrinsic and extrinsic parameters). |
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33 | % This function is called within the minimization loop. |
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34 | |
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35 | |
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36 | |
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37 | |
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38 | Np = size(m,2); |
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39 | |
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40 | if size(m,1)<3, |
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41 | m = [m;ones(1,Np)]; |
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42 | end; |
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43 | |
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44 | if size(M,1)<3, |
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45 | M = [M;ones(1,Np)]; |
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46 | end; |
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47 | |
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48 | |
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49 | m = m ./ (ones(3,1)*m(3,:)); |
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50 | M = M ./ (ones(3,1)*M(3,:)); |
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51 | |
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52 | % Prenormalization of point coordinates (very important): |
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53 | % (Affine normalization) |
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54 | |
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55 | ax = m(1,:); |
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56 | ay = m(2,:); |
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57 | |
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58 | mxx = mean(ax); |
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59 | myy = mean(ay); |
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60 | ax = ax - mxx; |
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61 | ay = ay - myy; |
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62 | |
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63 | scxx = mean(abs(ax)); |
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64 | scyy = mean(abs(ay)); |
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65 | |
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66 | |
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67 | Hnorm = [1/scxx 0 -mxx/scxx;0 1/scyy -myy/scyy;0 0 1]; |
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68 | inv_Hnorm = [scxx 0 mxx ; 0 scyy myy; 0 0 1]; |
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69 | |
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70 | mn = Hnorm*m; |
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71 | |
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72 | % Compute the homography between m and mn: |
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73 | |
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74 | % Build the matrix: |
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75 | |
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76 | L = zeros(2*Np,9); |
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77 | |
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78 | L(1:2:2*Np,1:3) = M'; |
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79 | L(2:2:2*Np,4:6) = M'; |
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80 | L(1:2:2*Np,7:9) = -((ones(3,1)*mn(1,:)).* M)'; |
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81 | L(2:2:2*Np,7:9) = -((ones(3,1)*mn(2,:)).* M)'; |
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82 | |
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83 | if Np > 4, |
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84 | L = L'*L; |
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85 | end; |
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86 | |
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87 | [U,S,V] = svd(L); |
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88 | |
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89 | hh = V(:,9); |
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90 | hh = hh/hh(9); |
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91 | |
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92 | Hrem = reshape(hh,3,3)'; |
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93 | %Hrem = Hrem / Hrem(3,3); |
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94 | |
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95 | |
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96 | % Final homography: |
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97 | |
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98 | H = inv_Hnorm*Hrem; |
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99 | |
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100 | if 0, |
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101 | m2 = H*M; |
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102 | m2 = [m2(1,:)./m2(3,:) ; m2(2,:)./m2(3,:)]; |
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103 | merr = m(1:2,:) - m2; |
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104 | end; |
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105 | |
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106 | %keyboard; |
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107 | |
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108 | %%% Homography refinement if there are more than 4 points: |
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109 | |
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110 | if Np > 4, |
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111 | |
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112 | % Final refinement: |
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113 | hhv = reshape(H',9,1); |
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114 | hhv = hhv(1:8); |
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115 | |
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116 | for iter=1:10, |
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117 | |
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118 | |
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119 | |
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120 | mrep = H * M; |
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121 | |
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122 | J = zeros(2*Np,8); |
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123 | |
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124 | MMM = (M ./ (ones(3,1)*mrep(3,:))); |
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125 | |
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126 | J(1:2:2*Np,1:3) = -MMM'; |
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127 | J(2:2:2*Np,4:6) = -MMM'; |
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128 | |
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129 | mrep = mrep ./ (ones(3,1)*mrep(3,:)); |
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130 | |
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131 | m_err = m(1:2,:) - mrep(1:2,:); |
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132 | m_err = m_err(:); |
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133 | |
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134 | MMM2 = (ones(3,1)*mrep(1,:)) .* MMM; |
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135 | MMM3 = (ones(3,1)*mrep(2,:)) .* MMM; |
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136 | |
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137 | J(1:2:2*Np,7:8) = MMM2(1:2,:)'; |
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138 | J(2:2:2*Np,7:8) = MMM3(1:2,:)'; |
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139 | |
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140 | MMM = (M ./ (ones(3,1)*mrep(3,:)))'; |
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141 | |
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142 | hh_innov = inv(J'*J)*J'*m_err; |
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143 | |
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144 | hhv_up = hhv - hh_innov; |
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145 | |
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146 | H_up = reshape([hhv_up;1],3,3)'; |
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147 | |
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148 | %norm(m_err) |
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149 | %norm(hh_innov) |
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150 | |
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151 | hhv = hhv_up; |
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152 | H = H_up; |
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153 | |
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154 | end; |
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155 | |
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156 | |
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157 | end; |
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158 | |
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159 | if 0, |
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160 | m2 = H*M; |
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161 | m2 = [m2(1,:)./m2(3,:) ; m2(2,:)./m2(3,:)]; |
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162 | merr = m(1:2,:) - m2; |
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163 | end; |
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164 | |
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165 | return; |
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166 | |
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167 | %test of Jacobian |
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168 | |
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169 | mrep = H*M; |
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170 | mrep = mrep ./ (ones(3,1)*mrep(3,:)); |
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171 | |
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172 | m_err = mrep(1:2,:) - m(1:2,:); |
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173 | figure(8); |
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174 | plot(m_err(1,:),m_err(2,:),'r+'); |
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175 | std(m_err') |
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