1 | function [omckk,Tckk,Rckk] = compute_extrinsic_init(x_kk,X_kk,fc,cc,kc,alpha_c), |
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2 | |
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3 | %compute_extrinsic |
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4 | % |
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5 | %[omckk,Tckk,Rckk] = compute_extrinsic_init(x_kk,X_kk,fc,cc,kc,alpha_c) |
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6 | % |
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7 | %Computes the extrinsic parameters attached to a 3D structure X_kk given its projection |
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8 | %on the image plane x_kk and the intrinsic camera parameters fc, cc and kc. |
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9 | %Works with planar and non-planar structures. |
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10 | % |
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11 | %INPUT: x_kk: Feature locations on the images |
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12 | % X_kk: Corresponding grid coordinates |
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13 | % fc: Camera focal length |
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14 | % cc: Principal point coordinates |
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15 | % kc: Distortion coefficients |
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16 | % alpha_c: Skew coefficient |
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17 | % |
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18 | %OUTPUT: omckk: 3D rotation vector attached to the grid positions in space |
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19 | % Tckk: 3D translation vector attached to the grid positions in space |
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20 | % Rckk: 3D rotation matrices corresponding to the omc vectors |
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21 | % |
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22 | %Method: Computes the normalized point coordinates, then computes the 3D pose |
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23 | % |
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24 | %Important functions called within that program: |
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25 | % |
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26 | %normalize_pixel: Computes the normalize image point coordinates. |
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27 | % |
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28 | %pose3D: Computes the 3D pose of the structure given the normalized image projection. |
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29 | % |
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30 | %project_points.m: Computes the 2D image projections of a set of 3D points |
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31 | |
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32 | |
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33 | |
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34 | if nargin < 6, |
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35 | alpha_c = 0; |
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36 | if nargin < 5, |
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37 | kc = zeros(5,1); |
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38 | if nargin < 4, |
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39 | cc = zeros(2,1); |
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40 | if nargin < 3, |
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41 | fc = ones(2,1); |
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42 | if nargin < 2, |
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43 | error('Need 2D projections and 3D points (in compute_extrinsic.m)'); |
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44 | return; |
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45 | end; |
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46 | end; |
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47 | end; |
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48 | end; |
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49 | end; |
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50 | |
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51 | |
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52 | %keyboard; |
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53 | |
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54 | % Compute the normalized coordinates: |
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55 | |
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56 | xn = normalize_pixel(x_kk,fc,cc,kc,alpha_c); |
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57 | |
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58 | |
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59 | |
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60 | Np = size(xn,2); |
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61 | |
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62 | %% Check for planarity of the structure: |
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63 | %keyboard; |
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64 | |
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65 | X_mean = mean(X_kk')'; |
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66 | |
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67 | Y = X_kk - (X_mean*ones(1,Np)); |
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68 | |
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69 | YY = Y*Y'; |
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70 | |
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71 | [U,S,V] = svd(YY); |
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72 | |
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73 | r = S(3,3)/S(2,2); |
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74 | |
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75 | %keyboard; |
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76 | |
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77 | |
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78 | if (r < 1e-3)|(Np < 5), %1e-3, %1e-4, %norm(X_kk(3,:)) < eps, % Test of planarity |
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79 | |
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80 | %fprintf(1,'Planar structure detected: r=%f\n',r); |
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81 | |
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82 | % Transform the plane to bring it in the Z=0 plane: |
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83 | |
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84 | R_transform = V'; |
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85 | |
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86 | %norm(R_transform(1:2,3)) |
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87 | |
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88 | if norm(R_transform(1:2,3)) < 1e-6, |
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89 | R_transform = eye(3); |
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90 | end; |
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91 | |
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92 | if det(R_transform) < 0, R_transform = -R_transform; end; |
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93 | |
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94 | T_transform = -(R_transform)*X_mean; |
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95 | |
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96 | X_new = R_transform*X_kk + T_transform*ones(1,Np); |
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97 | |
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98 | |
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99 | % Compute the planar homography: |
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100 | |
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101 | H = compute_homography(xn,X_new(1:2,:)); |
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102 | |
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103 | % De-embed the motion parameters from the homography: |
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104 | |
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105 | sc = mean([norm(H(:,1));norm(H(:,2))]); |
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106 | |
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107 | H = H/sc; |
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108 | |
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109 | % Extra normalization for some reasons... |
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110 | %H(:,1) = H(:,1)/norm(H(:,1)); |
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111 | %H(:,2) = H(:,2)/norm(H(:,2)); |
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112 | |
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113 | if 0, %%% Some tests for myself... the opposite sign solution leads to negative depth!!! |
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114 | |
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115 | % Case#1: no opposite sign: |
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116 | |
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117 | omckk1 = rodrigues([H(:,1:2) cross(H(:,1),H(:,2))]); |
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118 | Rckk1 = rodrigues(omckk1); |
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119 | Tckk1 = H(:,3); |
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120 | |
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121 | Hs1 = [Rckk1(:,1:2) Tckk1]; |
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122 | xn1 = Hs1*[X_new(1:2,:);ones(1,Np)]; |
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123 | xn1 = [xn1(1,:)./xn1(3,:) ; xn1(2,:)./xn1(3,:)]; |
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124 | e1 = xn1 - xn; |
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125 | |
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126 | % Case#2: opposite sign: |
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127 | |
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128 | omckk2 = rodrigues([-H(:,1:2) cross(H(:,1),H(:,2))]); |
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129 | Rckk2 = rodrigues(omckk2); |
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130 | Tckk2 = -H(:,3); |
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131 | |
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132 | Hs2 = [Rckk2(:,1:2) Tckk2]; |
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133 | xn2 = Hs2*[X_new(1:2,:);ones(1,Np)]; |
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134 | xn2 = [xn2(1,:)./xn2(3,:) ; xn2(2,:)./xn2(3,:)]; |
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135 | e2 = xn2 - xn; |
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136 | |
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137 | if 1, %norm(e1) < norm(e2), |
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138 | omckk = omckk1; |
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139 | Tckk = Tckk1; |
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140 | Rckk = Rckk1; |
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141 | else |
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142 | omckk = omckk2; |
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143 | Tckk = Tckk2; |
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144 | Rckk = Rckk2; |
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145 | end; |
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146 | |
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147 | else |
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148 | |
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149 | u1 = H(:,1); |
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150 | u1 = u1 / norm(u1); |
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151 | u2 = H(:,2) - dot(u1,H(:,2)) * u1; |
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152 | u2 = u2 / norm(u2); |
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153 | u3 = cross(u1,u2); |
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154 | RRR = [u1 u2 u3]; |
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155 | omckk = rodrigues(RRR); |
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156 | |
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157 | %omckk = rodrigues([H(:,1:2) cross(H(:,1),H(:,2))]); |
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158 | Rckk = rodrigues(omckk); |
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159 | Tckk = H(:,3); |
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160 | |
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161 | end; |
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162 | |
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163 | |
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164 | |
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165 | %If Xc = Rckk * X_new + Tckk, then Xc = Rckk * R_transform * X_kk + Tckk + T_transform |
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166 | |
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167 | Tckk = Tckk + Rckk* T_transform; |
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168 | Rckk = Rckk * R_transform; |
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169 | |
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170 | omckk = rodrigues(Rckk); |
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171 | Rckk = rodrigues(omckk); |
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172 | |
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173 | |
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174 | else |
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175 | |
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176 | %fprintf(1,'Non planar structure detected: r=%f\n',r); |
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177 | |
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178 | % Computes an initial guess for extrinsic parameters (works for general 3d structure, not planar!!!): |
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179 | % The DLT method is applied here!! |
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180 | |
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181 | J = zeros(2*Np,12); |
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182 | |
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183 | xX = (ones(3,1)*xn(1,:)).*X_kk; |
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184 | yX = (ones(3,1)*xn(2,:)).*X_kk; |
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185 | |
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186 | J(1:2:end,[1 4 7]) = -X_kk'; |
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187 | J(2:2:end,[2 5 8]) = X_kk'; |
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188 | J(1:2:end,[3 6 9]) = xX'; |
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189 | J(2:2:end,[3 6 9]) = -yX'; |
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190 | J(1:2:end,12) = xn(1,:)'; |
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191 | J(2:2:end,12) = -xn(2,:)'; |
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192 | J(1:2:end,10) = -ones(Np,1); |
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193 | J(2:2:end,11) = ones(Np,1); |
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194 | |
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195 | JJ = J'*J; |
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196 | [U,S,V] = svd(JJ); |
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197 | |
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198 | RR = reshape(V(1:9,12),3,3); |
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199 | |
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200 | if det(RR) < 0, |
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201 | V(:,12) = -V(:,12); |
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202 | RR = -RR; |
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203 | end; |
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204 | |
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205 | [Ur,Sr,Vr] = svd(RR); |
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206 | |
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207 | Rckk = Ur*Vr'; |
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208 | |
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209 | sc = norm(V(1:9,12)) / norm(Rckk(:)); |
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210 | Tckk = V(10:12,12)/sc; |
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211 | |
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212 | omckk = rodrigues(Rckk); |
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213 | Rckk = rodrigues(omckk); |
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214 | |
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215 | end; |
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