1 | function [xp,dxpdom,dxpdT,dxpdf,dxpdc,dxpdk,dxpdalpha] = project_points2(X,om,T,f,c,k,alpha) |
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2 | |
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3 | %project_points2.m |
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4 | % |
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5 | %[xp,dxpdom,dxpdT,dxpdf,dxpdc,dxpdk] = project_points2(X,om,T,f,c,k,alpha) |
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6 | % |
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7 | %Projects a 3D structure onto the image plane. |
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8 | % |
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9 | %INPUT: X: 3D structure in the world coordinate frame (3xN matrix for N points) |
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10 | % (om,T): Rigid motion parameters between world coordinate frame and camera reference frame |
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11 | % om: rotation vector (3x1 vector); T: translation vector (3x1 vector) |
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12 | % f: camera focal length in units of horizontal and vertical pixel units (2x1 vector) |
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13 | % c: principal point location in pixel units (2x1 vector) |
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14 | % k: Distortion coefficients (radial and tangential) (4x1 vector) |
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15 | % alpha: Skew coefficient between x and y pixel (alpha = 0 <=> square pixels) |
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16 | % |
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17 | %OUTPUT: xp: Projected pixel coordinates (2xN matrix for N points) |
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18 | % dxpdom: Derivative of xp with respect to om ((2N)x3 matrix) |
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19 | % dxpdT: Derivative of xp with respect to T ((2N)x3 matrix) |
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20 | % dxpdf: Derivative of xp with respect to f ((2N)x2 matrix if f is 2x1, or (2N)x1 matrix is f is a scalar) |
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21 | % dxpdc: Derivative of xp with respect to c ((2N)x2 matrix) |
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22 | % dxpdk: Derivative of xp with respect to k ((2N)x4 matrix) |
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23 | % |
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24 | %Definitions: |
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25 | %Let P be a point in 3D of coordinates X in the world reference frame (stored in the matrix X) |
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26 | %The coordinate vector of P in the camera reference frame is: Xc = R*X + T |
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27 | %where R is the rotation matrix corresponding to the rotation vector om: R = rodrigues(om); |
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28 | %call x, y and z the 3 coordinates of Xc: x = Xc(1); y = Xc(2); z = Xc(3); |
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29 | %The pinehole projection coordinates of P is [a;b] where a=x/z and b=y/z. |
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30 | %call r^2 = a^2 + b^2. |
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31 | %The distorted point coordinates are: xd = [xx;yy] where: |
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32 | % |
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33 | %xx = a * (1 + kc(1)*r^2 + kc(2)*r^4 + kc(5)*r^6) + 2*kc(3)*a*b + kc(4)*(r^2 + 2*a^2); |
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34 | %yy = b * (1 + kc(1)*r^2 + kc(2)*r^4 + kc(5)*r^6) + kc(3)*(r^2 + 2*b^2) + 2*kc(4)*a*b; |
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35 | % |
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36 | %The left terms correspond to radial distortion (6th degree), the right terms correspond to tangential distortion |
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37 | % |
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38 | %Finally, convertion into pixel coordinates: The final pixel coordinates vector xp=[xxp;yyp] where: |
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39 | % |
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40 | %xxp = f(1)*(xx + alpha*yy) + c(1) |
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41 | %yyp = f(2)*yy + c(2) |
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42 | % |
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43 | % |
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44 | %NOTE: About 90 percent of the code takes care fo computing the Jacobian matrices |
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45 | % |
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46 | % |
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47 | %Important function called within that program: |
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48 | % |
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49 | %rodrigues.m: Computes the rotation matrix corresponding to a rotation vector |
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50 | % |
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51 | %rigid_motion.m: Computes the rigid motion transformation of a given structure |
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52 | |
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53 | |
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54 | if nargin < 7, |
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55 | alpha = 0; |
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56 | if nargin < 6, |
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57 | k = zeros(5,1); |
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58 | if nargin < 5, |
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59 | c = zeros(2,1); |
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60 | if nargin < 4, |
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61 | f = ones(2,1); |
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62 | if nargin < 3, |
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63 | T = zeros(3,1); |
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64 | if nargin < 2, |
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65 | om = zeros(3,1); |
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66 | if nargin < 1, |
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67 | error('Need at least a 3D structure to project (in project_points.m)'); |
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68 | return; |
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69 | end; |
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70 | end; |
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71 | end; |
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72 | end; |
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73 | end; |
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74 | end; |
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75 | end; |
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76 | |
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77 | |
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78 | [m,n] = size(X); |
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79 | |
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80 | if nargout > 1, |
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81 | [Y,dYdom,dYdT] = rigid_motion(X,om,T); |
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82 | else |
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83 | Y = rigid_motion(X,om,T); |
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84 | end; |
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85 | |
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86 | |
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87 | inv_Z = 1./Y(3,:); |
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88 | |
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89 | x = (Y(1:2,:) .* (ones(2,1) * inv_Z)) ; |
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90 | |
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91 | |
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92 | bb = (-x(1,:) .* inv_Z)'*ones(1,3); |
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93 | cc = (-x(2,:) .* inv_Z)'*ones(1,3); |
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94 | |
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95 | if nargout > 1, |
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96 | dxdom = zeros(2*n,3); |
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97 | dxdom(1:2:end,:) = ((inv_Z')*ones(1,3)) .* dYdom(1:3:end,:) + bb .* dYdom(3:3:end,:); |
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98 | dxdom(2:2:end,:) = ((inv_Z')*ones(1,3)) .* dYdom(2:3:end,:) + cc .* dYdom(3:3:end,:); |
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99 | |
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100 | dxdT = zeros(2*n,3); |
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101 | dxdT(1:2:end,:) = ((inv_Z')*ones(1,3)) .* dYdT(1:3:end,:) + bb .* dYdT(3:3:end,:); |
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102 | dxdT(2:2:end,:) = ((inv_Z')*ones(1,3)) .* dYdT(2:3:end,:) + cc .* dYdT(3:3:end,:); |
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103 | end; |
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104 | |
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105 | |
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106 | % Add distortion: |
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107 | |
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108 | r2 = x(1,:).^2 + x(2,:).^2; |
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109 | |
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110 | if nargout > 1, |
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111 | dr2dom = 2*((x(1,:)')*ones(1,3)) .* dxdom(1:2:end,:) + 2*((x(2,:)')*ones(1,3)) .* dxdom(2:2:end,:); |
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112 | dr2dT = 2*((x(1,:)')*ones(1,3)) .* dxdT(1:2:end,:) + 2*((x(2,:)')*ones(1,3)) .* dxdT(2:2:end,:); |
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113 | end; |
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114 | |
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115 | |
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116 | r4 = r2.^2; |
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117 | |
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118 | if nargout > 1, |
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119 | dr4dom = 2*((r2')*ones(1,3)) .* dr2dom; |
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120 | dr4dT = 2*((r2')*ones(1,3)) .* dr2dT; |
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121 | end |
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122 | |
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123 | r6 = r2.^3; |
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124 | |
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125 | if nargout > 1, |
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126 | dr6dom = 3*((r2'.^2)*ones(1,3)) .* dr2dom; |
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127 | dr6dT = 3*((r2'.^2)*ones(1,3)) .* dr2dT; |
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128 | end; |
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129 | |
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130 | % Radial distortion: |
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131 | |
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132 | cdist = 1 + k(1) * r2 + k(2) * r4 + k(5) * r6; |
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133 | |
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134 | if nargout > 1, |
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135 | dcdistdom = k(1) * dr2dom + k(2) * dr4dom + k(5) * dr6dom; |
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136 | dcdistdT = k(1) * dr2dT + k(2) * dr4dT + k(5) * dr6dT; |
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137 | dcdistdk = [ r2' r4' zeros(n,2) r6']; |
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138 | end; |
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139 | |
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140 | xd1 = x .* (ones(2,1)*cdist); |
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141 | |
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142 | if nargout > 1, |
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143 | dxd1dom = zeros(2*n,3); |
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144 | dxd1dom(1:2:end,:) = (x(1,:)'*ones(1,3)) .* dcdistdom; |
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145 | dxd1dom(2:2:end,:) = (x(2,:)'*ones(1,3)) .* dcdistdom; |
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146 | coeff = (reshape([cdist;cdist],2*n,1)*ones(1,3)); |
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147 | dxd1dom = dxd1dom + coeff.* dxdom; |
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148 | |
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149 | dxd1dT = zeros(2*n,3); |
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150 | dxd1dT(1:2:end,:) = (x(1,:)'*ones(1,3)) .* dcdistdT; |
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151 | dxd1dT(2:2:end,:) = (x(2,:)'*ones(1,3)) .* dcdistdT; |
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152 | dxd1dT = dxd1dT + coeff.* dxdT; |
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153 | |
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154 | dxd1dk = zeros(2*n,5); |
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155 | dxd1dk(1:2:end,:) = (x(1,:)'*ones(1,5)) .* dcdistdk; |
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156 | dxd1dk(2:2:end,:) = (x(2,:)'*ones(1,5)) .* dcdistdk; |
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157 | end; |
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158 | |
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159 | |
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160 | % tangential distortion: |
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161 | |
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162 | a1 = 2.*x(1,:).*x(2,:); |
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163 | a2 = r2 + 2*x(1,:).^2; |
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164 | a3 = r2 + 2*x(2,:).^2; |
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165 | |
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166 | delta_x = [k(3)*a1 + k(4)*a2 ; |
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167 | k(3) * a3 + k(4)*a1]; |
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168 | |
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169 | |
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170 | %ddelta_xdx = zeros(2*n,2*n); |
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171 | aa = (2*k(3)*x(2,:)+6*k(4)*x(1,:))'*ones(1,3); |
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172 | bb = (2*k(3)*x(1,:)+2*k(4)*x(2,:))'*ones(1,3); |
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173 | cc = (6*k(3)*x(2,:)+2*k(4)*x(1,:))'*ones(1,3); |
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174 | |
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175 | if nargout > 1, |
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176 | ddelta_xdom = zeros(2*n,3); |
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177 | ddelta_xdom(1:2:end,:) = aa .* dxdom(1:2:end,:) + bb .* dxdom(2:2:end,:); |
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178 | ddelta_xdom(2:2:end,:) = bb .* dxdom(1:2:end,:) + cc .* dxdom(2:2:end,:); |
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179 | |
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180 | ddelta_xdT = zeros(2*n,3); |
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181 | ddelta_xdT(1:2:end,:) = aa .* dxdT(1:2:end,:) + bb .* dxdT(2:2:end,:); |
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182 | ddelta_xdT(2:2:end,:) = bb .* dxdT(1:2:end,:) + cc .* dxdT(2:2:end,:); |
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183 | |
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184 | ddelta_xdk = zeros(2*n,5); |
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185 | ddelta_xdk(1:2:end,3) = a1'; |
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186 | ddelta_xdk(1:2:end,4) = a2'; |
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187 | ddelta_xdk(2:2:end,3) = a3'; |
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188 | ddelta_xdk(2:2:end,4) = a1'; |
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189 | end; |
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190 | |
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191 | |
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192 | xd2 = xd1 + delta_x; |
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193 | |
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194 | if nargout > 1, |
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195 | dxd2dom = dxd1dom + ddelta_xdom ; |
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196 | dxd2dT = dxd1dT + ddelta_xdT; |
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197 | dxd2dk = dxd1dk + ddelta_xdk ; |
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198 | end; |
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199 | |
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200 | |
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201 | % Add Skew: |
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202 | |
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203 | xd3 = [xd2(1,:) + alpha*xd2(2,:);xd2(2,:)]; |
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204 | |
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205 | % Compute: dxd3dom, dxd3dT, dxd3dk, dxd3dalpha |
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206 | if nargout > 1, |
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207 | dxd3dom = zeros(2*n,3); |
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208 | dxd3dom(1:2:2*n,:) = dxd2dom(1:2:2*n,:) + alpha*dxd2dom(2:2:2*n,:); |
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209 | dxd3dom(2:2:2*n,:) = dxd2dom(2:2:2*n,:); |
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210 | dxd3dT = zeros(2*n,3); |
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211 | dxd3dT(1:2:2*n,:) = dxd2dT(1:2:2*n,:) + alpha*dxd2dT(2:2:2*n,:); |
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212 | dxd3dT(2:2:2*n,:) = dxd2dT(2:2:2*n,:); |
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213 | dxd3dk = zeros(2*n,5); |
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214 | dxd3dk(1:2:2*n,:) = dxd2dk(1:2:2*n,:) + alpha*dxd2dk(2:2:2*n,:); |
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215 | dxd3dk(2:2:2*n,:) = dxd2dk(2:2:2*n,:); |
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216 | dxd3dalpha = zeros(2*n,1); |
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217 | dxd3dalpha(1:2:2*n,:) = xd2(2,:)'; |
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218 | end; |
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219 | |
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220 | |
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221 | |
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222 | % Pixel coordinates: |
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223 | if length(f)>1, |
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224 | xp = xd3 .* (f(:) * ones(1,n)) + c(:)*ones(1,n); |
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225 | if nargout > 1, |
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226 | coeff = reshape(f(:)*ones(1,n),2*n,1); |
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227 | dxpdom = (coeff*ones(1,3)) .* dxd3dom; |
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228 | dxpdT = (coeff*ones(1,3)) .* dxd3dT; |
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229 | dxpdk = (coeff*ones(1,5)) .* dxd3dk; |
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230 | dxpdalpha = (coeff) .* dxd3dalpha; |
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231 | dxpdf = zeros(2*n,2); |
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232 | dxpdf(1:2:end,1) = xd3(1,:)'; |
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233 | dxpdf(2:2:end,2) = xd3(2,:)'; |
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234 | end; |
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235 | else |
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236 | xp = f * xd3 + c*ones(1,n); |
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237 | if nargout > 1, |
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238 | dxpdom = f * dxd3dom; |
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239 | dxpdT = f * dxd3dT; |
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240 | dxpdk = f * dxd3dk; |
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241 | dxpdalpha = f .* dxd3dalpha; |
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242 | dxpdf = xd3(:); |
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243 | end; |
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244 | end; |
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245 | |
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246 | if nargout > 1, |
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247 | dxpdc = zeros(2*n,2); |
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248 | dxpdc(1:2:end,1) = ones(n,1); |
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249 | dxpdc(2:2:end,2) = ones(n,1); |
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250 | end; |
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251 | |
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252 | |
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253 | return; |
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254 | |
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255 | % Test of the Jacobians: |
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256 | |
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257 | n = 10; |
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258 | |
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259 | X = 10*randn(3,n); |
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260 | om = randn(3,1); |
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261 | T = [10*randn(2,1);40]; |
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262 | f = 1000*rand(2,1); |
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263 | c = 1000*randn(2,1); |
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264 | k = 0.5*randn(5,1); |
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265 | alpha = 0.01*randn(1,1); |
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266 | |
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267 | [x,dxdom,dxdT,dxdf,dxdc,dxdk,dxdalpha] = project_points2(X,om,T,f,c,k,alpha); |
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268 | |
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269 | |
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270 | % Test on om: OK |
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271 | |
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272 | dom = 0.000000001 * norm(om)*randn(3,1); |
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273 | om2 = om + dom; |
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274 | |
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275 | [x2] = project_points2(X,om2,T,f,c,k,alpha); |
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276 | |
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277 | x_pred = x + reshape(dxdom * dom,2,n); |
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278 | |
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279 | |
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280 | norm(x2-x)/norm(x2 - x_pred) |
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281 | |
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282 | |
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283 | % Test on T: OK!! |
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284 | |
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285 | dT = 0.0001 * norm(T)*randn(3,1); |
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286 | T2 = T + dT; |
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287 | |
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288 | [x2] = project_points2(X,om,T2,f,c,k,alpha); |
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289 | |
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290 | x_pred = x + reshape(dxdT * dT,2,n); |
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291 | |
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292 | |
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293 | norm(x2-x)/norm(x2 - x_pred) |
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294 | |
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295 | |
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296 | |
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297 | % Test on f: OK!! |
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298 | |
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299 | df = 0.001 * norm(f)*randn(2,1); |
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300 | f2 = f + df; |
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301 | |
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302 | [x2] = project_points2(X,om,T,f2,c,k,alpha); |
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303 | |
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304 | x_pred = x + reshape(dxdf * df,2,n); |
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305 | |
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306 | |
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307 | norm(x2-x)/norm(x2 - x_pred) |
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308 | |
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309 | |
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310 | % Test on c: OK!! |
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311 | |
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312 | dc = 0.01 * norm(c)*randn(2,1); |
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313 | c2 = c + dc; |
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314 | |
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315 | [x2] = project_points2(X,om,T,f,c2,k,alpha); |
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316 | |
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317 | x_pred = x + reshape(dxdc * dc,2,n); |
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318 | |
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319 | norm(x2-x)/norm(x2 - x_pred) |
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320 | |
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321 | % Test on k: OK!! |
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322 | |
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323 | dk = 0.001 * norm(k)*randn(5,1); |
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324 | k2 = k + dk; |
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325 | |
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326 | [x2] = project_points2(X,om,T,f,c,k2,alpha); |
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327 | |
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328 | x_pred = x + reshape(dxdk * dk,2,n); |
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329 | |
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330 | norm(x2-x)/norm(x2 - x_pred) |
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331 | |
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332 | |
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333 | % Test on alpha: OK!! |
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334 | |
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335 | dalpha = 0.001 * norm(k)*randn(1,1); |
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336 | alpha2 = alpha + dalpha; |
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337 | |
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338 | [x2] = project_points2(X,om,T,f,c,k,alpha2); |
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339 | |
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340 | x_pred = x + reshape(dxdalpha * dalpha,2,n); |
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341 | |
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342 | norm(x2-x)/norm(x2 - x_pred) |
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