MATLAB: Calculating bivariate polynomial coefficients in Matlab!

bivariate polynomialcamera distortion modelleast squares

Hello all,

I have the following bivariate polynomial system:

Xa= ax1 + ax2*X + ax3*Y + ax4*X^2 + ax5*Y^2 + ax6*XY

and

Ya= ay1 + ay2*X + ay3*Y + ay4*X^2 + ay5*Y^2 + ay6*XY

This system acts as a distortion model in pixel coordinate to real world coordinate camera system. I know the pixel coordinates and the actual distances of the calibration pattern from the camera and I need to extract the coefficients (ax1 … ax6) (ay1 … ay6) of the bivariate polynomial system using least squares. Since I have not find any ready made function for bivariate polynomial fitting, could you please have any ideas on how to do that?

Thank you in advance!

Best Answer

download polyfitn from the file exchange.

Related Solutions

MATLAB: How to find polynomial roots in Simulink

By using roots() on symbolic variables, you can get four closed form expressions for the roots. They occur in pairs, A+/-B and P+/-Q where B and Q are sqrt(), so by detecting whether the sqrt() involve imaginary quantities you can eliminate conjugate pairs as you wanted.

The closed form expressions are long, but if you are willing to run some code overnight, you can matlabFunction() writing to a 'File' with 'optimize' turned on. That produces a serquence of simple MATLAB code to calculate the expression.

You indicated that you do not want to use scripts, just Simulink blocks. Each of the statements from the optimized code is simple enough to make it practical to implement as a small number of Math blocks (or similar).

It would be a bit tedious to create these expressions manually, but it it only about 100 of them. And you would get exact solutions without iteration.

t2 = C2.*3.0;
t3 = C3.*2.0;
t4 = C4.*2.0;
t5 = C5.*3.0;
t9 = 1.0./C5;
t13 = sqrt(3.0);
t14 = sqrt(6.0);
t6 = -t3;
t7 = -t4;
t8 = -t5;
t10 = t9.^2;
t11 = t9.^3;
t15 = C2.*t9;
t12 = t10.^2;
t16 = C1+C6+t6;
t17 = C1+C6+t7;
t18 = t15.*1.2e+1;
t19 = t2+t8;
t20 = -t18;
t21 = t17.^2;
t22 = t17.^3;
t24 = t9.*t16;
t25 = t9.*t19;
t26 = (t9.*t17)./4.0;
t32 = t10.*t16.*t17.*3.0;
t35 = (t10.*t16.*t17)./4.0;
t37 = (t10.*t17.*t19)./2.0;
t23 = t21.^2;
t27 = t10.*t21.*(3.0./8.0);
t28 = (t11.*t22)./8.0;
t36 = -t35;
t38 = t11.*t19.*t21.*(3.0./4.0);
t39 = (t11.*t19.*t21)./1.6e+1;
t29 = -t27;
t30 = -t28;
t31 = t12.*t23.*(3.0./2.56e+2);
t33 = t12.*t23.*(9.0./6.4e+1);
t40 = -t39;
t34 = -t33;
t41 = t25+t29;
t47 = t24+t30+t37;
t53 = t15+t31+t36+t40;
t42 = t41.^2;
t43 = t41.^3;
t48 = t47.^2;
t54 = t53.^2;
t55 = t53.^3;
t58 = t41.*t53.*(4.0./3.0);
t59 = t41.*t53.*7.2e+1;
t44 = t42.^2;
t45 = t43.*2.0;
t46 = t43./2.7e+1;
t49 = t48.^2;
t50 = t48.*2.7e+1;
t52 = t48./2.0;
t56 = t55.*2.56e+2;
t57 = t43.*t48.*4.0;
t61 = t42.*t54.*1.28e+2;
t62 = t41.*t48.*t53.*1.44e+2;
t51 = t49.*2.7e+1;
t60 = t44.*t53.*1.6e+1;
t63 = t51+t56+t57+t60+t61+t62;
t64 = sqrt(t63);
t65 = t13.*t64.*3.0;
t66 = (t13.*t64)./1.8e+1;
t67 = t45+t50+t59+t65;
t69 = t46+t52+t58+t66;
t68 = sqrt(t67);
t70 = t69.^(1.0./3.0);
t72 = 1.0./t69.^(1.0./6.0);
t71 = t70.^2;
t74 = t41.*t70.*6.0;
t76 = t14.*t47.*t68.*3.0;
t73 = t71.*9.0;
t75 = -t74;
t77 = -t76;
t78 = t20+t32+t34+t38+t42+t73+t75;
t79 = sqrt(t78);
t80 = 1.0./t78.^(1.0./4.0);
t81 = t42.*t79;
t83 = t53.*t79.*1.2e+1;
t84 = t73.*t79;
t85 = t71.*t79.*-9.0;
t86 = (t72.*t79)./6.0;
t88 = t41.*t70.*t79.*1.2e+1;
t82 = -t81;
t87 = -t86;
t89 = -t88;
t90 = t76+t82+t83+t85+t89;
t91 = t77+t82+t83+t85+t89;
t92 = sqrt(t90);
t93 = sqrt(t91);
t94 = (t72.*t80.*t92)./6.0;
t95 = (t72.*t80.*t93)./6.0;
GG = [t26+t87-t94; t26+t87+t94; t26+t86-t95; t26+t86+t95];

MATLAB: Unexpected MATLAB expression when making a table

Try this:

T= [ax ay subs(D, {x y},{ax ay}) subs(fxx,{x,y},{ax ay})]

Note the extra space I added before subs.

Best Answer

Related Solutions

MATLAB: How to find polynomial roots in Simulink

MATLAB: Unexpected MATLAB expression when making a table

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