I am fitting different 2D geometric transformations by least-squares.
For instance quadratic,
$$\begin{cases}a'x^2+b'xy+c'y^2+d'x+e'y+f'=X\\a''x^2+b''xy+c''y^2+d''x+e''y+f''=Y\end{cases}$$
which is linear in the unknowns $a',b',\cdots f''$.
I also solve homographic transformations
$$\begin{cases}\dfrac{a'x+b'y+c'}{dx+ey+1}=X\\\dfrac{a''x+b''y+c''}{dx+ey+1}=Y\end{cases}$$
by linearization,
$$\begin{cases}a'x+b'y+c'-dxX-eyX=X\\a''x+b''y+c''-dxY-eyY=Y.\end{cases}$$
This works fine.
But when I switch to the more complex rational model
$$\begin{cases}\dfrac{a'x^2+b'xy+c'y^2+d'x+e'y+f'}{gx+hy+1}=X\\\dfrac{a''x^2+b''xy+c''y^2+d''x+e''y+f''}{gx+hy+1}=Y\end{cases}$$
on the same points, the resolution by linearization always results in a solution such that the singular line $gx+hy+1=0$ crosses the $(x,y)$ point cloud and makes a poor fitting. I had expected that the fit would improve, as the number of adjustable parameters is larger.
Do you have a mathematical explanation of this phenomenon ?
Typical sample data ($x,y,X,Y$):
155,277,0,0
191,271,0,2
265,261,0,6
340,259,0,10
376,259,0,12
415,258,0,14
452,260,0,16
725,261,0,30
804,266,0,34
841,264,0,36
878,273,0,38
173,245,1,1
209,247,1,3
247,242,1,5
321,237,1,9
358,234,1,11
433,233,1,15
509,228,1,19
586,235,1,23
624,231,1,25
706,234,1,29
823,240,1,35
859,244,1,37
192,228,2,2
303,224,2,8
415,210,2,14
452,215,2,16
490,216,2,18
528,213,2,20
567,212,2,22
686,215,2,28
765,224,2,32
804,216,2,34
878,226,2,38
248,207,3,5
285,206,3,7
395,198,3,13
432,191,3,15
508,185,3,19
547,193,3,21
586,189,3,23
706,193,3,29
746,193,3,31
823,201,3,35
192,188,4,2
229,189,4,4
302,179,4,8
339,179,4,10
451,171,4,16
490,175,4,18
528,175,4,20
567,175,4,22
686,170,4,28
726,174,4,30
764,174,4,32
876,180,4,38
247,165,5,5
285,162,5,7
321,161,5,9
359,158,5,11
395,154,5,13
547,151,5,21
584,149,5,23
705,152,5,29
746,155,5,31
783,153,5,33
823,155,5,35
192,151,6,2
228,147,6,4
264,149,6,6
414,138,6,14
488,131,6,18
528,135,6,20
566,134,6,22
605,135,6,24
686,134,6,28
804,138,6,34
840,140,6,36
173,134,7,1
283,122,7,7
321,122,7,9
357,117,7,11
395,119,7,13
547,114,7,21
624,114,7,25
665,118,7,27
744,113,7,31
821,114,7,35
858,124,7,37
191,107,8,2
303,103,8,8
339,98,8,10
376,98,8,12
450,91,8,16
528,92,8,20
645,94,8,26
685,90,8,28
725,92,8,30
803,96,8,34
875,97,8,38
173,92,9,1
245,86,9,5
320,82,9,9
394,76,9,13
431,77,9,15
469,77,9,17
546,73,9,21
664,73,9,27
704,70,9,29
857,77,9,37
155,74,10,0
190,69,10,2
227,67,10,4
302,60,10,8
413,56,10,14
527,50,10,20
565,48,10,22
604,51,10,24
644,50,10,26
724,49,10,30
763,51,10,32
802,51,10,34
839,55,10,36
