Difficulty fitting translation and rotation between two matrices of coordinate points

My current effort seems like a simple one, but for the life of me I cannot get a working version, with or without builtin functions.

In this example, I have two sets of points where one is a simple rotation of the other. My real world application will have to worry about rotation, translation, and potentially minor radial distortion issues (scaling based on radial distance). I need to be able to fit one set to another and extract (at least) the rotation and x and y translation of the transform. Later I hope to check non-linear radial distortions.

I tried using regionprop stuff via Centroid and Orientation to rotate around, but translation seems to get messed up, since the centroid and the rotaiton isn't about the center of the "image".

Over the weekend I learned about fitgeotrans and am trying to get that, or some similar family of functions to work. (Looks like having 2022b would give me more potentially useful tools, but alas, I have 2020b). Using fitgeotrans gives me a tform that includes a significant scaling factor, which isn't real. There should just be a rotation of ~30 degrees and maybe a translation of a few pixels.

ROTFID = [140,45;-96,-32;-58,186;60,-184;111,96;-76,-67;127,-146;-126,148;63,123;-43,-88;167,-83;-165,85];
FID = [99,109;-67,-76;-143,132;145,-130;49,138;-32,-96;184,-63;-182,65;-6,138;7,-98;-185,-9;187,11];

tform = tform = fitgeotrans(ROFID,FID, 'similarity');

%Regardless of Forward,Inverse, Fid or ROFID, the answer is clearly wrong
%with an unwanted and incorrect scale factor

answ = transformPointsForward(rform,ROFID);

I tried using imwrap and similar, with and without a reference via imref2d. I also tried a suggestion involving worldToIntrinsic which I did not understand and did not work.

I'd be happy to share my regionprop centroid/orientation half-solution, but I'm confident that solution will not proof robust in final application.

To show the above code gives a poor result:

figure
p1 = scatter(FID(:,1),FID(:,2));
hold;
p2 = scatter(ROFID(:,1),ROFID(:,2));
p3 = scatter(answ(:,1),answ(:,2));

p1 and p2 are rotations of eachother about 0,0. p3 has a strange scaling effect. I cannot seem to suppress it. I cannot seem to extract just the rotation and translation and apply it to the points (for visualizing purposes)

I can use something like this to get the angle, and translation from the matrix:

u = [0 1];
v = [0 0];
[x, y] = transformPointsForward(tform,u,v);
dx = x(2) - x(1);
dy = y(2) - y(1);
angle = (180/pi) *atan2(dy,dx);

translation = [tform.T(3,1) tform.T(3,2)];
%scale can also be gotten here, which is how I know it is wrong.

%scale
scale = 1 / sqrt(dx^2+dy^2);

There could be some shear in here I have not calculated, but the true answer should be 0. All I did was rotate the points in another program.

I hope I have been thorough in my end-goal and efforts so far. Thank you for reading and thinking.

Answer 1

[Matt J]

This FEX download handles your posted data pretty well,

https://www.mathworks.com/matlabcentral/fileexchange/26186-absolute-orientation-horn-s-method?s_tid=srchtitle

ROTFID = [140,45;-96,-32;-58,186;60,-184;111,96;-76,-67;127,-146;-126,148;63,123;-43,-88;167,-83;-165,85];
FID = [99,109;-67,-76;-143,132;145,-130;49,138;-32,-96;184,-63;-182,65;-6,138;7,-98;-185,-9;187,11];

[reg,FIDFit,err]=absor(ROTFID',FID');

Certainly the rotation angle reg.theta looks close to what it should be,

reg =

struct with fields:

R: [2×2 double]

t: [2×1 double]

s: 1

M: [3×3 double]

theta: 29.9978

[Matt J]

It looks, however, like you have mixed up the order in the final two data points of either ROTFID or FID. Once that is fixed, the registration error is uniformly quite low.

>> err

err =

struct with fields:

errlsq: 1.4915

errmax: 0.6026

>> [FID, FIDFit']

ans =

99.0000 109.0000 99.4280 108.6687

-67.0000 -76.0000 -66.5757 -75.9059

-143.0000 132.0000 -142.5297 131.9378

145.0000 -130.0000 144.4889 -129.6809

49.0000 138.0000 48.8328 138.3693

-32.0000 -96.0000 -31.7686 -96.2397

184.0000 -63.0000 183.5563 -63.2977

-182.0000 65.0000 -182.4636 65.0551

-6.0000 138.0000 -6.2368 137.7883

7.0000 -98.0000 7.3091 -97.9520

187.0000 11.0000 186.7465 11.2599

-185.0000 -9.0000 -184.7874 -9.0030

fitgeotrans also works pretty well after that fix, and with similar resutls

tform = fitgeotrans(ROTFID,FID, 'nonreflectivesimilarity')

>> tform.T

ans =

0.8671 0.4998 0

-0.4998 0.8671 0

0.6094 -0.2457 1.0000

[Max]

Wow what a huge mistake on my part. I have fixed the ordering in the generation functions for these points. The points match up now, but I am still getting a different tform.T than you.

New, reordered ROTFID (I reorded ROTFID rather than FID, but in my testing it doesn't seem to matter)

ROTFID = [140,45;-58,186;-96,-32;60,-184;111,96;-76,-67;127,-146;-126,148;63,123;-43,-88;-165,85;167,-83];
FID =[99,109;-67,-76;-143,132;145,-130;49,138;-32,-96;184,-63;-182,65;-6,138;7,-98;-185,-9;187,11];

tform = fitgeotrans(ROTFID,FID,'similarity');

>> tform.T

ans =

1.0292 0.5933 0

-0.5933 1.0292 0

-0.1494 -2.1777 1.0000

changing to nonreflective doesn't matter either.

tform = fitgeotrans(ROTFID,FID,'nonreflectivesimilarity');

>> tform.T

ans =

1.0292 0.5933 0

-0.5933 1.0292 0

-0.1494 -2.1777 1.0000

using your absor function I did get similar results which shows promise, but I'm concerned my fitgeotrans still isn't outputting the same. If this continues to fail, I may rely on your absor function, but I'd like to try to stick to matlab defaults.

>> reg

reg =

struct with fields:

R: [2×2 double]

t: [2×1 double]

s: 1

M: [3×3 double]

theta: 29.9627

>> reg.R

ans =

0.8664 -0.4994

0.4994 0.8664

>> reg.t

ans =

0.6127

-0.2374

>> reg.s

ans =

1

[Matt J]

Your reordering didn't work. I still see huge errors

err =

struct with fields:

errlsq: 313.0658

errmax: 221.5083

It appears that the second and third points are interchanged this time,

>> [FID,FID_Fit']

ans =

99.0000 109.0000 99.4271 108.6695

-67.0000 -76.0000 -142.5308 131.9365

-143.0000 132.0000 -66.5749 -75.9065

145.0000 -130.0000 144.4901 -129.6796

49.0000 138.0000 48.8317 138.3697

-32.0000 -96.0000 -31.7676 -96.2400

184.0000 -63.0000 183.5570 -63.2961

-182.0000 65.0000 -182.4640 65.0535

-6.0000 138.0000 -6.2379 137.7882

7.0000 -98.0000 7.3101 -97.9520

-185.0000 -9.0000 -184.7872 -9.0047

187.0000 11.0000 186.7465 11.2615

Unfortuantely also, fitgeotrans does not let you do a scaling-free registration, i.e., both nonreflectivesimilarity and similarity both estimate an unknown global stretch factor. Conversely, ABSOR is scaling-free by default, but you can ask absor to estimate a global scaling factor as well with the the input option doScale=true.

[Max]

I didn't reorder BOTH data sets with my new algorithm. I have applied it to both and redone the tform. I now get the same result as you.

There are z values (and anothe property) to each of these that I have not shared. I know sort them all based on zvalue before saving, using and comparing them. So far so good.

I assumed fitgeotrans tried to fit element to element before, but didn't apply that to my troubleshooting. Thank you for the troubleshooting assistance and the new tool, Matt!