How to Convert Euler Angles (Z-X-Z Convention) to Axis Angles for Image Stack Rotation

I have Eularian angles ( based on Z-X-Z convention) that I would like to convert to axis angles prior to using the function "imrotate3" to rotate a 3-D image stack. I have searched different resources for this, including Martin John Baker (http://www.euclideanspace.com/maths/geometry/rotations/conversions/eulerToAngle/index.htm), and I know there is an option to use "angle2quat", but I'd like to skip purchasing the package just for that.

Below, I have included my attempt; however, I am not getting the correct output. I really appreciate your input.

% Note: The input angles are: -330.592, -351.265, and -330.767 based on Proper Euler angles (i.e. Z-X-Z convention)

alpha = deg2rad(-330.592);

beta = deg2rad(-351.265);

gamma = deg2rad(-330.767);

% Construct Individual Rotation Matrix

z1_rot_Mat = [cos(alpha) -sin(alpha) 0;

sin(alpha) cos(alpha) 0;

0 0 1];

x_rot_Mat = [1 0 0

0 cos(beta) -sin(beta);

0 sin(beta) cos(beta)];

z2_rot_Mat = [cos(gamma) -sin(gamma) 0;

sin(gamma) cos(gamma) 0;

0 0 1];

% Combined Rotation Matrix

R = z1_rot_Mat * x_rot_Mat * z2_rot_Mat;

theta = acos((trace(R) - 1)/2);

multi = 1/(2 * sin(theta));

rx = multi * (R(3, 2) - R(2, 3)) * theta;

ry = multi * (R(1, 3) - R(3, 1)) * theta;

rz = multi * (R(2, 1) - R(1, 2)) * theta;

% Convert to degree

angle1 = rad2deg(rx);

angle2 = rad2deg(ry);

angle3 = rad2deg(rz);

% Rotate image stack via angle axis

rot1 = imrotate3(dataset,-angle1,[1 0 0],'crop'); % rot about Z

rot2 = imrotate3(rot1,-angle2,[1 0 0],'crop'); % rot about X

rot3 = imrotate3(rot2,-angle3,[0 0 1],'crop'); % rot about Z

Answer 1

[Matt J]

If you're going to use imrotate3 why not obtain rot3 in a single rotation of the dataset?

[V,d]=eig(R,'vector');
 [~,i]=min(abs(d-1));
W=V(:,i);

N=null(W.'); N=N(:,1);
angle=atan2d( det([W,N,R*N]) , N.'*R*N)

rot3=imrotate3(dataset,angle,W);

Alternatively, if you insist on decomposing into 3 rotations why not do them with the Z-X-Z rotation angles that you already have?

% Rotate image stack via angle axis
rot1 = imrotate3(dataset,alpha,[0 0 1],'crop'); % rot about Z
rot2 = imrotate3(rot1,beta,[1 0 0],'crop'); % rot about X
rot3 = imrotate3(rot2,gamma,[0 0 1],'crop'); % rot about Z

[Alborz Jelvani]

Thanks Matt J.

So, I get these angles from a different software (DataViewer) which uses a Z-X-Z convention. It spits out these 3 numbers after I perform my desired rotation of the object.

Then, when I use these same angles to perform the following operations (as you suggested), Matlab does not give me the same results.

% Rotate image stack via angle axis
rot1 = imrotate3(dataset,alpha,[0 0 1],'crop'); % rot about Z
rot2 = imrotate3(rot1,beta,[1 0 0],'crop'); % rot about X
rot3 = imrotate3(rot2,gamma,[0 0 1],'crop'); % rot about Z

I figured that the reason is because imrotate3 uses axis angle and that is why I can't input the exact same angles I took from DataViewer.

I just tried that one more time, but no luck.

[Matt J]

The only thing I can think of is that maybe the Z-X-Z angles from DataViewer are intrinsic rotation angles?? That would be very strange, though. If possible, I would see if DataViewer can output the 3x3 rotation matrix directly.

Answer 2

[Matt J]

If angle2 is a rotation in y, I think you really meant

rot2 = imrotate3(rot1,-angle2,[0 1 0],'crop'); % rot about y

Answer 3

[James Tursa]

An Euler Angle conversion routine by John Fuller that handles all possible conventions can be found here:

https://www.mathworks.com/matlabcentral/fileexchange/20696-function-to-convert-between-dcm-euler-angles-quaternions-and-euler-vectors

It will convert between Euler Angles, Direction Cosine Matrices, and Quaternions.