Transformation use cases

Use cases for defining and using transforms on images.

We should be very careful to only use the terms x, y, z to refer to physical space. For voxels, we should use i, j, k, or i', j', k' (i prime, j prime k prime).

I have an image Img.

Image Orientation

I would like to know what the voxel sizes are.

I would like to determine whether it was acquired axially, coronally or sagittally. What is the brain orientation in relation to the voxels? Has it been acquired at an oblique angle? What are the voxel dimensions?:

img = load_image(file)
cm = img.coordmap
print cm

input_coords axis_i:
             axis_j:
             axis_k:

             effective pixel dimensions
                            axis_i: 4mm
                            axis_j: 2mm
                            axis_k: 2mm

input/output mapping
               <Affine Matrix>




                   x   y   z
                 ------------
               i|  90  90   0
               j|  90   0  90
               k| 180  90  90

               input axis_i maps exactly to output axis_z
               input axis_j maps exactly to output axis_y
               input axis_k maps exactly to output axis_x flipped 180

output_coords axis0: Left -> Right
              axis1: Posterior -> Anterior
              axis2: Inferior -> Superior

In the case of a mapping that does not exactly align the input and output axes, something like:

...
input/output mapping
               <Affine Matrix>

               input axis0 maps closest to output axis2
               input axis1 maps closest to output axis1
               input axis2 maps closest to output axis0
...

If the best matching axis is reversed compared to input axis:

...
input axis0 maps [closest|exactly] to negative output axis2

and so on.

Creating transformations / co-ordinate maps

I have an array pixelarray that represents voxels in an image and have a matrix/transform mat which represents the relation between the voxel coordinates and the coordinates in scanner space (world coordinates). I want to associate the array with the matrix:

img = load_image(infile)
pixelarray = np.asarray(img)

(pixelarray is an array and does not have a coordinate map.):

pixelarray.shape
(40,256,256)

So, now I have some arbitrary transformation matrix:

mat = np.zeros((4,4))
mat[0,2] = 2 # giving x mm scaling
mat[1,1] = 2 # giving y mm scaling
mat[2,0] = 4 # giving z mm scaling
mat[3,3] = 1 # because it must be so
# Note inverse diagonal for zyx->xyz coordinate flip

I want to make an Image with these two:

coordmap = voxel2mm(pixelarray.shape, mat)
img = Image(pixelarray, coordmap)

The voxel2mm function allows separation of the image array from the size of the array, e.g.:

coordmap = voxel2mm((40,256,256), mat)

We could have another way of constructing image which allows passing of mat directly:

img = Image(pixelarray, mat=mat)

or:

img = Image.from_data_and_mat(pixelarray, mat)

but there should be “only one (obvious) way to do it”.

Composing transforms

I have two images, img1 and img2. Each image has a voxel-to-world transform associated with it. (The “world” for these two transforms could be similar or even identical in the case of an fmri series.) I would like to get from voxel coordinates in img1 to voxel coordinates in img2, for resampling:

imgA = load_image(infile_A)
vx2mmA = imgA.coordmap
imgB = load_image(infile_B)
vx2mmB = imgB.coordmap
mm2vxB = vx2mmB.inverse
# I want to first apply transform implied in
# cmA, then the inverse of transform implied in
# cmB.  If these are matrices then this would be
# np.dot(mm2vxB, vx2mmA)
voxA_to_voxB = mm2vxB.composewith(vx2mmA)

The (matrix) multiply version of this syntax would be:

voxA_to_voxB = mm2vxB * vx2mmA

Composition should be of form Second.composewith(First) - as in voxA_to_voxB = mm2vxB.composewith(vx2mmA) above. The alternative is First.composewith(Second), as in voxA_to_voxB = vx2mmA.composewith(mm2vxB). We choose Second.composewith(First) on the basis that people need to understand the mathematics of function composition to some degree - see wikipedia_function_composition.

Real world to real world transform

We remind each other that a mapping is a function (callable) that takes coordinates as input and returns coordinates as output. So, if M is a mapping then:

[i',j',k'] = M(i, j, k)

where the i, j, k tuple is a coordinate, and the i’, j’, k’ tuple is a transformed coordinate.

Let us imagine we have somehow come by a mapping T that relates a coordinate in a world space (mm) to other coordinates in a world space. A registration may return such a real-world to real-world mapping. Let us say that V is a useful mapping matching the voxel coordinates in img1 to voxel coordinates in img2. If img1 has a voxel to mm mapping M1 and img2 has a mm to voxel mapping of inv_M2, as in the previous example (repeated here):

imgA = load_image(infile_A)
vx2mmA = imgA.coordmap
imgB = load_image(infile_B)
vx2mmB = imgB.coordmap
mm2vxB = vx2mmB.inverse

then the registration may return the some coordinate map, T such that the intended mapping V from voxels in img1 to voxels in img2 is:

mm2vxB_map = mm2vxB.mapping
vx2mmA_map = vx2mmA.mapping
V = mm2vxB_map.composewith(T.composedwith(vx2mmA_map))

To support this, there should be a CoordinateMap constructor that looks like this:

T_coordmap = mm2mm(T)

where T is a mapping, so that:

V_coordmap = mm2vxB.composewith(T_coordmap.composedwith(vx2mmA))

I have done a coregistration between two images, img1 and img2. This has given me a voxel-to-voxel transformation and I want to store this transformation in such a way that I can use this transform to resample img1 to img2. Resampling use cases

I have done a coregistration between two images, img1 and img2. I may want this to give me a worldA-to-worldB transformation, where worldA is the world of voxel-to-world for img1, and worldB is the world of voxel-to-world of img2.

My img1 has a voxel to world transformation. This transformation may (for example) have come from the scanner that acquired the image - so telling me how the voxel positions in img1 correspond to physical coordinates in terms of the magnet isocenter and millimeters in terms of the primary gradient orientations (x, y and z). I have the same for img2. For example, I might choose to display this image resampled so each voxel is a 1mm cube.

Now I have these transformations: ST(img1-V2W), and ST(img2-V2W) (where ST is scanner tranform as above, and V2W is voxel to world).

I have now done a coregistration between img1 and img2 (somehow) - giving me, in addition to img1 and img2, a transformation that registers img1 and img2. Let’s call this tranformation V2V(img1, img2), where V2V is voxel-to-voxel.

In actuality img2 can be an array of images, such as series of fMRI images and I want to align all the img2 series to img1 and then take these voxel-to-voxel aligned images (the img1 and img2 array) and remap them to the world space (voxel-to-world). Since remapping is an interpolation operation I can generate errors in the resampled pixel values. If I do more than one resampling, error will accumulate. I want to do only a single resampling. To avoid the errors associated with resampling I will build a composite transformation that will chain the separate voxel-to-voxel and voxel-to-world transformations into a single transformation function (such as an affine matrix that is the result of multiplying the several affine matrices together). With this single composite transformatio I now resample img1 and img2 and put them into the world coordinate system from which I can make measurements.