Coordinate Systems and Spatial Transforms

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Every neuro object in neuroim2 — a NeuroVol, a NeuroVec, an ROI — carries a NeuroSpace that answers one question: where in the scanner room does this voxel sit? Get that mapping right and anatomical coordinates, atlas overlays, and multi-subject registration all fall into place. Get it wrong and your activations end up in the wrong hemisphere.

This vignette builds the mental model from the ground up: what a NeuroSpace is, how the affine transform works, and how to move fluently between voxel indices, grid coordinates, and millimetre world coordinates.

What is a NeuroSpace?

A NeuroSpace is the spatial reference frame attached to every neuro image. It bundles:

Grid dimensions — how many voxels along each axis (dim)
Voxel spacing — physical size of each voxel in millimetres (spacing)
Origin — world coordinates of voxel (1, 1, 1) in mm (origin)
Affine transform — the 4 × 4 matrix that maps voxel indices to mm (trans)
Axis orientation — which anatomical direction each image axis points (axes)

sp <- NeuroSpace(
  dim     = c(64L, 64L, 40L),
  spacing = c(2, 2, 2),
  origin  = c(-90, -126, -72)
)
sp
#> <NeuroSpace> [3D] 
#> ── Geometry ──────────────────────────────────────────────────────────────────── 
#>   Dimensions    : 64 x 64 x 40
#>   Spacing       : 2 x 2 x 2 mm
#>   Origin        : -90, -126, -72
#>   Orientation   : RAS
#>   Voxels        : 163,840

dim(sp)       # grid dimensions
#> [1] 64 64 40
spacing(sp)   # voxel sizes in mm
#> [1] 2 2 2
origin(sp)    # world coords of voxel (1,1,1)
#> [1]  -90 -126  -72
ndim(sp)      # number of spatial dimensions
#> [1] 3

The NeuroSpace is also the bridge to any volume or time-series built on top of it. Call space(x) on any neuro object to retrieve it.

The Affine Transform

The affine transform is a 4 × 4 homogeneous matrix stored in trans(sp). It maps a voxel position (expressed as a zero-based column vector) to millimetre world coordinates:

[x_mm]   [M  t] [i]
[y_mm] = [      [j]
[z_mm]    0  1] [k]
[ 1  ]          [1]

where M is the 3 × 3 linear part (encodes spacing, rotation, and possible shear) and t is the 3-element translation (the origin in mm).

For an axis-aligned image built from spacing and origin, M is diagonal:

trans(sp)
#>      [,1] [,2] [,3] [,4]
#> [1,]    2    0    0  -90
#> [2,]    0    2    0 -126
#> [3,]    0    0    2  -72
#> [4,]    0    0    0    1

The translation column (trans(sp)[1:3, 4]) recovers the origin, and the diagonal of the linear block recovers the voxel sizes:

# Translation column = origin
trans(sp)[1:3, 4]
#> [1]  -90 -126  -72

# Diagonal of linear block = voxel sizes
diag(trans(sp)[1:3, 1:3])
#> [1] 2 2 2

The inverse affine (world -> voxel) is cached and accessible via inverse_trans(sp):

inverse_trans(sp)
#>      [,1] [,2] [,3] [,4]
#> [1,]  0.5  0.0  0.0   45
#> [2,]  0.0  0.5  0.0   63
#> [3,]  0.0  0.0  0.5   36
#> [4,]  0.0  0.0  0.0    1

Passing an explicit affine

You can supply a full 4 × 4 affine directly to NeuroSpace(). neuroim2 will derive spacing and origin from it automatically:

aff <- diag(c(3, 3, 4, 1))          # 3 × 3 × 4 mm voxels
aff[1:3, 4] <- c(-90, -126, -72)    # origin

sp_aff <- NeuroSpace(dim = c(60L, 60L, 35L), trans = aff)
spacing(sp_aff)   # derived from column norms of linear block
#> [1] 3 3 4
origin(sp_aff)    # derived from translation column
#> [1]  -90 -126  -72

When the affine is oblique (rotated relative to the scanner axes), spacing() returns the column norms of the linear block — the true physical voxel sizes — while the diagonal of the linear block would be smaller. See the Oblique affines section below.

Voxel vs World Coordinates

neuroim2 uses 1-based grid indices everywhere, matching R’s array conventions. The affine convention is therefore:

grid_to_coord subtracts 1 from each 1-based index before applying the affine, placing voxel (1, 1, 1) at the origin.

There are two flavours of voxel addressing:

Address type	Range	Description
Linear index	`1 ... prod(dim)`	Single integer; column-major (as in R arrays)
Grid index	`(1...d1, 1...d2, 1...d3)`	3-tuple of 1-based integers

And one world-coordinate system: millimetres, defined by the affine.

The three addressing schemes and the functions that convert between them.

Coordinate Conversions

All conversion functions accept a NeuroSpace (or any neuro object, via its attached space) as the first argument.

Grid <-> linear index

# Which linear index does grid position (10, 12, 5) map to?
grid_to_index(sp, matrix(c(10, 12, 5), nrow = 1))
#> [1] 17098

# And back to grid
index_to_grid(sp, 8394L)
#>      [,1] [,2] [,3]
#> [1,]   10    4    3

Grid -> world coordinates

grid_to_coord subtracts 1 from the 1-based grid before applying the affine, so grid (1, 1, 1) maps exactly to the origin:

grid_to_coord(sp, matrix(c(1, 1, 1), nrow = 1))   # should equal origin(sp)
#>      [,1] [,2] [,3]
#> [1,]  -90 -126  -72
grid_to_coord(sp, matrix(c(10, 12, 5), nrow = 1))
#>      [,1] [,2] [,3]
#> [1,]  -72 -104  -64

Multiple points are passed as a matrix with one row per point:

pts <- matrix(c(
   1,  1,  1,
  32, 32, 20,
  64, 64, 40
), ncol = 3, byrow = TRUE)

grid_to_coord(sp, pts)
#>      [,1] [,2] [,3]
#> [1,]  -90 -126  -72
#> [2,]  -28  -64  -34
#> [3,]   36    0    6

World -> grid coordinates

coord_to_grid(sp, c(0, 0, 0))          # near isocenter
#> [1] 46 64 37
coord_to_grid(sp, matrix(c(0, 0, 0,
                           10, -20, 8), ncol = 3, byrow = TRUE))
#>      [,1] [,2] [,3]
#> [1,]   46   64   37
#> [2,]   51   54   41

Convenience: linear index <-> world

# Linear index 1 should land at the origin (pass integer)
index_to_coord(sp, 1L)
#>      [,1] [,2] [,3]
#> [1,]  -89 -125  -71

# World coord back to linear index
coord_to_index(sp, matrix(c(-90, -126, -72), nrow = 1))
#> [1] -2079

A concrete round-trip

idx   <- 12345L
grid  <- index_to_grid(sp, idx)
world <- grid_to_coord(sp, grid)
back  <- coord_to_index(sp, world)

cat("index:", idx, "-> grid:", grid, "-> world:", round(world, 2),
    "-> index:", back, "\n")
#> index: 12345 -> grid: 57 1 4 -> world: 22 -126 -66 -> index: 10264

Orientation Codes

Neuroimaging images can be stored in many orientations. The orientation code (also called axis codes or RAS codes) tells you which anatomical direction each image axis points:

Letter	Anatomical direction
R	increasing -> Right
L	increasing -> Left
A	increasing -> Anterior
P	increasing -> Posterior
S	increasing -> Superior
I	increasing -> Inferior

A code of "RAS" means: axis 1 runs left-to-right, axis 2 runs posterior-to-anterior, axis 3 runs inferior-to-superior. This is the standard NIfTI/MNI convention. "LPI" (sometimes called “neurological”) reverses all three.

affine_to_axcodes() reads the orientation directly from the affine matrix:

affine_to_axcodes(trans(sp))
#> [1] "R" "A" "S"

The default axis-aligned NeuroSpace built above uses the nearest-anatomy convention inferred from the affine. You can also inspect the axes slot directly:

axes(sp)
#> <AxisSet3D> 
#>   Axis 1        : Left-to-Right
#>   Axis 2        : Posterior-to-Anterior
#>   Axis 3        : Inferior-to-Superior

Reorienting a space

reorient() flips and permutes the affine to match a target orientation string:

sp_ras <- reorient(sp, c("R", "A", "S"))
affine_to_axcodes(trans(sp_ras))
#> [1] "L" "P" "I"

Creating NeuroSpaces

From dim + spacing + origin (axis-aligned)

The simplest case: an isotropic or anisotropic grid with no rotation.

# 2 mm isotropic, MNI-ish origin
sp_mni <- NeuroSpace(
  dim     = c(91L, 109L, 91L),
  spacing = c(2, 2, 2),
  origin  = c(-90, -126, -72)
)
sp_mni
#> <NeuroSpace> [3D] 
#> ── Geometry ──────────────────────────────────────────────────────────────────── 
#>   Dimensions    : 91 x 109 x 91
#>   Spacing       : 2 x 2 x 2 mm
#>   Origin        : -90, -126, -72
#>   Orientation   : RAS
#>   Voxels        : 902,629

From an explicit affine

When you have a NIfTI sform/qform matrix, pass it directly:

# Oblique affine: slight off-diagonal terms (scanner tilt)
aff_obl <- matrix(c(
   2.0,  0.2,  0.0,  -90,
   0.0,  2.0,  0.1, -126,
   0.0,  0.0,  2.0,  -72,
   0.0,  0.0,  0.0,    1
), nrow = 4, byrow = TRUE)

sp_obl <- NeuroSpace(dim = c(91L, 109L, 91L), trans = aff_obl)

# spacing() returns column norms — the true physical voxel sizes
spacing(sp_obl)
#> [1] 2.00000 2.00998 2.00250

# diagonal is not exactly 2 mm when the image is tilted
diag(aff_obl[1:3, 1:3])
#> [1] 2 2 2

When does spacing() differ from the affine diagonal?

spacing() always returns the column norms of the 3 × 3 linear block — the physical length of each voxel edge. For a pure diagonal affine these equal the diagonal entries. For a rotated or oblique affine they differ. Always use spacing() for physical voxel sizes; never rely on the diagonal directly.

You can also compute voxel sizes from any affine matrix with voxel_sizes():

voxel_sizes(aff_obl)
#> [1] 2.000000 2.009975 2.002498

Dimension Manipulation

Adding a time dimension: add_dim

add_dim() extends a 3D NeuroSpace to 4D by appending a new dimension. The spatial affine is preserved unchanged:

sp_3d <- NeuroSpace(c(64L, 64L, 40L), spacing = c(2, 2, 2),
                    origin = c(-90, -126, -72))
sp_4d <- add_dim(sp_3d, 200)   # 200 time points
dim(sp_4d)
#> [1]  64  64  40 200
trans(sp_4d)                   # 4x4 spatial affine unchanged
#>      [,1] [,2] [,3] [,4]
#> [1,]    2    0    0  -90
#> [2,]    0    2    0 -126
#> [3,]    0    0    2  -72
#> [4,]    0    0    0    1

Dropping the time dimension: drop_dim

drop_dim() removes the last (or a named) dimension:

sp_back <- drop_dim(sp_4d)
dim(sp_back)
#> [1] 64 64 40
all.equal(trans(sp_back), trans(sp_3d))   # affine preserved
#> [1] TRUE

This round-trip is used internally whenever a 4D volume is sliced to a single time point.

Resampling and Reorientation

resample — change voxel size, keep geometry

resample() resamples a NeuroVol into a target space (or another volume’s space). To demonstrate, build a small volume and resample it to a coarser grid:

vol_mni <- NeuroVol(array(rnorm(prod(dim(sp_mni))), dim(sp_mni)), sp_mni)
sp_coarse <- NeuroSpace(c(45L, 54L, 45L), spacing = c(4, 4, 4),
                        origin = origin(sp_mni))
vol_coarse <- resample(vol_mni, sp_coarse)
dim(vol_coarse)
#> [1] 45 54 45
spacing(space(vol_coarse))
#> [1] 4 4 4

deoblique — remove scanner tilt

Many scanners acquire data with a slight tilt relative to the MNI axes. The resulting affine has non-zero off-diagonal elements (an oblique affine). deoblique() builds an axis-aligned output space that encloses the original field of view, using the minimum input voxel size by default (AFNI-style):

sp_deob <- deoblique(sp_obl)
affine_to_axcodes(trans(sp_deob))   # now axis-aligned
#> [1] "R" "A" "S"
obliquity(trans(sp_deob))           # near zero
#> [1] 0 0 0

When passed a NeuroVol, deoblique() also resamples the image data into the new space.

Common Gotchas

1. Oblique affines and spacing() If diag(trans(sp)[1:3, 1:3]) does not match spacing(sp), the affine is oblique. Use obliquity(trans(sp)) to quantify the tilt angle (in radians) per axis.

obliquity(aff_obl)        # non-zero: image is slightly tilted
#> [1] 0.00000000 0.09966865 0.04995840
obliquity(trans(sp_mni))  # near zero: axis-aligned
#> [1] 0 0 0

2. 1-based grid indices R uses 1-based indexing. grid_to_coord() handles this by subtracting 1 before applying the affine, so voxel (1, 1, 1) lands at origin(sp). If you ever use raw affine arithmetic, remember to subtract 1 from your 1-based grid coordinates first.

3. NIfTI sform vs qform NIfTI files store two possible affines (sform and qform). read_vol() and read_vec() follow the NIfTI priority rules: sform_code > 0 -> use sform; otherwise use qform. The resulting trans(space(img)) reflects whichever was chosen. You can inspect the header with read_header() if you need to see both.

4. Float32 precision NIfTI stores affine coefficients as 32-bit floats. neuroim2 rounds the affine to 7 significant figures on construction (signif(trans, 7)) to match this precision. Round-trip coordinates through index_to_coord / coord_to_index may therefore show sub-voxel floating-point noise at the ~0.001 mm level.

Quick Reference

Function	Input -> Output	Typical use
`grid_to_index(sp, mat)`	grid -> linear index	looking up voxel data
`index_to_grid(sp, idx)`	linear index -> grid	converting R array subscripts
`grid_to_coord(sp, mat)`	grid -> mm	overlay on anatomy
`coord_to_grid(sp, mat)`	mm -> grid	atlas lookup
`index_to_coord(sp, idx)`	linear index -> mm	shortcut past grid
`coord_to_index(sp, mat)`	mm -> linear index	mask extraction
`affine_to_axcodes(aff)`	affine -> `"RAS"` etc.	orientation check
`voxel_sizes(aff)`	affine -> mm vector	physical voxel size
`obliquity(aff)`	affine -> radians	tilt check
`add_dim(sp, n)`	3D space -> 4D space	attach time axis
`drop_dim(sp)`	4D space -> 3D space	strip time axis
`resample(sp, spacing)`	space -> resampled space	change resolution
`reorient(sp, codes)`	space -> reoriented space	standardise orientation
`deoblique(sp)`	oblique space -> aligned space	remove scanner tilt

Where to go next

vignette("ImageVolumes") — creating and manipulating NeuroVol objects
vignette("NeuroVector") — working with 4D time-series (NeuroVec)
vignette("Resampling") — image resampling in depth
vignette("regionOfInterest") — ROI construction and coordinate-based queries
?NeuroSpace, ?affine_to_axcodes, ?deoblique — function reference pages

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