BIAP9 - The Coordinate Image API

Author

Chris Markiewicz

Status

Draft

Type

Standards

Created

2021-09-16

Background

Surface data is generally kept separate from geometric metadata

In contrast to volumetric data, whose geometry can be fully encoded in the shape of a data array and a 4x4 affine matrix, data sampled to a surface require the location of each sample to be explicitly represented by a coordinate. In practice, the most common approach is to have a geometry file and a data file.

A geometry file consists of a vertex coordinate array and a triangle array describing the adjacency of vertices, while a data file is an n-dimensional array with one axis corresponding to vertex.

Keeping these files separate is a pragmatic optimization to avoid costly reproductions of geometric data, but presents an administrative burden to direct consumers of the data.

Terminology

For the purposes of this BIAP, the following terms are used:

  • Coordinate - a triplet of floating point values in RAS+ space

  • Vertex - an index into a table of coordinates

  • Triangle (or face) - a triplet of adjacent vertices (A-B-C); the normal vector for the face is (\(\overline{AB}\times\overline{AC}\))

  • Topology - vertex adjacency data, independent of vertex coordinates, typically in the form of a list of triangles

  • Geometry - topology + a specific set of coordinates for a surface

  • Parcel - a subset of vertices; can be the full topology. Special cases include: * Patch - a connected parcel * Decimated mesh - a parcel that has a desired density of vertices

  • Parcel sequence - an ordered set of parcels

  • Data array - an n-dimensional array with one axis corresponding to the vertices (typical) OR faces (more rare) in a patch sequence

Currently supported surface formats

  • FreeSurfer
  • GIFTI: GiftiImage
    • Every image contains a collection of data arrays, which may be coordinates, topology, or data (further subdivided by type and intent)

  • CIFTI-2: Cifti2Image
    • Pure data array, with image header containing flexible axes

    • The BrainModelAxis is a subspace sequence including patches for each hemisphere (cortex without the medial wall) and subcortical structures defined by indices into three-dimensional array and an affine matrix

    • Geometry referred to by an associated wb.spec file (no current implementation in NiBabel)

    • Possible to have one with no geometric information, e.g., parcels x time

Other relevant formats

  • MNE’s STC (source time course) format. Contains:
    • Subject name (resolvable with a FreeSurfer SUBJECTS_DIR)

    • Index arrays into left and right hemisphere surfaces (subspace sequence)

    • Data, one of: * ndarray of shape (n_verts, n_times) * tuple of ndarrays of shapes (n_verts, n_sensors) and (n_sensors, n_times)

    • Time start

    • Time step

Desiderata for an API supporting surfaces

The following are provisional guiding principles:

  1. A surface image (data array) should carry a reference to geometric metadata that is easily transferred to a new image.

  2. Partial images (data only or geometry only) should be possible. Absence of components should have a well-defined signature, such as a property that is None or a specific Exception is raised.

  3. All arrays (coordinates, triangles, data arrays) should be proxied to avoid excess memory consumption

  4. Selecting among coordinates (e.g., gray/white boundary, inflated surface) for a single topology should be possible.

  5. Combining multiple brain structures (canonically, left and right hemispheres) in memory should be easy; serializing to file may be format-specific.

  6. Splitting a data array into independent patches that can be separately operated on and serialized should be possible.

Prominent use cases

We consider the following use cases for working with surface data. A good API will make retrieving the components needed for each use case straightforward, as well as storing the results in new images.

  • Arithmetic/modeling - per-vertex mathematical operations

  • Smoothing - topology/geometry-respecting smoothing

  • Plotting - paint the data array as a texture on a surface

  • Decimation - subsampling a topology (possibly a subset, possibly with interpolated vertex locations)

  • Resampling to a geometrically-aligned surface * Downsampling by decimating, smoothing, resampling * Inter-subject resampling by using ?h.sphere.reg

  • Interpolation of per-vertex and per-face data arrays

When possible, we prefer to expose NumPy ndarrays and allow use of numpy, scipy, scikit-learn. In some cases, it may make sense for NiBabel to provide methods.

Proposal

A CoordinateImage is an N-dimensional array, where one axis corresponds to a sequence of points in one or more parcels.

class CoordinateImage:
    """
    Attributes
    ----------
    header : a file-specific header
    coordaxis : ``CoordinateAxis``
    dataobj : array-like
    """

class CoordinateAxis:
    """
    Attributes
    ----------
    parcels : list of ``Parcel`` objects
    """

    def load_structures(self, mapping):
        """
        Associate parcels to ``Pointset`` structures
        """

    def __getitem__(self, slicer):
        """
        Return a sub-sampled CoordinateAxis containing structures
        matching the indices provided.
        """

    def get_indices(self, parcel, indices=None):
        """
        Return the indices in the full axis that correspond to the
        requested parcel. If indices are provided, further subsample
        the requested parcel.
        """

class Parcel:
    """
    Attributes
    ----------
    name : str
    structure : ``Pointset``
    indices : object that selects a subset of coordinates in structure
    """

To describe coordinate geometry, the following structures are proposed:

class Pointset:
    @property
    def n_coords(self):
        """ Number of coordinates """

    def get_coords(self, name=None):
        """ Nx3 array of coordinates in RAS+ space """


class TriangularMesh(Pointset):
    @property
    def n_triangles(self):
        """ Number of faces """

    def get_triangles(self, name=None):
        """ Mx3 array of indices into coordinate table """

    def get_mesh(self, name=None):
        return self.get_coords(name=name), self.get_triangles(name=name)

    def get_names(self):
        """ List of surface names that can be passed to
        ``get_{coords,triangles,mesh}``
        """

    def decimate(self, *, n_coords=None, ratio=None):
        """ Return a TriangularMesh with a smaller number of vertices that
        preserves the geometry of the original """
        # To be overridden when a format provides optimization opportunities


class NdGrid(Pointset):
    """
    Attributes
    ----------
    shape : 3-tuple
        number of coordinates in each dimension of grid
    """
    def get_affine(self, name=None):
        """ 4x4 array """

The NdGrid class allows raveled volumetric data to be treated the same as triangular mesh or other coordinate data.

Finally, a structure for containing a collection of related geometric files is defined:

class GeometryCollection:
    """
    Attributes
    ----------
    structures : dict
        Mapping from structure names to ``Pointset``
    """

    @classmethod
    def from_spec(klass, pathlike):
        """ Load a collection of geometries from a specification. """

The canonical example of a geometry collection is a left hemisphere mesh, right hemisphere mesh.

Here we present common use cases:

Modeling

from nilearn.glm.first_level import make_first_level_design_matrix, run_glm

bold = CoordinateImage.from_filename("/data/func/hemi-L_bold.func.gii")
dm = make_first_level_design_matrix(...)
labels, results = run_glm(bold.get_fdata(), dm)
betas = CoordinateImage(results["betas"], bold.coordaxis, bold.header)
betas.to_filename("/data/stats/hemi-L_betas.mgz")

In this case, no reference to the surface structure is needed, as the operations occur on a per-vertex basis. The coordinate axis and header are preserved to ensure that any metadata is not lost.

Here we assume that CoordinateImage is able to make the appropriate translations between formats (GIFTI, MGH). This is not guaranteed in the final API.

Smoothing

bold = CoordinateImage.from_filename("/data/func/hemi-L_bold.func.gii")
bold.coordaxis.load_structures({"lh": "/data/anat/hemi-L_midthickness.surf.gii"})
# Not implementing networkx weighted graph here, so assume we have a function
# that retrieves a graph for each structure
graphs = get_graphs(bold.coordaxis)
distances = distance_matrix(graphs['lh'])  # n_coords x n_coords matrix
weights = normalize(gaussian(distances, sigma))
# Wildly inefficient smoothing algorithm
smoothed = CoordinateImage(weights @ bold.get_fdata(), bold.coordaxis, bold.header)
smoothed.to_filename(f"/data/func/hemi-L_smooth-{sigma}_bold.func.gii")

Plotting

Nilearn currently provides a plot_surf function. With the proposed API, we could interface as follows:

def plot_surf_img(img, surface="inflated"):
    from nilearn.plotting import plot_surf
    coords, triangles = img.coordaxis.parcels[0].get_mesh(name=surface)

    data = img.get_fdata()

    return plot_surf((triangles, coords), data)

tstats = CoordinateImage.from_filename("/data/stats/hemi-L_contrast-taskVsBase_tstat.mgz")
# Assume a GeometryCollection that reads a FreeSurfer subject directory
fs_subject = FreeSurferSubject.from_spec("/data/subjects/fsaverage5")
tstats.coordaxis.load_structures(fs_subject.get_structure("lh"))
plot_surf_img(tstats)

Subsampling CIFTI-2

img = nb.load("sub-01_task-rest_bold.dtseries.nii")  # Assume CIFTI CoordinateImage
parcel = nb.load("sub-fsLR_hemi-L_label-DLPFC_mask.label.gii") # GiftiImage
structure = parcel.meta.metadata['AnatomicalStructurePrimary'] # "CortexLeft"
vtx_idcs = np.where(parcel.agg_data())[0]
dlpfc_idcs = img.coordaxis.get_indices(parcel=structure, indices=vtx_idcs)

# Subsampled coordinate axes will override any duplicate information from header
dlpfc_img = CoordinateImage(img.dataobj[dlpfc_idcs], img.coordaxis[dlpfc_idcs], img.header)

# Now load geometry so we can plot
wbspec = CaretSpec("fsLR.wb.spec")
dlpfc_img.coordaxis.load_structures(wbspec)
...