Euler Characteristic

Count the unique cells of the union of closed voxel cubes and compute V−E+F−C; edge/corner contacts share cells, unlike a six-connected component convention.

Description

Euler characteristic summarizes topology by alternating counts of vertices, edges, faces and solid cells.

Euler characteristic summarizes topology by alternating counts of vertices, edges, faces and solid cells. Occupied voxels are treated as closed unit cubes sharing geometric boundary elements.

The tool returns vertexCount (vertex count); edgeCount (edge count); faceCount (face count); cubeCount (cube count); eulerCharacteristic (euler characteristic). Input measurements and model parameters are supplied by the caller; the tool computes the stated summary rather than acquiring those measurements.

When to use Euler Characteristic

Use Euler Characteristic when you already have binaryMask, dimensions and want vertexCount, edgeCount, faceCount, cubeCount, eulerCharacteristic. This is useful for checking a supplied calculation, documenting a measurement convention, or comparing observations prepared on the same basis.

How to use Euler Characteristic and understand its results

Enter binaryMask, dimensions using the field table below. Match the units, ordering and data boundaries described in the calculation.
Run the tool after checking the stated bounds and matching paired observations or categories. A malformed request gives a tool error rather than a usable measurement.
Read vertexCount, edgeCount, faceCount, cubeCount, eulerCharacteristic. Use the calculation and interpretation below to distinguish amounts, ratios, signs and decision flags where applicable.
Review the supplied parameters and assumptions before using the result in a report or comparing it with another observation.

Input field	Meaning	Value type and bounds	Requirement
binaryMask	Binary mask	list of number	Required
dimensions	Dimensions	list of number	Required

Traverse a binary mask in x-fastest order and deduplicate shared boundary elements. Dimensions must match the mask, with at most 4096 voxels.

χ = vertices − edges + faces − cubes

The symbols refer to the supplied quantities or generated values described above; unit conversions and comparison boundaries follow the stated implementation.

json

{
  "input": {
    "binaryMask": [
      1,
      1
    ],
    "dimensions": [
      2,
      1,
      1
    ]
  },
  "output": {
    "vertexCount": 12,
    "edgeCount": 20,
    "faceCount": 11,
    "cubeCount": 2,
    "eulerCharacteristic": 1
  }
}

Limitations and assumptions

This measures the union of closed cubes, including edge and corner contacts; it is distinct from counting six-connected components.

Numerical inputs must be finite. The tool checks its declared input shape and calculation constraints; those checks do not verify the quality, calibration or representativeness of the supplied observations.

Alternative or Complementary methods

Cluster Extent: Find face-connected 3D suprathreshold voxel components using inclusive thresholding, x-fastest flattening and a supplied voxel volume; no inferential cluster p-values are produced. TFCE Value: Approximate nonnegative TFCE by bounded midpoint height integration over face-connected voxel clusters; extent is voxel count, not physical volume, and inference requires an external permutation null.

Methodological background: Euler number of a binary image.^[1]

References

Euler number of a binary image — scikit-image developers

Similar or alternative tools

Cluster Extent
Find face-connected 3D suprathreshold voxel components using inclusive thresholding, x-fastest flattening and a supplied voxel volume; no inferential cluster p-values are produced.
TFCE Value
Approximate nonnegative TFCE by bounded midpoint height integration over face-connected voxel clusters; extent is voxel count, not physical volume, and inference requires an external permutation null.