Figure 2
From: Fluorescence microscopy tensor imaging representations for large-scale dataset analysis
Covariance matrix and tensor imaging representation. 3D renderings of a representative volume of lectin-TRITC stained heart. (a) Image processed fluorescence data, (b) corresponding skeleton (white) and graph nodes (red), (c) nodes (white circles) and their connecting segments (white lines). Bounding box, 250 × 205 × 340 μm. (d) 3D representation of the covariance ellipsoid. All feature descriptors (segments, for the microvasculature case) present within the red sampling volume ε (corresponding to a voxel in the final 3D image) are plotted and rendered with a directionally encoded color, defined in the microscopy acquisition frame. (e) By calculating the covariance Cov of the vascular tracts we can obtain the entries of the variance-covariance matrix CM (3 × 3), which is a tensor of rank 2. Here the matrix’s diagonal entries are the variances, while the others are the covariances. (f) Image representation of the tensor field components of the mid axial plane of the representative volume of lectin-TRITC stained heart shown in (a), using a pseudo-color map. Off diagonal components rescaled for better contrast. (g) Rotating the reference frame makes it possible to diagonalize the CM tensor with the positive eigenvalues λ1, λ2, λ3 related to the spread (variance) in the eigenvectors directions v1, v2, v3. (h) In the new principal axis frame, the principal eigenvector v1 gives the main vascular directionality within every voxel. The square root of the eigenvalues is proportional to the ellipsoid radii. The eigenvectors to their orientations. (i) The 3D ellipsoid glyph-based visualization of the mid axial plane of the representative volume of lectin-TRITC stained heart shown in (a), describes the tensor information at any point along an axial plane of the microvasculature. Ellipsoid colors are rendered using a directional color-encoded map (X, red; Y, green; Z, blue). (j) 3D vector-field representation of the entire vascular volume in (a). (k) Voxel maps of the principal eigenvector component along the lab reference frame, using a pseudo-color map, for the mid axial plane of the representative volume of lectin-TRITC stained heart shown in (a). (l) Directional color-encoded maps of the CM eigenvectors for the same mid axial plane shown in (k). RGB components are defined as the absolute values of the eigenvectors’ XYZ components (see inset). Scale bars in (f,k,l), 50 μm.
