Fig 1.
Illustration of topological and non-topological maps.
(A) A polygon patch on an inflated cortical surface (source domain). (B) A topological mapping of (A) in the target domain (retina space). It preserves the structure of the source polygon. (C) A non-topological mapping of (A). It breaks the structure of the source polygon: fi is moved out of the edge , resulting in what is called a flipped triangle Δfifjfk. (d) A typical retinotopic map of V1. Flipped triangles are labeled with red color.
Fig 2.
Beltrami coefficient and topological condition: (A) source triangle (triangle ΔPiPjPk), (B) three target triangles Δfifjfk from three different types of mappings, with different Beltrami coefficients.
Fig 3.
(A) Visual stimuli. (B) Coordinate system of the visual field. (C) Structural (anatomical) MR image. (D) BOLD fMRI scans. (E) Preprocessed fMRI signals. (F) Reconstructed cortical surface with projected fMRI activations. (G) Retinotopic maps from the pRF model. (H) Level set of the raw retinotopic maps. (I) Level set of the smoothed retinotopy maps.
Fig 4.
Illustration of (A) the visual field space, (B) the cortical surface space, and (C) the parametric space.
Fig 5.
Illustration of a mapping function and the divergence computation.
(A) An illustration of the mapping function in the discrete domain, and (B) the divergence approximation for a vertex.
Fig 6.
Removing phase jumping and phase reversal.
(A, B) Phase jumping and the shift scheme used to remove it. (C) Phase reversal and the transformation used to remove it.
Table 1.
Polar angle transformations are used to remove phase-jumping and changes of visual field sign for the left and right hemispheres in multiple visual areas.
Fig 7.
Illustration of the synthetic data.
(A) The visual field domain. (B) Mapping without noise. (C) Mapping with a small amount of noise (PSNR = 10). (D) Mapping with a large amount of noise (PSNR = 5).
Table 2.
Comparison of different smoothing methods based on three metrics: value deviation, angle distortion, and number of flipped triangles with a small amount of noise (PSNR = 10).
Table 3.
Comparison of different smoothing methods based on three metrics: value deviation, angle distortion, and number of flipped triangles with a large amount of noise (PSNR = 5).
Fig 8.
Smoothing results for the synthetic data: (A)-(D) are the results with PSNR = 10 for (A) Average smoothing, (B) Median smoothing, (C) Laplacian smoothing, and (D) proposed topological smoothing; (E)-(H) are the results with the same order as (A)-(D) but with PSNR = 5.
Table 4.
Comparison of different smoothing methods based on noisy fMRI decoding with about 22% of flipping triangles.
Table 5.
Comparison of different methods in fitting the fMRI time series.
Fig 9.
Results on the first observer: (A) the raw retinotopic map on the 2D parametric domain; (B)-(E) results from the Average, Median, Laplacian, and proposed methods, respectively; (F) the whole inflated left mesh; (G) the raw map on the cortical surface; (H)-(K) results on the inflated surface, in the same order of (B)-(E). In (A)-(E) the x axis is the visual x coordinate, and the y axis is the visual y coordinate.
Fig 10.
The retinotopic mapping of the V1/V2/V3 complex of the left hemisphere of the first observer: (A) the raw retinotopic map in the eccentricity-extended-polar coordinate space (with 232 flipped triangles), (B)-(E) the smoothing results of the Average (with 55 flipped triangles), Median (with 85 flipped triangles), Laplacian (with 151 flipped triangles), and the proposed smooth (no flipped triangles), respectively, (F) the entire left inflated mesh in medial view, (G) an enlarged graph of the raw retinotopic map on the inflated mesh, (H)-(K) smoothing results on the inflated surface of the four smoothing methods, in the same order of (B)-(E). In (A)-(E) the x axis is the eccentricity v(1), and the y axis is the extended polar angle v(2), in (H)-(K), the green and blue curves are levels sets, i.e., the contours of eccentricity v(1) and extended polar angle v(2), respectively.
Fig 11.
Visual area boundary delineation of the V1/V2/V3 complex of the first observer.
The colors of the surface represent manually labeled visual areas from the average data. The purple curves indicate boundaries between visual areas of V3d/V2d/V1d/V1v/V2v/V3v (from up to down), respectively. The green curves are eccentricity contours. (A) boundaries inferred from the raw data, and (B)-(E) boundaries after average, median, Laplacian, and topological smoothing.
Fig 12.
Visual coordinate change after smoothing.
(A) The visual change is rendered on the inflated cortical surface (B) The visual change is rendered on the parametric surface for the first observer. Similarly, (C) is the visual change for the second observer. (D) is the histogram of average visual change for the HCP dataset (N = 181).
Fig 13.
Comparison of two boundary approaches.
(A) The result of fixed boundary visual coordinates. (B) The result of changeable boundary visual coordinates. (C) Result of (A) on the inflated cortex. (D) Result of (B) on the inflate cortex.
Fig 14.
Smoothing is robust to the inaccurate interior boundary: (A) the raw visual polar angle v(2) is rendered on the parametric unit disk (i.e., radius is one), (B) v(2) as a function on the cropped region, (C) visual polar angle v(2) along the white arc by topological smoothing, (D) similar result as (C) but the curve is from Laplacian smoothing.
Fig 15.
(A, B, C) Perturbations of the V1/V2 boundary, where (B) was obtained after three rounds of expansion and (C) was obtained after five rounds of expansion. (D) Visual coordinate changes with different degrees of perturbations.
Fig 16.
Smoothing extended to the retinotopic map of V3B (the blue color region) of the first four observers.