are a compact multi-resolution image pyramid data structure that sparsely encodes pre-computed pixel neighborhood probability density functions (pdfs) for all pixels in the pyramid. They enable the accurate, anti-aliased evaluation of non-linear image operators directly at any output resolution. A variety of operators can be computed at run time from the same pre-computed data structure in a way that scales to gigapixel images, such as local Laplacian filters for (b,d) detail enhancement or (c,e) smoothing, (f) median filters, (g) dominant mode filters, (h) maximum mode filters, (i) bilateral filters. The original image (a) has resolution 16, 898 × 14, 824 (250 Mpixels).
(Top row) Gray-scale 21, 601 × 10, 801 (233 MPixels) bathymetry image from the NASA Blue Marble collection [NASA 2005]. (Center row) Anti-aliased color mapping computed from the sPDF-map; (Bottom row) Standard pre-filtering and downsampling followed by color mapping: coarser resolutions introduce wrong colors, and whole structures are changing or disappearing.
Local Laplacian filtering with an sPDF-map in O(n) time.
(Top row) Night scene of resolution 47, 908 × 7, 531 (361 Mpixels). The top third of the image is shown with detail enhancement (σr = 0.2, α = 0.25), the center third is the original image, and the bottom third is shown with smoothing (σr = 0.2, α = 3.0). (Bottom row) Images used by Paris et al. : left-hand image of each pair with detail enhancement (σr = 0.4, α = 0.25), right-hand image with smoothing (σr = 0.2, α = 2.0). RGB color channels were computed separately.
Local Laplacian smoothing.
sPDF-map results vs. the original implementation of Paris et al. (σr = 0.2, α = 2.0). (a,b) level 0 (1, 744 × 1, 160); (c,d) level 3 (218 × 145). (a,c) Paris et al.; (b,d) sPDF-map. Luminance PSNR [dB] between (a,b) 35, (c,d) 36.
Local Laplacian detail enhancement.
sPDF-map re- sults vs. the original implementation of Paris et al. (σr = 0.2, α = 0.5). Flower (800 × 533) level 0: (a) Paris et al. (b) sPDF-maps. Luminance PSNR between (a,b) 37 dB.
Smoothed local histogram filtering.
(a,b) Beach image in (a) dominant mode-filtered (luminance only) in (b). (f,g) Rock image in HSV color model: Standard downsampling introduces strong haloes of the wrong color around the rock (f), whereas dominant mode-filtering the H and V channels correctly preserves the circular domain of the hue channel (g). (c,d,e) and (h,i,j): Median filtering (luminance only) using sPDF-maps vs. downsampling and then filtering. (c,h) Original zoom-ins of the Night Scene (c) and the Machu Picchu (h) images. (d,i) Median filtering with sPDF-maps prevents over-smoothing by properly preserving the non-linearity of the image operation. (e,j) Median filtering after downsampling introduces strong over-smoothing that cannot be reversed by the median filter applied directly at the coarser resolution.
(a) Original image (320 × 428) with salt and pepper noise. (b) Ground truth 5 × 5 median applied to level 0, then downsampled to level 1. (g) Naive equivalent median computed in level 1. sPDF-map where Wj is a Gaussian of size (c,d,e) 3 × 3, (h,i,j) 5 × 5. Median from sPDF-map with (c,h) 1 coefficient chunk, (d,i) 2 chunks. (f) Gaussian pyramid level 1 (160 × 214). (e,j) E[Xp] of sPDF-map level 1 (top half: 1 coefficient chunk, bottom half: 2 chunks). PSNR [dB] between (g,b): 30, (c,b): 30, (d,b): 36, (h,b): 38, (i,b): 39.
(Top row) Zoom-ins of original 16, 898 × 14, 824 image (a) at level 4: (b) Ground truth bilateral, (c) sPDF-map bilateral, (d) Naive bilateral. Luminance PSNR [dB] between (c,b) 43, (d,b) 41. (Bottom row) (e) Original image (512 × 512). The sPDF-map of (e) uses a Wj of size 3 × 3 and two coefficient chunks. Zoom-ins from (f,g,h,i) level 1, (j,k,l,m) level 2. (f,j) Downsampled image, no bilateral filtering. (g,k) Ground truth bilateral. (h,l) sPDF-map bilateral. (i,m) Naive bilateral. PSNR [dB] between (h,g) 37, (i,g) 35, (l,k) 38, (m,k) 37.