Spectral Remapping for Image Downscaling

Supplementary Materials

Eduardo S. L. Gastal
eslgastal@inf.ufrgs.br

and

Manuel M. Oliveira
oliveira@inf.ufrgs.br

For up-to-date information refer to our Project Website:
http://inf.ufrgs.br/~eslgastal/SpectralRemapping

ACM Transactions on Graphics.
Volume 36 (2017), Number 4, Proceedings of SIGGRAPH 2017, Article 145.

This is the supplementary material for our Spectral Remapping paper (SIGGRAPH 2017). If you wish to compute the following examples in your machine, go to our Project Website and look for our Julia source code.

As mentioned in the paper, Spectral Remapping is performed at the original image resolution, transforming the input image s onto its spectrally-remapped counterpart s̊ (note the small circle ∘ on top of s). Subsequently, a suitable resampling strategy may be employed to downscale the image, without the risk of introducing aliasing or losing high-frequency structured detail. The captions of the figures below follow the standard used in the paper, where each resampling algorithm is denoted by a specific letter:

Z is Lanczos filtering followed by resampling with a cubic B-Spline;
D is the Cubic O-MOMS described by Nehab and Hoppe (2014) and Blue et al. (2001);
K is the algorithm of Kopf et al. (2013);
O is the algorithm of Öztireli and Gross (2015);
W is the algorithm of Weber et al. (2016);
III is nearest-neighbor subsampling.

Thus, for example, Z(s) indicates Lanczos downscaling applied to the original image s, while Z(s̊) indicates Lanczos downscaling applied to its spectrally-remapped counterpart s̊.

Important: the images below are best viewed in Google's Chrome or Chromium web browsers, as Internet Explorer (IE) cannot resize the images properly, and the zoom buttons do not work in Mozilla's Firefox.

Figures from the paper

Additional examples

Graph Cut

Synthetic circular test patterns with spatial frequency increasing towards the center present a challenge for all downscaling algorithms. This is illustrated in the starchart figure. For this example, the original image on the left was reduced from 4,725 x 4,725 to 236 x 236 pixels (R=20 and \sigma=20). Previous approaches either discard high-frequency information (D(s) and W(s)), and/or introduce aliasing artifacts (O(s) and K(s)). Our wave alignment, as described in Section 4.3 of our paper, also fails in this case due to the impossibility of properly aligning such a circular wave of constant remapped frequency 0.4/R. However, by manually introducing a cut in the graph topology illustrated in Figure 5 of our paper, our wave alignment system returns a solution that retains the convergence of the stripes at the cost of a visual discontinuity, as shown by the results in the starchartcut example. Neither our spectral-remapping nor existing approaches produce optimal results for this challenging example.

The two illustrations below show the graph topology for the two cited examples. On the image to the right, note the absence of an edge connecting the following pairs of wave sets: W_32 to W_33, W_42 to W_43, and W_52 to W_53. Thus, two waves, each belonging to one of these non-connected sets, will not be aligned by the wave-alignment linear system.

Graph topology for s̊ in figure starchart

Graph topology for s̊ in figure starchartcut

Table of Contents

Figures from the paper

barbara (R = 4.00, σ = 3.00) — click to zoom: 1x 1.5x 2x 4x — back to top

radialchirp (R = 4.00, σ = 3.50) — click to zoom: 1x 1.5x 2x 4x — back to top

vittore (R = 6.00, σ = 4.50) — click to zoom: 1x 1.5x 2x 4x — back to top

fish3 (R = 8.00, σ = 6.00) — click to zoom: 1x 1.5x 2x 4x — back to top

texturedshirt (R = 8.00, σ = 8.00) — click to zoom: 1x 1.5x 2x 4x — back to top

jeans (R = 5.00, σ = 5.00) — click to zoom: 1x 1.5x 2x 4x — back to top

michel (R = 8.00, σ = 6.00) — click to zoom: 1x 1.5x 2x 4x — back to top

blinds (R = 20.00, σ = 12.00) — click to zoom: 1x 1.5x 2x 4x — back to top

shirt2 (R = 8.00, σ = 6.00) — click to zoom: 1x 1.5x 2x 4x — back to top

brothers (R = 9.00, σ = 9.75) — click to zoom: 1x 1.5x 2x 4x — back to top

lizard (R = 6.00, σ = 8.00) — click to zoom: 1x 1.5x 2x 4x — back to top

romanesco (R = 10.00, σ = 7.00) — click to zoom: 1x 1.5x 2x 4x — back to top

kodim19 (R = 5.00, σ = 4.00) — click to zoom: 1x 1.5x 2x 4x — back to top

reschart2 (R = 15.00, σ = 15.00) — click to zoom: 1x 1.5x 2x 4x — back to top

roof (R = 12.00, σ = 10.00) — click to zoom: 1x 1.5x 2x 4x — back to top

starchart (R = 20.00, σ = 20.00) — click to zoom: 1x 1.5x 2x 4x — back to top

starchartcut (R = 20.00, σ = 20.00) — click to zoom: 1x 1.5x 2x 4x — back to top

Additional examples

battery (R = 6.00, σ = 4.00) — click to zoom: 1x 1.5x 2x 4x — back to top

rose (R = 6.00, σ = 5.00) — click to zoom: 1x 1.5x 2x 4x — back to top

ruler (R = 4.00, σ = 4.00) — click to zoom: 1x 1.5x 2x 4x — back to top

brickpalace (R = 12.00, σ = 12.00) — click to zoom: 1x 1.5x 2x 4x — back to top

cloth (R = 13.00, σ = 10.00) — click to zoom: 1x 1.5x 2x 4x — back to top

shade (R = 8.00, σ = 6.00) — click to zoom: 1x 1.5x 2x 4x — back to top

pagoda (R = 6.00, σ = 4.00) — click to zoom: 1x 1.5x 2x 4x — back to top

yamaha (R = 12.00, σ = 10.00) — click to zoom: 1x 1.5x 2x 4x — back to top

macaw (R = 8.00, σ = 8.00) — click to zoom: 1x 1.5x 2x 4x — back to top

kodim04 (R = 5.00, σ = 5.00) — click to zoom: 1x 1.5x 2x 4x — back to top

bird (R = 5.50, σ = 4.00) — click to zoom: 1x 1.5x 2x 4x — back to top

fish1 (R = 12.00, σ = 10.00) — click to zoom: 1x 1.5x 2x 4x — back to top

weaver (R = 24.30, σ = 17.00) — click to zoom: 1x 1.5x 2x 4x — back to top

turing (R = 8.00, σ = 8.00) — click to zoom: 1x 1.5x 2x 4x — back to top

chair (R = 10.00, σ = 10.00) — click to zoom: 1x 1.5x 2x 4x — back to top

indian (R = 4.00, σ = 4.00) — click to zoom: 1x 1.5x 2x 4x — back to top

blinds2 (R = 17.30, σ = 10.00) — click to zoom: 1x 1.5x 2x 4x — back to top

shirt4 (R = 8.00, σ = 8.00) — click to zoom: 1x 1.5x 2x 4x — back to top

tree (R = 10.00, σ = 7.50) — click to zoom: 1x 1.5x 2x 4x — back to top

reschart (R = 15.00, σ = 10.00) — click to zoom: 1x 1.5x 2x 4x — back to top

Graph Cut