The Filter Echo: A General Tool for Filter Visualisation

Abstract

To select suitable filters for a task or to improve existing filters, a deep understanding of their inner workings is vital. Diffusion echoes, which are space-adaptive impulse responses, are useful to visualise the effect of nonlinear diffusion filters. However, they have received little attention in the literature. There may be two reasons for this: Firstly, the concept was introduced specifically for diffusion filters, which might appear too limited. Secondly, diffusion echoes have large storage requirements, which restricts their practicality. This work addresses both problems. We introduce the filter echo as a generalisation of the diffusion echo and use it for applications beyond adaptive smoothing, such as image inpainting, osmosis, and variational optic flow computation. We provide a framework to visualise and inspect echoes from various filters with different applications. Furthermore, we propose a compression approach for filter echoes, which reduces storage requirements by a factor of 20 to 100.

Figures (16)

• Figure 1: Inpainting example using the test image peppers. $5\%$ randomly selected pixels are stored and the rest is discarded. Then the image is reconstructed using EED inpainting with the Charbonnier diffusivity ($\lambda=0.8$, $\sigma=1.0$).
• Figure 2: Osmosis evolution in the compatible case with semi-implicit scheme visualised at different stopping times $T$. Test image square is the initial image, and head the guidance image. The initial image is rescaled, such that its average grey value matches that of the guidance image. For $t \to \infty$ the guidance image is recovered.
• Figure 3: Estimation of the optic flow field (colour-coded) using the Horn--Schunck method ($\alpha=10000$) on frame 12 and frame 13 of the Urban test sequence from the Middlebury flow data set BSLR+11 and the used colour code.
• Figure 4: Drain echo comparison for different smoothing filters. (a) original image. (b) nonlinear diffusion with Weickert diffusivity (NLD, $t=150$, $\lambda=0.3$, $\sigma=0.0$) echo. (c) bilateral filtering (BIL, $\sigma_t = 30$, $\sigma_s = 10$) echo. (d) NL means (NLM, patch radius $3$, $\sigma=10$) echo. The echo location is marked by the red dot. The three echoes are rescaled jointly, i.e. the largest echo value among all three echoes is mapped to 255. The nonlinear diffusion echo uses only data from the same segment. Bilateral filtering also includes data from tonally similar, unconnected segments, but reduces weights for distant pixels. Nonlocal means uses information from the entire image, if the local neighbourhood is similar.
• Figure 5: Nonlinear diffusion echoes for image segmentation. (a) original image. (b) filtered by nonlinear diffusion with the Weickert diffusivity ($t=15000$, $\lambda=5.0$, $\sigma=0.5$). (c) source echo in $(120, 72)$. (d) source echo in $(168, 136)$. The echo locations are marked in red. The diffusivity creates a segmentation-like result. By computing the source echo of a pixel in a segment we extract the segment from the filtered result.