Implicit Filtering for Image and Shape Processing Alex Belyaev - PowerPoint PPT Presentation

Vision, Image & Signal Processing (VISP) Implicit Filtering for Image and Shape Processing Alex Belyaev Electrical, Electronic & Computer Engineering School of Engineering & Physical Sciences Heriot-Watt University Edinburgh

Implicit filtering and its applications  A. Belyaev, “On implicit image derivatives and their applications.” BMVC 2011, Dundee, Scotland, UK, August 2011.  A. Belyaev and H. Yamauchi, “Implicit filtering for image and shape processing.” VMV 2011, Berlin, Germany, October 2011.  A. Belyaev, B. Khesin, and S. Tabachnikov, “Discrete speherical means of directional derivatives and Veronese maps.” Journal of geometry and Physics , 2011. Accepted.

Implicit vs explicit Discrete f f i signal i 1 f sampled i 1 regularly with spacing h h h 1 f f f Standard explict finite difference scheme i i 1 i 1 2 h An implicit finite 1 1 f w f f f f i 1 i i 1 i 1 i 1 difference scheme w 2 2 h 2 2 f x h f x h h d 4 f x f x O h 2 2 h 6 dx w= 4 gives a higher 2 h 1 4 approximation order f x f x h 2 f x f x h O h 2 6 h for small h . 1 4 f x h 4 f x f x h O h , h 1 6

A DSP approach to estimating Image Derivatives j x L e H frequency response, j 1 h 1 - d d j x j x e j e j dx dx Delivers a good approximation f x 1 f x 1 of j ω for small ω only j sin 2 Implicit finite differences: an example 3 4 f x h 4 f x f x h f x h f x h O h h  Non-causal IIR filters in the DSP language 3sin H j  Rational (Pade) approximations in Maths 2 cos

A DSP approach to estimating Image Derivatives 1 1 f w f f f f i 1 i i 1 i 1 i 1 w 2 2 h w 2 H j sin w 2cos Let us introduce  implicit Scharr scheme w= 10/3  implicit Bickley scheme w= 4 A 6-order Pade scheme: sin 28 2cos H j f x h 3 f x f x h 6 3 2cos 1 6 f x 2 h 28 f x h 28 f x h f x 2 h O h 12 h

Commonly used discrete gradients & Laplacians Rotation-invariant differential quantities (operators) used widely in Image Processing and Computer Vision: 2 2 2 2 2 2 2 2 2 2 x , , , 2 2 2 2 2 x y y x y x x y y Need for accurate discrete approximations. The standard discrete approximations are not sufficiently accurate. 1 0 1 1 1 w 1 1 0 4 1 w w w w w 2 x 2 h w 2 h w 2 1 0 1 1 1 w w 1 (Prewitt, 1970) w 1 (Gonzalez &Woods) 2 (Sobel, 1970) w w 2 (Kamgar-Parsi & Resenfeld, 1999) w 10 3 (Scharr, 2000) w 4 Mehrstellen Laplacian w 4 (Bickley, 1947) w standard 5-point stencil w simple symmetric f.d.

Approximation Accuracy and Rotational Invariance 0 0 0 1 0 1 2 1 1 h 2 4 1 0 1 O h 4 0 4 1 O h 2 h x 12 h 12 x 0 0 0 1 0 1 Optimally rotation-invatiant? 0 1 0 1 4 1 2 4 4 2 1 h 1 h 4 2 4 1 4 1 O h 4 20 4 O h 2 4 4 2 h 12 x y 6 h 12 0 1 0 1 4 1 (Horn, Robot Vision ) Two natural questions to ask:  Is it possible to achieve a better approximation accuray for the same computational cost?  Why shuld we assume that the grid spacing (pixel size) h tends to 0 ?

Estimating the gradient direction and maginude gradient direction error Sobel Scharr implicit Scharr Bickley implicit Bickley gradient magnitude error Explicit schemes and their implicit counterparts deliver remarkably similar estimates of the gradient direction field.

Explicit vs. Implicit 1 f w f f 1 0 1 i 1 i 1 i 1 w 2 1 0 w w 1 x 2 h w 2 f f 1 0 1 i 1 i 1 2 h Smoothing introduced by Smoothing introduced by [-1 0 1]/2 in x -direction is [-1 0 1]/2 in x -direction is compensated by applying comensated by applying [1 w 1]/(w+2) smoothing [1 w 1]/(w+2) smoothing in y -direction to the derivative. Given an explicit scheme and its implicit counterpart, both the schemes produce similar estimates of the gradient direction, however the implicit scheme does a better job in estimating the gradient magnitude.

High-resolution schemes S. K. Lele, “Compact finite difference schemes with spectral like resolution.” Journal of Computational Physics, 1992. f f f f f i 2 i 1 i i 1 i 2 a b c f f f f f f i 1 i 1 i 2 i 2 i 3 i 3 2 4 6 a sin b 2 sin 2 3 sin3 H j 1 2 cos 2 cos2 0.5771439, 0.0896406 Lele scheme: a 1.302566, b 0.99355, c 0.03750245

Fourier-Pade-Galerkin approximations 1 Space or trigonometric Rational Fourier series polynomials of degree N R P Q kl k l F jn span e : N n N F F P , Q N k k l l f R , , kl F Q f P g W d 0 g l k k l W is a properly chosen weighting function. It gives a system of k+l lnear equations with k+l unknowns. In our case, k= 3 and l= 2. P a sin b 2 sin 2 3 sin3 3 Q 1 2 cos 2 cos2 2

Fourier-Pade-Galerkin approximations 2 A system of k+l linear equations with k+l unknowns. k= 3 and l= 2. W 1

Fourier-Pade-Galerkin approximations 3 A system of k+l linear equations with k+l unknowns. k= 3 and l= 2. 1 0 0.9 W 0 0.9

Fourier-Pade-Galerkin approximations 4 Lele scheme 1 W 1 0 0.9 W 0 0.9

Applications: edge detection (Canny edge detection)

Applications: deblurring Gaussian blur I x y t , , I x y t , , t A higly unstable process. The idea is to use a discrete Laplacian which dumps high frequences Restored image

Applications: unsharp masking Standard unsharp masking oversharpens high-frequency details I x y , I x y , I x y , sharp Implicit filtereing does a good job in supressing oversharpened high-frequency details

Implicit frequency 1 2 p H 1 tan , p 1,2,3, response filtering , p 2 function 2 p 2 2 p 1 2 O as 0 H 2 p 1 , p 2 p 2 O as 2 1 ˆ ˆ ˆ f f f i 1 i i 1 1 2 1 f 2 f f i 1 i i 1 4 1 2 1 cos H 1 cos 2 1 2 p 1, 1 2

Stabilized inverse diffusion I x y t , , dt low-pass I x y t , , dt I x y t , , h

Implicit filtering and approximation subdivision 1 1 ˆ ˆ ˆ f f f f 2 f f i 1 i i 1 i 1 i i 1 1 2 4 1 2 1 cos H 1 cos 2 1 k k 1 k k 1 k 1 u v , u v v 2 i i 2 i 1 i i 1 2 1 1 1 k k k k k k v v v u u u i 1 i i 1 i i 1 i 1 1 2 2 4

Curve subdivision

Implicit filtering and interpolatory subdivision a b ˆ ˆ ˆ f f f f f f f i 1 i i 1 i 1 2 i 1 2 i 3 2 i 3 2 2 2 a cos 2 b cos 3 2 H 1 2 cos Dyn-Levin- Gregory: α = 0, a= 1/16, b= -1/9 Kobbelt K2 variational subdivision scheme: α = 1/6, a= 4/3, b= 0

Implicit filtering and interpolatory subdivision

Implicit subdivision  Implicit subdivision schemes were introduced by Kobbelt [1996,1998] in the case of interpolatory subdivision from a variational standpoint.  Sabin [2010] does not mention them at all in his book (althought he cited that paper of Kobbelt).  Peters and Reif [2008] devoted to variational subdivision only two sentences where the authors acknowledged its existence but wrongly stated that more or less nothing was known about the underlying theoretical properties of variational subdivision schemes.

Future research  Weighted (non-iniform) implicit filtering schems  edge- aware image filtering (in a hope to beat results of Gastal & Oliveira, Siggraph 2011).  Extending to mesh processing (in a hope to beat results of Chuang & Kazhdan, Siggraph 2011).