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

1 1 1 1 1 1

1 2 1 1 2 2

i i i i i i i i

f f f h f w f f f f w h

1 i

f

i

f

1 i

f

h

h Discrete signal sampled regularly with spacing h Standard explict finite difference scheme An implicit finite difference scheme

2 2 4 2 2 4 2 4

2 6 1 2 6 1 4 , 1 6 f x h f x h h d f x f x O h h dx h f x f x h f x f x h O h h f x h f x f x h O h h

w=4 gives a higher approximation order for small h.

A DSP approach to estimating Image Derivatives

frequency response, 1 1

• 1

1 sin 2

j x j x j x

L e H j h d d e j e j dx dx f x f x j

Delivers a good approximation

• f jω for small ω only

Implicit finite differences: an example

4

3 4 f x h f x f x h f x h f x h O h h

• Non-causal IIR filters in the DSP language
• Rational (Pade) approximations in Maths

3sin 2 cos H j

A DSP approach to estimating Image Derivatives

6

3 1 2 28 28 2 12 f x h f x f x h f x h f x h f x h f x h O h h

1 1 1 1

1 1 2 2 2 sin 2cos

i i i i i

f w f f f f w h w H j w

sin 28 2cos 6 3 2cos H j

A 6-order Pade scheme: Let us introduce

• implicit Scharr scheme w=10/3
• implicit Bickley scheme w=4

2 2 2 2 2 2 2 2 2 2 2 2 2 2

, , , 2 x y x y x y x x y y

Commonly used discrete gradients & Laplacians

Rotation-invariant differential quantities (operators) used widely in Image Processing and Computer Vision: Need for accurate discrete approximations. The standard discrete approximations are not sufficiently accurate.

1 1 1 2 2 1 1 w w x h w 1 (Prewitt, 1970) 2 (Sobel, 1970) 10 3 (Scharr, 2000) 4 (Bickley, 1947) simple symmetric f.d. w w w w w 1 (Gonzalez &Woods) 2 (Kamgar-Parsi & Resenfeld, 1999) 4 Mehrstellen Laplacian standard 5-point stencil w w w w

2

1 1 1 4 1 2 1 1 w w w w h w w

Approximation Accuracy and Rotational Invariance

2 2 4

1 1 1 1 1 1 4 4 1 2 12 12 1 1 h O h O h h x h x

2 4 4 2 4 2 4 2 4 4 2

1 1 4 1 1 1 1 4 1 4 20 4 12 6 12 1 1 4 1 h h O h O h h x y h

(Horn, Robot Vision) Optimally rotation-invatiant?

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 1 1 2 2 1 1 w w x h w

1 1 1 1 1

1 2 1 2

i i i i i

f w f f w f f h Smoothing introduced by [-1 0 1]/2 in x-direction is compensated by applying [1 w 1]/(w+2) smoothing in y-direction Smoothing introduced by [-1 0 1]/2 in x-direction is comensated by applying [1 w 1]/(w+2) smoothing 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

2 1 1 2 1 1 2 2 3 3

2 4 6 sin 2 sin 2 3 sin3 1 2 cos 2 cos2

i i i i i i i i i i i

f f f f f a b c f f f f f f a b H j

• S. K. Lele, “Compact finite difference

schemes with spectral like resolution.” Journal of Computational Physics, 1992. Lele scheme:

0.5771439, 0.0896406 1.302566, 0.99355, 0.03750245 a b c

Fourier-Pade-Galerkin approximations 1

Space or trigonometric polynomials of degree N

span :

jn N

e N n N F

Rational Fourier series

,

kl k l k k l l

R P Q P Q F F , , is a properly chosen weighting function.

kl l k k l

f R Q f P g W d g W F

It gives a system of k+l lnear equations with k+l unknowns. In our case, k=3 and l=2.

3 2

sin 2 sin 2 3 sin3 1 2 cos 2 cos2 P a b Q

Fourier-Pade-Galerkin approximations 2

A system of k+l linear equations with k+l unknowns. k=3 and l=2.

1 W

Fourier-Pade-Galerkin approximations 3

A system of k+l linear equations with k+l unknowns. k=3 and l=2.

1 0.9 0.9 W

Fourier-Pade-Galerkin approximations 4

Lele scheme

1 W

1 0.9 0.9 W

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

sharp

, , , I x y I x y I x y

Standard unsharp masking oversharpens high-frequency details Implicit filtereing does a good job in supressing

• versharpened

high-frequency details

Implicit filtering

1 1 1 1

1 ˆ ˆ ˆ 1 2 1 2 4 1 2 1 cos 1 cos 2 1 2 1, 1 2

i i i i i i

f f f f f f H p frequency response function

1 2 , 2 2 2 2 , 2 2

1 tan , 1,2,3, 2 1 2 as 1 as 2

p p p p p p p

H p O H O

Stabilized inverse diffusion

, , low-pass , , , ,

h

I x y t dt I x y t dt I x y t

Implicit filtering and approximation subdivision

1 1 1 1

1 1 ˆ ˆ ˆ 2 1 2 4 1 2 1 cos 1 cos 2

i i i i i i

f f f f f f H

1 1 1 2 2 1 1 1 1 1 1

1 , 2 1 1 1 1 2 2 4

k k k k k i i i i i k k k k k k i i i i i i

u v u v v v v v u u u

Curve subdivision

Implicit filtering and interpolatory subdivision

1 1 1 2 1 2 3 2 3 2

ˆ ˆ ˆ 2 2 cos 2 cos 3 2 1 2 cos

i i i i i i i

a b f f f f f f f a b H 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

• r 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).