Best apertures from entropy minimization

Transcription

1 Best apertures from entropy minimization Bart M. ter Haar Romeny, PhD Eindhoven University of Technology Excerpt from the book "Front-End Vision & Multi-Scale Image Analysis", Kluwer Academic Publishers, << MathVisionTools`; Frame > FalseD; PlotRange AllD; MathVisionTools 2008 Biomedical Image-Analysis, Technische Universiteit Eindhoven, the Netherlands This Mathematica add-on, version (July 23rd, 2010) is for academic use only. Please, contact Dr. Markus van Almsick or Prof. Bart ter Haar Romeny for bug reports and commercial use. MathVisionTools` is a package written at Eindhoven University of Technology. It contains the functions gd and rasterplot, used below. It can be downloaded from Derivation The derivation is based on: M. Nielsen, "From paradigm to algorithms in computer vision". PhD thesis, Datalogisk Institut Kopenhagen University, Denmark, Dept. of Computer Science, Universitetsparken 1, DK-2100 Kopenhagen 0, Denmark, ISSN We do a measurement Ø finite aperture. All locations treated similar Ø translation invariance. Measurement is linear Ø superposition principle. Ø The observation must be a convolution: hhxl=ÿ - LHaL ghx -al a. LHxL is the luminance, ghxl is our aperture, hhxl the result of our measurement. A. The aperture function ghxl should be a normalized filter: Ÿ - ghxl x = 1. B. The mean of the filter ghxl is Ÿ - x ghxl x = x 0 = 0. C. The width is the variance: Ÿ - x 2 ghxl x=s 2. The information or entropy of our filter is by definition: H = Ÿ - -ghxl ln ghxl x. We look for the ghxl for which the entropy is minimal, given the constraints: Ÿ - ghxl x = 1, Ÿ - x ghxl x= 0 andÿ - x 2 ghxl x= s 2. When we want to find a minimum under a set of given constraints, we apply a standard mathematical technique named variational methods. The energy E becomes: E = Ÿ - -ghxl ln ghxl x + l 1Ÿ - ghxl x+l 2Ÿ - x ghxl x +l 3 Ÿ - x 2 ghxl x and is minimum when E = 0. The l's are the Lagrange multipliers. g

2 2 << VariationalMethods`; var = VariationalDA g@xdlog@g@xdd+λ 1 g@xd+λ 2 xg@xd+λ 3 x 2 g@xd, g@xd, xe 1 Log@g@xDD+λ 1 +x λ 2 +x 2 λ 3 Solve@var == 0, g@xdd ::g@xd 1+λ 1 +x λ 2 +x2 λ 3 >> g@x_d = First@g@xD ê. Solve@var == 0, g@xddd 1+λ 1 +x λ 2 +x2 λ 3 An important first finding is that g is an exponential function. l 3 must be negative, otherwise the function explodes, which is physically unrealistic. The constraint equations are now: eqn1 = SimplifyB g@xd x == 1, λ3 < 0F λ 3 λ 1 λ λ 3 π eqn2 = SimplifyB xg@xd x 0, λ3 < 0F λ 1 λ2 2 4 λ 3 λ2 0 eqn3 = SimplifyB x 2 g@xd x == σ 2, λ 3 < 0F 1+λ 1 λ2 2 4 λ 3 π Iλ2 2 2 λ 3 M 4 H λ 3 L 5ê2 σ 2 Now we can solve for all three l's: solution = Solve@8eqn1, eqn2, eqn3<, 8λ 1, λ 2, λ 3 <D ::λ LogB 4 4 π 2 σ 4F, λ 2 0, λ σ 2>> g@x_, σ_d = Simplify@g@xD ê. Flatten@solutionD, σ > 0D x 2 2 σ 2 2 π σ which is the Gaussian function. A beautiful result. We have found the Gaussian as the unique solution to the set of constraints, which in principle are a formal statement of the uncommittment of the observation.

3 3 There are 11 known axiomatic derivations of the Gaussian kernel as the optimal aperture for uncommitted observations [Weickert 2002]. Relaxing the constraints gives other families, e.g.: - separability: Poisson scale-space [Duits, Felsberg] - tuned to spatial frequency: Gabor kernels Plot@g@x, 1D, 8x, 4, 4<D Derivatives of sampled, observed data All partial derivatives of the Gaussian kernel are solutions too of the diffusion equation. g := 1 x2 +y2 ExpB F; σ = 1; 2 π σ2 2 σ 2 Plot3D@g, 8x, 3.5, 3.5<, 8y, 3.5, 3.5<D

4 4 x gd, 8x, 3.5, 3.5<, 8y, 3.5, 3.5<D Plot3DAEvaluateA y ge, 8x, 3.5, 3.5<, 8y, 3.5, 3.5<E Plot3DAEvaluateA x I y gme, 8x, 3.5, 3.5<, 8y, 3.5, 3.5<E

5 5 Plot3DAEvaluateA x,x,x,x I y,y gme, 8x, 3.5, 3.5<, 8y, 3.5, 3.5<E The derivative of the observed data is 8L 0Hx, yl GHx, y; sl<, which can be rewritten as L 0 Hx, yl GHx, y; sl. x x Differentiation of discrete data is convolution with a Gaussian derivative æ Differentiation and observation are done in a single step: convolution with a Gaussian derivative kernel. æ Differentiation is now done by integration, namely by the convolution integral. æ The Gaussian kernel is the physical analogue of a mathematical point, the Gaussian derivative kernels are the physical analogons of the mathematical differential operators. Equivalence is reached for the limit when the scale of the Gaussian goes to zero: The function gd[im,nx,ny,σ] implements a convolution with a Gaussian derivative: im = Table@If@80 < x < 170&&80 < y < 170, 1, 0D, 8y, 1, 256<, 8x, 1, 256<D; σ = 1; im x = gd@im, 1, 0, σd; im y = gd@im, 0, 1, σd; grad = im x 2 +im y 2 ; p1 = RasterPlotê@9im, im x, im y, grad=; GraphicsGrid@8p1<, ImageSize 300D FrontEndVision Version 2.0 for Mathematica 6 The first order derivative of an image gives edges. Left: original test image LHx, yl, resolution Second: the derivative with respect to x: L at scale x s = 1 pixel. Note the positive and negative edges. Third: the derivative with respect to y: L at scale s = 1 pixel. Right: the gradient J L scale of s = 1 pixel which gives all edges. y x N2 + J L x N2 at a im = ColorSeparate@Import@"mr256.gif"DDP1, 1T;

6 6 p1 = RasterPlot@imD Lxx = gd@im, 2, 0, 1D; Lyy = gd@im, 0, 2, 1D; RasterPlot@Lxx + LyyD

7 7 The Laplacian of the image, as seen by a center-surround cell of scale s=1 pixel. p2 = ListContourPlot@Reverse@Lxx + LyyD, Contours 80<, ContourStyle Red, ContourShading None, Frame NoneD Show@p1, p2d æ The Gaussian kernel is the unique kernel that generates no spurious resolution. It is the blown-up physical point operator, the Gaussian derivatives are the blown-up physical multi-scale derivative operators.

8 8 up ddx.jpg", ImageSize 300D æ Because convolving is an integration, the Gaussian kernel has by definition a strong regularizing effect. æ Recently some interesting papers have shown the complete equivalence of Gaussian scale space regularization with a number of other methods for regularization such as splines, thin plate splines, graduated convexity etc. [Scherzer2000a, Nielsen1996b, Nielsen1997b]. æ The set of Gaussian derivative kernels (including the zeroth order derivative: the Gaussian kernel itself) forms a complete set of derivatives. This set is sometimes referred to as the N-jet. A scale-space is a stack of 2D images, where scale is the third dimension. And here are - a scale-space of the intensity (no derivatives, only blurred); - a scale-space of the Laplacian ( 2 L x L y 2 ). im = ColorSeparate@Import@"mr256.gif"DDP1, 1T; GraphicsGrid@8Table@RasterPlot@gD@im, 0, 0, E τ DD, 8τ, 0, 2.1,.3<D, Table@RasterPlot@gD@im, 2, 0, E τ D+gD@im, 0, 2, E τ DD, 8τ, 0, 2.1,.3<D<, ImageSize 450D