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Scalespaceviz: α-Scale Spaces in Practice View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector IMAGE PROCESSING, ANALYSIS, RECOGNITION, AND UNDERSTANDING ScaleSpaceViz: a-Scale Spaces in Practice¶ F. Kanters, L. Florack, R. Duits, B. Platel, and B. ter Haar Romeny Eindhoven University of Technology Department of Biomedical Engineering Den Dolech 2, Postbus 513 5600 MB Eindhoven, Netherlands e-mail: {F.M.W.Kanters, L.M.J.Florack, R.Duits, B.Platel, B.M.terHaarRomeny}@tue.nl Abstract—Kernels of the so-called α-scale space have the undesirable property of having no closed-form rep- resentation in the spatial domain, despite their simple closed-form expression in the Fourier domain. This obstructs spatial convolution or recursive implementation. For this reason an approximation of the 2D α-kernel in the spatial domain is presented using the well-known Gaussian kernel and the Poisson kernel. Experiments show good results, with maximum relative errors of less than 2.4%. The approximation has been successfully implemented in a program for visualizing α-scale spaces. Some examples of practical applications with scale space feature points using the proposed approximation are given. DOI: 10.1134/S1054661807010129 1. INTRODUCTION Witkin [2] and Koenderink [3] in the 1980s. A more recent introduction to scale space can be found in Scale is an essential parameter in computer vision, monograph by ter Haar Romeny [4]. It is, however, since it is an immediate consequence of the process of shown by Duits [5] that, for reasonable axioms, there observation, of measurements. Observations are always exists a complete class of kernels leading to the so- done by integrating some physical property with a called α-scale spaces, of which the Gaussian scale measurement device—for example, integration (over a space is a special case (α = 1). spatial area and a time frame)—of reflected light inten- sity of an object with a CCD sensor in a digital camera Scale space theory makes it possible to look at (spa- or a photoreceptor in our eyes. The range of possibili- tial) derivatives of the image in a mathematically well- ties of observing certain sizes of objects is bounded on posed way. This is the basis of many well-known fea- two sides: there is a minimal size, about the size of the ture detectors at the present time, such as blob detectors smallest aperture, and there is a maximal size, about the or edge detectors. It is also shown that the human front- size of the whole detector array. end visual system takes derivatives up to at least fourth order at various scales. It seems that the differential The front end of the human visual system (the very structure of an image is important for detecting objects, first few layers of the visual system) is able to “detect” for human beings as well as for machines. at different apertures. A good example for this necessity is an image mosaic as shown in Fig. 1. To see the details For a better understanding of image analysis algo- of each patch we use a small aperture, which prevents rithms and the differential structure of images and to explore α-scale spaces, it is important to have a tool us from seeing the global structure. At a large aperture α we are able to see the global structure, but not the that can calculate and visualize these -scale spaces in details of each patch. Objects are only detected at a cer- tain scale. Since one cannot know a priori at which scales objects are present in a scene, it is necessary to detect at multiple scales simultaneously. Scale space theory is the theory of the apertures through which machines and we observe the world. For computer vision systems, the notion of an aperture can be introduced as blurring the high-resolution image with a kernel of a certain width. In 1962, Taizo Iijima derived the Gaussian kernel for this purpose from a set of basic axioms in a Japanese paper [1]. Later, papers on linear or Gaussian scale space followed from various other authors, including ¶ The text was submitted by the authors in English. Received April 13, 2005 Fig. 1. Image mosaic of the Mona Lisa. ISSN 1054-6618, Pattern Recognition and Image Analysis, 2007, Vol. 17, No. 1, pp. 106–116. © Pleiades Publishing, Ltd., 2007. SCALESPACEVIZ: α-SCALE SPACES IN PRACTICE 107 Fig. 2. A scale space of a 2D image as a continuous 3D vol- ume (left) and the same scale space as a stack of sequen- tially blurred images (right). an efficient way. Since α kernels are only known in closed form in the Fourier domain, we present a fast and accurate approximation of the kernels in the spatial Fig. 3. An example image blurred by different kernels at domain, based on the well-known Gaussian and Pois- various scales. Top row: α = 1, center row: α = 3/4, bottom son kernels. These approximations are implemented in row: α = 1/2. a tool for visualizing α-scale spaces: ScaleSpaceViz. The remainder of this paper is organized as follows: 1 In Section 2 we will present the general framework of lution process of the Gaussian scale space. For α = --- the α scale spaces and the properties of the α-kernels. 2 Next we will present an approximation of these kernels we obtain the evolution process of the Poisson scale in the spatial domain in Section 3. In Section 4 we space, which has some special properties and can be present the results of our approximation and compare it extended to a monogenic scale space, as described by with the original kernel. In section 5 we will introduce Felsberg [6]. Figure 3 shows an example of an image an application of the approximated kernels. Finally, we blurred with three different α-kernels. provide our conclusions and ideas for future research in Section 6. Recall that the solution of the evolution process in (2) is given by ()α (), ()()α ∗ () 2. α-SCALE SPACES u xs = ks f x , (3) A scale space of an image contains an extra dimen- which is called an α-scale space1, with the α-kernel sion, the scale. In a Gaussian scale space the image is ()α 2 k : ޒ ޒ given in the Fourier domain by blurred according to the evolution process s α –s ω 2 ⎧ ∂ Ᏺ()ω()α () x ----- u = ∆u ks x = e , (4) ⎪∂s ⎨ (1) α ≤ α ⎪ (), () ތ ()ޒd with 0 < 1. Note that for = 1 one obtains the well- limu · s = f · in 2 -sense, α ⎩ s↓0 known Gaussian kernel and for = 1/2 the Poisson ker- nel. These so-called α-kernels (with 0 < α ≤ 1) are easy which leads to convolution of the image with a Gauss- to obtain in the Fourier domain, but unfortunately in ian kernel. A scale space of a 2D image is in fact a 3D general there is no closed-form representation in the volume with the scale s as the third axis. In practice, spatial domain. In the case of a Gaussian or Poisson however, a scale space is often seen as a stack of images kernel, one can easily apply an inverse Fourier trans- sequentially blurred by convolution with a kernel. Fig- form to obtain the kernel in the spatial domain, but in ure 2 shows an example of a scale space of an image. general this is not possible. For a fixed α, however, an The α-scale spaces are created using a different evolu- expression can be found for the kernel in the spatial tion process, domain using the Hankel transform of the kernel in polar coordinates. Using ⎧ ∂ α ----- u = –()–∆ u ()φ, φ ⎪∂s xr= cos rsin , (5) ⎨ α ∈ (01, ], (2) ⎪ (), () ތ ()ޒd ωρϕ(), ρϕ limu · s = f · in 2 -sense = cos sin , (6) ⎩ s↓0 the kernel in polar coordinates becomes which leads to convolution with a different, so-called α-kernel. Note that for α = 1, we indeed obtain the evo- 1 Also known as α-stable Lévy processes in probability theory. PATTERN RECOGNITION AND IMAGE ANALYSIS Vol. 17 No. 1 2007 108 KANTERS et al. 2α Ᏺ()ω()α () –sρ ρω Note that a, b, sg (Gauss scale), and sp (Poisson ks ==e , . (7) scale) are constants, which may depend on α. In order ω ∈ ތ ޒ2 to obtain a correct amplitude for = 0, we must have Any 2D function f 2( ) in polar coordinates can a + b = 1, which results in be decomposed as follows: 2α ()ωα 2 ()ωα –s ω –sg –sp ∞ e ≈ a()α e + ()1 – a()α e . (15) (), φ ()–imφ fr = ∑ f m r e . (8) An exact result is desired for α = 1 and α = 1/2, m = 0 which results in the following constraints: () () The Fourier transform of a product of a radial func- a 1 ==1, sg 1 s (16) tion with a harmonic function can be written in terms of () () the Hankel transform [7], a 1/2 ==0, s p 1/2 s. ∞ For dimensionality reasons and using these con- Ᏺ()ρϕ(), imϕᏴ ()ρ() straints, it is natural to assume f = ∑ e m f m , (9) α ()α m = 0 st==, and sg c1 t, (17) with Ᏺ(f)(ρ, ϕ) the Fourier transform of f and and s = c ()α t Ᏼ ρ p 2 m(fm)( ) the Hankel transform of fm defined by with constraints ∞ () () Ᏼ ()ρ() m () ()ρ c1 1 ==1 and c2 1/2 1, (18) m f = i ∫ frrJm r dr, (10) 0 which results in 1 with Jν(z) the Bessel function defined by --- ()α ()α ()α α sg ==c1 tc1 s ∞ ν k (19) ⎛⎞z –1 ⎛⎞z 2k 1 () ------α- Jν z = ⎝⎠--- ∑ ----------------------------------- ⎝⎠--- .
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