DESIGN OF STEERABLE FILTERS FOR THE DETECTION OF MICRO-PARTICLES C´ edric Vonesch, Fr´ ed´ eric Stauber and Michael Unser Ecole Polytechnique F´ ed´ erale de Lausanne, Switzerland ABSTRACT This paper presents two contributions. We first introduce a continu- ous-domain version of Principal-Component Analysis (PCA) for de- signing steerable filters so that they best approximate a given set of image templates. We exploit the fact that steerability does not need to be enforced explicitly if one extends the set of templates by in- corporating all their rotations. Our results extend previous work by Perona to multiple templates. We then apply our framework to the automatic detection and classification of micro-particles that carry biochemical probes for molecular diagnostics. Our continuous-domain PCA formalism is particularly well adapted in this context because the geometry of the carriers is known analytically. In addition, the steerable structure of our filters allows for a fast FFT-based recognition of the type of probe. Index Terms— Steerable filters, Principal-Component Analysis (PCA), micro-particle detection, molecular diagnostics. 1. INTRODUCTION 1.1. Motivation The current evolution towards personalized medicine is creating a need for cost-effective, high-throughput yet patient-specific diagnos- tics solutions. In this context, the Swiss company Biocartis has de- veloped a micro-fluidic system that can simultaneously test a given sample for the presence of a large number of different biological markers. The system is built around circular micro-particles (see Fig. 1) that carry suitable biochemical probes in their central part. The type of probe is encoded at the periphery of each micro-carrier through a series of perforations. The readout of the assay is performed in time-lapse microscopy using two complementary optical modalities: the binding process between the probes and the markers is moni- tored using fluorescence imaging, while the particles are tracked us- ing brightfield imaging. Here we concentrate on the latter problem. Specifically, our goal is to detect each particle and to decode its perforations so as to de- termine the type of probe it carries. To this end, the orientation of each particle must be determined in an accurate and computationally efficient way. We have thus chosen to develop an algorithm based on steerable filters, because they can provide non-discretized directional information using only a finite number of correlation measurements. Our approach can be divided into two parts, corresponding to Section 2 and Section 3 of the present paper. We first design a family This work was funded in part by the Center for Biomedical Imaging and ERC Grant ERC-2010-AdG 267439-FUN-SP. of steerable filters that is tailored to the patterns of interest based on Principal-Component Analysis (PCA). Then we use the obtained filters within a fast convolution-based algorithm for determining the location, orientation and encoding of each particle. 1.2. Existing work and contributions of this paper Steerable filters have been characterized from two different mathe- matical perspectives: functional analysis [1, 2, 3] and the theory of Lie groups [4, 5]. While the angular part of steerable filters is rel- atively constrained (it must be a finite Fourier series), its design for optimal orientation selectivity has been the subject of recent work in the context of wavelet design [6, 7]. There is more freedom when de- signing the radial part of steerable filters, which we do in the present paper. The idea of approximating a single template in a steerable ba- sis has been studied independently by Perona [1, 2] and Hel-Or & Teo [8], using continuous-domain formalisms; others have favored purely discrete formulations [3, 9]. Note that [9] covers the case of multiple templates, but only in combination with a discrete set of ro- tations. Here we consider a completely isotropic formulation in the continuous domain. The originality of our work stems from the following contribu- tions. 1. We extend Perona’s continuous-domain approach [1, 2] to multiple templates. 2. We establish the connection between the isotropic PCA prob- lem that underlies [9] and our steerability-constrained PCA. Moreover our functional perspective puts the emphasis on the rotation invariance of the principal-component subspaces and on their application to the efficient directional analysis of im- ages. 3. We apply our framework to the specific problem in molec- ular diagnostics described above, where the templates are known analytically. This constitutes a strong motivation for our continuous-domain formalism and allows for an analytic design. 2. DESIGN OF THE STEERABLE FILTERS In this paper we use (r, θ) to denote polar coordinates in R 2 . We will also refer to spaces of finite-energy functions using no- tations of the form L2(domain, codomain). For scalar-valued func- tions (codomain = C) the inner product is defined as 〈f,g〉 =  f (s) g(s)ds, where the integral is taken over the domain and the bar denotes complex conjugation. 2013 IEEE 10th International Symposium on Biomedical Imaging: From Nano to Macro San Francisco, CA, USA, April 7-11, 2013 978-1-4673-6454-6/13/$31.00 ©2013 IEEE 922