SIMILARITY-BASED SHAPE PRIORS FOR 2D SPLINE SNAKES Daniel Schmitter and Michael Unser Biomedical Imaging Group, ´ Ecole polytechnique f´ ed´ erale de Lausanne (EPFL), Switzerland ABSTRACT We present a new formulation of a shape space containing all continuously defined 2D spline curves up to a similarity trans- form of a reference shape. We are able to measure a distance between an arbitrary curve and the shape space itself. Our contribution is an explicit formula for this distance measure in the continuous domain. This allows us to define efficient snake energies based on shape-dependent prior knowledge to facilitate segmentation in bioimaging. The spline-based al- gorithm that we propose allows us to implement continuous- domain solutions with no additional computational cost com- pared to the case where curves are described by a discrete set of landmarks. The proposed implementation is freely avail- able in the public domain. Index Terms— shape space, splines, spline snakes, active contours, similarity. 1. INTRODUCTION The geometric transform called ”similarity” is defined as the combination of translation, rotation, and isotropic scaling. It is particularly useful whenever one wants to segment known structures (e.g., cells, bacteria) but does not know beforehand their orientation and location in an image, a task which is of- ten encountered in natural scenes (see Figure 1). It represents an important class of transformations if we want to character- ize structures of interest [1]. In this paper, we present a formu- lation of the space that contains precisely all the shapes that are described by such a transformation of a reference shape. Our formulation facilitates segmentation tasks because it al- lows us to formulate energy terms that combine a data term with the distance between a given curve and the closest sim- ilarity transform of the reference shape that defines the shape space. The classical approach to define shape spaces is to con- sider I shapes that are described by an ordered set of N points or landmarks in R 2 [2]. Each shape is itself represented as one large vector x i P R 2N , where i P I . It is geometrically normalized by aligning it to a common reference in order to remove some effects of rigid transformations. Considering This work was funded by the Swiss National Science Foundation under Grant 200020-144355. Fig. 1. Similarity occuring in nature. Top row (macroscopic world): Nautilus shell (left) and fractal-like trees (middle and right); Bottom row (microscopic world): rod-shaped (left) and oval (right) yeast cells which are all similar to each other. a reference shape x such an alignment is achieved by com- puting the normalized shapes as y i “ min A,b } x ´ Ax i ´ b} 2 , where A is a transformation matrix and b a translation vector. Using the normalized shapes, modes of variation of the shape collection can be computed to construct a shape space. Aside from only operating with discrete data, the classical approach has a second drawback which is the geometric nor- malization. By normalizing, a bias is introduced in the model because computing distances between normalized shapes usually does not yield the same result as for non-normalized shapes. Our alternative proposal is to define a continuous-domain shape space that contains all possible similarity transforms of some reference shape. Then, searching for the minimum dis- tance between an arbitrary shape and the shape space defined by some reference shape requires one to find the reference shape up to a similarity transform that is closest to the arbi- trary shape. This idea is illustrated in Figure 2. The advan- tages of our method compared to the traditional approach are three-fold: 1) no normalization step is required prior to the definition of the shape space. Hence, no bias due to normal- 978-1-4799-2374-8/15/$31.00 ©2015 IEEE 1216