Finding Mirror Symmetry via Registration

Symmetry is prevalent in nature and a common theme in man-made designs. Both the human visual system and computer vision algorithms can use symmetry to facilitate object recognition and other tasks. Detecting mirror symmetry in images and data is, therefore, useful for a number of applications. Here, we demonstrate that the problem of fitting a plane of mirror symmetry to data in any Euclidian space can be reduced to the problem of registering two datasets. The exactness of the resulting solution depends entirely on the registration accuracy. This new Mirror Symmetry via Registration (MSR) framework involves (1) data reflection with respect to an arbitrary plane, (2) registration of original and reflected datasets, and (3) calculation of the eigenvector of eigenvalue -1 for the transformation matrix representing the reflection and registration mappings. To support MSR, we also introduce a novel 2D registration method based on random sample consensus of an ensemble of normalized cross-correlation matches. With this as its registration back-end, MSR achieves state-of-the-art performance for symmetry line detection in two independent 2D testing databases. We further demonstrate the generality of MSR by testing it on a database of 3D shapes with an iterative closest point registration back-end. Finally, we explore its applicability to examining symmetry in natural systems by assessing the degree of symmetry present in myelinated axon reconstructions from a larval zebrafish.

A convolutional approach to reflection symmetry

We present a convolutional approach to reflection symmetry detection in 2D. Our model, built on the products of complex-valued wavelet convolutions, simplifies previous edge-based pairwise methods. Being parameter-centered, as opposed to feature-centered, it has certain computational advantages when the object sizes are known a priori, as demonstrated in an ellipse detection application. The method outperforms the best-performing algorithm on the CVPR 2013 Symmetry Detection Competition Database in the single-symmetry case. Code and a new database for 2D symmetry detection is available.

Complex-Valued Hough Transforms for Circles

This paper advocates the use of complex variables to represent votes in the Hough transform for circle detection. Replacing the positive numbers classically used in the parameter space of the Hough transforms by complex numbers allows cancellation effects when adding up the votes. Cancellation and the computation of shape likelihood via a complex number's magnitude square lead to more robust solutions than the "classic" algorithms, as shown by computational experiments on synthetic and real datasets.

Quantum Pairwise Symmetry: Applications in 2D Shape Analysis

A pair of rooted tangents -- defining a quantum triangle -- with an associated quantum wave of spin 1/2 is proposed as the primitive to represent and compute symmetry. Measures of the spin characterize how "isosceles" or how "degenerate" these triangles are -- which corresponds to their mirror or parallel symmetry. We also introduce a complex-valued kernel to model probability errors in the parameter space, which is more robust to noise and clutter than the classical model.

A Geometric Descriptor for Cell-Division Detection

We describe a method for cell-division detection based on a geometric-driven descriptor that can be represented as a 5-layers processing network, based mainly on wavelet filtering and a test for mirror symmetry between pairs of pixels. After the centroids of the descriptors are computed for a sequence of frames, the two-steps piecewise constant function that best fits the sequence of centroids determines the frame where the division occurs.

On the Product Rule for Classification Problems

We discuss theoretical aspects of the product rule for classification problems in supervised machine learning for the case of combining classifiers. We show that (1) the product rule arises from the MAP classifier supposing equivalent priors and conditional independence given a class; (2) under some conditions, the product rule is equivalent to minimizing the sum of the squared distances to the respective centers of the classes related with different features, such distances being weighted by the spread of the classes; (3) observing some hypothesis, the product rule is equivalent to concatenating the vectors of features.