Convergence of the Mean Shift Using the Infinity Norm in Image Segmentation

Abstract

In this work a comparison between two algorithms for image segmentation via the mean shift is carried out. These algorithms apply recursively the mean shift filtering by using the euclidean and infinity norms in order to define pixel neighborhoods. In the conventional mean shift algorithm for image segmentation euclidean norm is used. Due to matrix representation of images and the rectangular windows used in the implementation of the algorithm, with the aim of evaluating pixels membership to neighborhoods, the use of euclidean norm always implies evaluate membership of pixels wich certainly cannot be inside each defined neighborhood. However, the use of the infinity norm ensures that each evaluated pixel could be inside the neighborhood. This fact avoids unnecessary calculations. In the work the convergence of the algorithm, by using the infinity norm, is proven. Additionaly, the runtimes of both algorithms were compared using different spatial and range bandwidth sizes. Through an extensive experimentation using standard images was evidenced that the use of the infinity norm, instead of the euclidean norm, decreases the runtime of the mean shift when the values of spatial and range bandwidths were increased.

Image Segmentation Through an Iterative Algorithm of the Mean Shift

“…The following theorem guarantees the convergence when it replaces the l 2 norm by the l ∞ norm. The proof is similar to the theorem proved in [Comaniciu, 2000] and it can be found in [Domínguez & Rodríguez, 2011].…”

“…The convergence of the algorithm by using the l ∞ norm was empirically shown through an extensive experimentation [Domínguez & Rodríguez, 2009]. In [Domínguez & Rodríguez, 2011] was proven a theorem which guarantees the convergence of the l ∞ norm instead of the l 2 norm in order to define the regionS h (x). The convergence of mean shift for discrete data was proved in [Comaniciu, 2000] using the l 2 norm for defining the hypersphereS h (x).…”

“…The iterative algorithm that is used in this chapter is based on the mean shift and in several works was previously introduced and applied [Rodríguez & Suarez, 2006;Rodríguez, 2008;Domínguez & Rodríguez, 2009;Domínguez & Rodríguez, 2011;Rodríguez et. al., 2011a;Rodríguez et.…”

Cuboid Colour Image Segmentation Using Intuitive Distance Measure

“…To address this issue, we propose to use infinity norm distance because the RGB colour space is orthogonal and human eyes perceive the difference of two colour by the maximum difference in the RGB value. In [2], instead of the Euclidean distance, infinity norm distance is used to measure the neighbourhood between two pixels thus reducing the calculation time.…”

“…Deterministic algorithms can be classified as supervised (e.g. Mean Shift (MS) [2], Fuzzy C-Means (FCM) [3], and K-Means) and unsupervised (e.g. pulse coupled neural network [4]).…”