An approximate Hahn–Banach theorem for positively homogeneous functions

This note provides an approximate version of the Hahn–Banach theorem for non-necessarily convex extended-real valued positively homogeneous functions of degree one. Given p : X → R∪{+∞} such a function defined on the real vector space X, and a linear function defined on a subspace V of X and dominated by p (i.e. (x) ≤ p(x) for all x ∈ V), we say that can approximately be p-extended to X, if is the pointwise limit of a net of linear functions on V, every one of which can be extended to a linear function defined on X and dominated by p. The main result of this note proves that can approximately be p-extended to X if and only if is dominated by p∗∗, the pointwise supremum over the family of all the linear functions on X which are dominated by p.

Football Video Annotation Based on Player Motion Recognition Using Enhanced Entropy

This paper presents a semi-parametric Algorithm for parsing football video structures. The approach works on a two interleaved based process that closely collaborate towards a common goal. The core part of the proposed method focus perform a fast automatic football video annotation by looking at the enhance entropy variance within a series of shot frames. The entropy is extracted on the Hue parameter from the HSV color system, not as a global feature but in spatial domain to identify regions within a shot that will characterize a certain activity within the shot period. The second part of the algorithm works towards the identification of dominant color regions that could represent players and playfield for further activity recognition. Experimental Results shows that the proposed football video segmentation algorithm performs with high accuracy.