The Generalization of the Decomposition of Functions by Energy Operators

This work starts with the introduction of a family of differential energy operators. Energy operators ($Psi_R^+$, $Psi_R^-$) were defined together with a method to decompose the wave equation in a previous work. Here the energy operators are defined following the order of their derivatives ($Psi_k^+$, $Psi_k^-$, k = {0,1,2,..}). The main part of the work is to demonstrate that for any smooth real-valued function f in the Schwartz space ($S^-(R)$), the successive derivatives of the n-th power of f (n in Z and n not equal to 0) can be decomposed using only $Psi_k^+$ (Lemma) or with $Psi_k^+$, $Psi_k^-$ (k in Z) (Theorem) in a unique way (with more restrictive conditions). Some properties of the Kernel and the Image of the energy operators are given along with the development. Finally, the paper ends with the application to the energy function.Comment: The paper was accepted for publication at Acta Applicandae Mathematicae (15/05/2013) based on v3. v4 is very similar to v3 except that we modified slightly Definition 1 to make it more readable when showing the decomposition with the families of energy operator of the derivatives of the n-th power of

Validating the detection of everyday concepts in visual lifelogs

The Microsoft SenseCam is a small lightweight wearable camera used to passively capture photos and other sensor readings from a user's day-to-day activities. It can capture up to 3,000 images per day, equating to almost 1 million images per year. It is used to aid memory by creating a personal multimedia lifelog, or visual recording of the wearer's life. However the sheer volume of image data captured within a visual lifelog creates a number of challenges, particularly for locating relevant content. Within this work, we explore the applicability of semantic concept detection, a method often used within video retrieval, on the novel domain of visual lifelogs. A concept detector models the correspondence between low-level visual features and high-level semantic concepts (such as indoors, outdoors, people, buildings, etc.) using supervised machine learning. By doing so it determines the probability of a concept's presence. We apply detection of 27 everyday semantic concepts on a lifelog collection composed of 257,518 SenseCam images from 5 users. The results were then evaluated on a subset of 95,907 images, to determine the precision for detection of each semantic concept and to draw some interesting inferences on the lifestyles of those 5 users. We additionally present future applications of concept detection within the domain of lifelogging. © 2008 Springer Berlin Heidelberg