Advances in Imaging and Electron Physics, Volume 178 by Peter W. Hawkes

By Peter W. Hawkes

Advances in Imaging and Electron Physics features state-of-the-art articles at the physics of electron units (especially semiconductor devices), particle optics at low and high energies, microlithography, photo technology and electronic photograph processing, electromagnetic wave propagation, electron microscopy, and the computing equipment utilized in a lot of these domain names.

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* Contributions from best experts * Informs and updates on the entire most modern advancements within the box

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Extra resources for Advances in Imaging and Electron Physics, Volume 178

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1995, Lindeberg 1996, Sporring et al. 1996, Florack 1997, Weickert et al. 1999, ter Haar Romeny 2003). The Gaussian function is also special in the following respects: • It minimizes the uncertainty relation (Folland & Sitaram 1997), which implies that in an N-dimensional space with f ˛L 2 ðRN Þ and with Z Z 2 2 xj f ðxÞj2 dx jx À xj j f ðxÞj dx N N 2 x˛R x˛R Z where x ¼ Z (61) hxi ¼ 2 j f ðxÞj dx j f ðxÞj2 dx x˛RN x˛RN and Z hui2 ¼ u˛RZN  2 ju À uj2  bf ðuÞ du u˛RN    bf ðuÞ2 du Z where u ¼ Zu˛R N  2 u bf ðuÞ du u˛RN ;    bf ðuÞ2 du (62) then it holds for any f that hxihui !

S1 and any two moments t2 ! t1 , where • The kernel U updates the internal state. • The kernel B incorporates new image data into the representation. • z˛Rþ is an integration variable referring to internal temporal buffers at different temporal scales. Note that this algebraic structure comprises increments over both time t2 ! t1 and spatial scales s2 ! s1 . K, temporal recursivity can, in combination with the requirement of a semigroup property over space, be expressed in terms of a spatio-temporal time-recursive structure of the form Z X K Lðx; t2 ; s2 ; kÞ ¼ Uðx À x; t2 À t1 ; s2 À s1 ; k; zÞLðx; t1 ; s1 ; zÞdz dx x˛RN z¼0 Z Zt2 Bðx À x; t2 À u; s2 ; kÞf ðx; uÞdx du; þ x˛RN u¼t1 where the kernel U updates the internal state, the kernel B incorporates new information, and z constitutes an index over the internal temporal scale 52 Tony Lindeberg levels.

S1 and s2 ! , a similar type of transformation as from the original data f. An image representation having these properties is referred to as a spatio-temporal multiscale representation. 2. Additional Scale-Space Axioms for Time-Dependent Image Data For spatio-temporal image data, the following covariance requirements are natural to impose motivated by the special nature of time and space-time. 47 Generalized Axiomatic Scale-Space Theory Temporal covariance If the same scene is observed by two different cameras that sample the spatiotemporal image data with different temporal sampling rates, or if a camera observes similar types of motion patterns that occur at different speeds, it seems natural that the visual system should be able to relate the spatiotemporal scale-space representations that are computed from the timedependent image data.

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