The Gaussian scale-space paradigm and the multiscale local jet

Abstract

A representation of local image structure is proposed which takes into account both the image's spatial structure at a given location, as well as its "deep structure", that is, its local behaviour as a function of scale or resolution (scale-space). This is of interest for several low-level image tasks. The proposed basis of scale-space, for example, enables a precise local study of interactions of neighbouring image intensities in the course of the blurring process. It also provides an extrapolation scheme for local image data, obtained at a given spatial location and resolution, to a finite scale-space neighbourhood. This is especially useful for the determination of sampling rates and for interpolation algorithms in a multilocal context. Another, particularly straightforward application is image enhancement or deblurring, which is an instance of data extrapolation in the high-resolution direction.A potentially interesting feature of the proposed local image parametrisation is that it captures a trade-off between spatial and scale extrapolations from a given interior point that do not exceed a given tolerance. This trade-off suggests the possibility of a fairly coarse scale sampling at the expense of a dense spatial sampling (large relative spatial overlap of scale-space kernels).The central concept developed in this paper is an equivalence class called the multiscale local jet, which is a hierarchical, local characterisation of the image in a full scale-space neighbourhood. For this local jet, a basis of fundamental polynomials is constructed that captures the scale-space paradigm at the local level up to any given order.

Computer aided diagnosis in chest radiography

“…without violating the causality principle [136]. As such, one can apply to LOIs the whole machinery of techniques that has been developed in the context of scale-space research, such as the N-jet for local image structure description [137,70,94], the construction of invariants and (oriented) filter families [138,139,140,71], non-linear diffusion schemes [195,93], scale selection methods [160,161], etc.…”

“…In this work we will not give a thorough discussion on how to construct a filter bank that is optimally tuned for the task of texture feature extraction, although this is an interesting and non-trivial matter. In the context of LOIs, which are a natural extension of the linear scale-space theory of image observations to histograms, the N-jet of derivatives up to a given order appears the natural choice for a filter bank [137,70], analogous to the use of a Taylor expansion to locally describe a function. This type of filter banks will be used, where the n-th order derivative in the direction θ will be indicated as L θ n .…”

“…The derivatives can be measured at various scales. The Taylor expansion of images is known as the local jet, or multi-scale local jet in the case of the Taylor expansion of the scale-space of the image [137,70].…”

Jet-Based Local Image Descriptors

“…To construct the jet descriptor we use the multi-scale k-jet as described by Florack et al [6]. We consider the scale normalized derivatives of the linear scale-space of a 2D image I(r) :…”

“…The introduction of PCA-SIFT inevitable leads to the question of whether the complex structure and relatively high dimensionality of most descriptors are in fact over-representations and perhaps much simpler, theoretically sound, and compact formulations exist. This observation has inspired us to investigate the local k-jet [6], a higher order differential descriptor, as a natural basis for encoding the geometry of an image patch. Thus, the descriptor we propose can, similar to PCA-SIFT, be interpreted as a basis representation of the image patch.…”