Preface Through many centuries physics has been one of the most fruitful sources of inspiration for mathematics. As a consequence, mathematics has become an economic language providing a few basic principles which allow to explain a large variety of physical phenomena. Many of them are described in terms of partial diierential equations (PDEs). In recent years, however, mathematics also has been stimulated by other novel elds such as image processing. Goals like image segmentation, multiscale image representation, or image restoration cause a lot of challenging mathematical questions. Nevertheless, these problems frequently have been tackled with a pool of heuristical recipes. Since the treatment of digital images requires very much computing power, these methods had to be fairly simple. With the tremendous advances in computer technology in the last decade, it has become possible to apply more sophisticated techniques such as PDE-based methods which have been inspired by physical processes. Among these techniques, parabolic PDEs have found a lot of attention for smoothing and restoration purposes, see e.g. 113]. To restore images these equations frequently arise from gradient descent methods applied to variational problems. Image smoothing by parabolic PDEs is closely related to the scale-space concept where one embeds the original image into a family of subsequently simpler , more global representations of it. This idea plays a fundamental role for extracting semantically important information. The pioneering work of Alvarez, Guichard, Lions and Morel 11] has demonstrated that all scale-spaces fulllling a few fairly natural axioms are governed by parabolic PDEs with the original image as initial condition. Within this framework, two classes can be justiied in a rigorous way as scale-spaces: the linear diiusion equation with constant dif-fusivity and nonlinear so-called morphological PDEs. All these methods satisfy a monotony axiom as smoothing requirement which states that, if one image is brighter than another, then this order is preserved during the entire scale-space evolution. An interesting class of parabolic equations which pursue both scale-space and restoration intentions is given by nonlinear diiusion lters. Methods of this type have been proposed for the rst time by Perona and Malik in 1987 190]. In v vi PREFACE order to smooth the image and to simultaneously enhance semantically important features such as edges, they apply a diiusion process whose diiusivity is steered by local image properties. These lters are diicult to analyse mathematically , as they may act locally like a backward diiusion process. …

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Anisotropic diffusion in image processing

An inherent property of objects in the world is that they only exist as meaningful entities over certain ranges of scale. If one aims at describing the structure of unknown real-world signals, then ...

/pdf/scale-space-theory-a-basic-tool-for-analysing-structures-at-5umoin3tlu.pdf

Scale-Space Theory : A Basic Tool for Analysing Structures at Different Scales

Many computer vision problems can be considered to consist of two main tasks: the extraction of image content descriptions and their subsequent matching. The appropriate choice of type and level of description is of course task dependent, yet it is generally accepted that the low-level or so called early vision layers in the Human Visual System are context independent.

This paper concentrates on the use of low-level approaches for solving computer vision problems and discusses three inter-related aspects of this: saliencys scale selection and content description. In contrast to many previous approaches which separate these tasks, we argue that these three aspects are intrinsically related. Based on this observation, a multiscale algorithm for the selection of salient regions of an image is introduced and its application to matching type problems such as tracking, object recognition and image retrieval is demonstrated.

/pdf/saliency-scale-and-image-description-19bs91yuos.pdf

Saliency, Scale and Image Description

When computing descriptors of image data, the type of information that can be extracted may be strongly dependent on the scales at which the image operators are applied. This article presents a systematic methodology for addressing this problem. A mechanism is presented for automatic selection of scale levels when detecting one-dimensional image features, such as edges and ridges.

A novel concept of a scale-space edge is introduced, defined as a connected set of points in scale-space at which: (i) the gradient magnitude assumes a local maximum in the gradient direction, and (ii) a normalized measure of the strength of the edge response is locally maximal over scales. An important consequence of this definition is that it allows the scale levels to vary along the edge. Two specific measures of edge strength are analyzed in detail, the gradient magnitude and a differential expression derived from the third-order derivative in the gradient direction. For a certain way of normalizing these differential descriptors, by expressing them in terms of so-called γ-normalized derivatives, an immediate consequence of this definition is that the edge detector will adapt its scale levels to the local image structure. Specifically, sharp edges will be detected at fine scales so as to reduce the shape distortions due to scale-space smoothing, whereas sufficiently coarse scales will be selected at diffuse edges, such that an edge model is a valid abstraction of the intensity profile across the edge.

Since the scale-space edge is defined from the intersection of two zero-crossing surfaces in scale-space, the edges will by definition form closed curves. This simplifies selection of salient edges, and a novel significance measure is proposed, by integrating the edge strength along the edge. Moreover, the scale information associated with each edge provides useful clues to the physical nature of the edge.

With just slight modifications, similar ideas can be used for formulating ridge detectors with automatic selection, having the characteristic property that the selected scales on a scale-space ridge instead reflect the width of the ridge.

It is shown how the methodology can be implemented in terms of straightforward visual front-end operations, and the validity of the approach is supported by theoretical analysis as well as experiments on real-world and synthetic data.

/pdf/edge-detection-and-ridge-detection-with-automatic-scale-3pkqae89ct.pdf

Edge Detection and Ridge Detection with Automatic Scale Selection

When extracting features from image data, the type of information that can be extracted may be strongly dependent on the scales at which the feature detectors are applied. This article presents a systematic methodology for addressing this problem. A mechanism is presented for automatic selection of scale levels when detecting one-dimensional features, such as edges and ridges. A novel concept of a scale-space edge is introduced, defined as a connected set of points in scale-space at which: (i) the gradient magnitude assumes a local maximum in the gradient direction, and (ii) a normalized measure of the strength of the edge response is locally maximal over scales. An important property of this definition is that it allows the scale levels to vary along the edge. Two specific measures of edge strength are analysed in detail. It is shown that by expressing these in terms of /spl gamma/-normalized derivatives, an immediate consequence of this definition is that fine scales are selected for sharp edges (so as to reduce the shape distortions due to scale-space smoothing), whereas coarse scales are selected for diffuse edges, such that an edge model constitutes a valid abstraction of the intensity profile across the edge. With slight modifications, this idea can be used for formulating a ridge detector with automatic scale selection, having the characteristic property that the selected scales on a scale-space ridge instead reflect the width of the ridge.

/pdf/edge-detection-and-ridge-detection-with-automatic-scale-3j59qjxp0l.pdf

Edge detection and ridge detection with automatic scale selection

Edge detection in a gray-scale image at a fine resolution typically yields noise and unnecessary detail, whereas edge detection at a coarse resolution distorts edge contours. We show that ``edge focusing'', i.e., a coarse-to-fine tracking in a continuous manner, combines high positional accuracy with good noise-reduction. This is of vital interest in several applications. Junctions of different kinds are in this way restored with high precision, which is a basic requirement when performing (projective) geometric analysis of an image for the purpose of restoring the three-dimensional scene. Segmentation of a scene using geometric clues like parallelism, etc., is also facilitated by the algorithm, since unnecessary detail has been filtered away. There are indications that an extension of the focusing algorithm can classify edges, to some extent, into the categories diffuse and nondiffuse (for example diffuse illumination edges). The edge focusing algorithm contains two parameters, namely the coarseness of the resolution in the blurred image from where we start the focusing procedure, and a threshold on the gradient magnitude at this coarse level. The latter parameter seems less critical for the behavior of the algorithm and is not present in the focusing part, i.e., at finer resolutions. The step length of the scale parameter in the focusing scheme has been chosen so that edge elements do not move more than one pixel per focusing step.

Edge Focusing

Climate warming in the Swedish sub-Arctic since 2000 has reached a level at which statistical analysis shows for the first time that current warming has exceeded that in the late 1930' s and early 1940' s, and has significantly crossed the 0 degrees C mean annual temperature threshold which causes many cryospheric and ecological impacts. The accelerating temperature increase trend has driven similar trends in the century-long increase in snow thickness, loss of lake ice, increases in active layer thickness, lake water TOC (total organic carbon) concentrations and the assemblages of diatoms, and changes in tree-line location and plant community structure. Some of these impacts were not evident in the first warm period of the 20th Century. Changes in climate are associated with reduced temperature variability, particularly loss of cold winters and cool summers, and an increase in extreme precipitation events that cause mountain slope instability and infrastructure failure. The long term records of multiple, local environmental factors compiled here for the first time provide detailed information for adaptation strategy development while dramatic changes in an environment particularly vulnerable to climate change highlight the need to adopt global mitigation strategies. Citation: Callaghan, T. V., F. Bergholm, T. R. Christensen, C. Jonasson, U. Kokfelt, and M. Johansson (2010), A new climate era in the sub-Arctic: Accelerating climate changes and multiple impacts, Geophys. Res. Lett., 37, L14705, doi:10.1029/2009GL042064.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2009GL042064

A new climate era in the sub-Arctic: Accelerating climate changes and multiple impacts

The present invention relates to the production of instantaneous or non-instantaneous multi-band images, to be transformed into multi- or hyperspectral images, comprising light collecting means (11), an image sensor (12) with at least one two dimensional sensor array (121), and an instantaneous colour separating means (123), positioned before the image sensor array (121) in the optical path (OP) of the arrangement (1), and first uniform spectral filters (13) in the optical path (OP), with the purpose of restricting imaging to certain parts of the electromagnetic spectrum. The present invention specifically teaches that a filter unit (FU) comprising colour or spectral filter mosaics and/or uniform colour or spectral filters mounted on filter wheels (114) or displayed by transmissive displays (115), is either permanently or interchangeably positioned before the colour separating means (123) in the optical path (OP) in, or close to, converged light (B). Each colour or spectral filter mosaic consists of a multitude of homogeneous filtering regions. The transmission curves (TC) of the filtering regions of a colour or spectral filter mosaic can be partly overlapping, in addition to overlap between these transmission curves and those belonging to the filtering regions of the colour separating means (123). The transmission curves (TC) of the colour or spectral filter mosaics and the colour separating means (123) are suitably spread out in the intervals of a spectrum to be studied. The combination of the colour separating means (123) and the spectral or colour or spectral filter mosaics produces different sets of linearly independent transmission curves (TC). The multiple-filter image captured by the image sensor (12) is demosaicked by identifying and segmenting the image regions that are affected by the regions of the multiple filter mosaic, and after an optional interpolation step, a multi-band image is obtained. The resulting multi-band image is transformed into a multi- or hyperspectral image.

A system for multi- and hyperspectral imaging

Signatures, in this work, are multi-scale representations of local gray-level information, tied to places in gray scale images where regional differences are locally maximal. The information may involve the regional differences themselves (called Gaussian differences or signed normalized gradient magnitudes, (Korn, 1988)), or, distance relations between edges (apparent width measurements), or, absence of edges in pulse edge pairs, at coarser scales. Using signatures involves the classical problem mentioned by Marr and others of relating information across scales. A novel result is that a fruitful way of doing this is to build scale paths from coarse-to-fine exploiting edge focusing and associate with pixel positions, along these paths, the three quantities Gaussian differences, apparent width and the binary information absence/presence of edges (in edge-pairs). Such a structure, if used together with proper conditional tests, serves the purpose of classifying edges with respect to profile-type, and can also be used for measuring global contrast, degree of diffuseness, deblurred line width, and qualitative labels such as diffuse versus sharp. The structure is used simultaneously for labelling tasks and quantitative measurements. Theory on apparent widths, absence/presence of edges in pulse edge pairs is developed. For measuring diffuseness and global contrast from Gaussian difference signatures a linear least squares approach is suggested. Extensive experimental results are presented. Possible applications are in image segementation, junction analysis, and depth-from-defocus. For the purpose of distinguishing between objects and illumination phenomena, such as diffuse shadow edges, classification of contours with respect to diffuseness seems useful.

Fredrik Bergholm

Papers

Edge Focusing

A new climate era in the sub-Arctic: Accelerating climate changes and multiple impacts

A system for multi- and hyperspectral imaging

Multi-Scale Blur Estimation and Edge Type Classification for Scene Analysis

Digital sieving-Matlab based 3-D image analysis