Scalespace theory is a framework for multiscale signal representation developed by the computer vision, image processing and signal processing communities with complementary motivations from physics and biological vision. It is a formal theory for handling image structures at different scales, by representing an image as a oneparameter family of smoothed images, the scalespace representation, parametrized by the size of the smoothing kernel used for suppressing finescale structures.^{[1]}^{[2]}^{[3]}^{[4]}^{[5]}^{[6]}^{[7]} The parameter in this family is referred to as the scale parameter, with the interpretation that image structures of spatial size smaller than about have largely been smoothed away in the scalespace level at scale .
The main type of scale space is the linear (Gaussian) scale space, which has wide applicability as well as the attractive property of being possible to derive from a small set of scalespace axioms. The corresponding scalespace framework encompasses a theory for Gaussian derivative operators, which can be used as a basis for expressing a large class of visual operations for computerized systems that process visual information. This framework also allows visual operations to be made scale invariant, which is necessary for dealing with the size variations that may occur in image data, because realworld objects may be of different sizes and in addition the distance between the object and the camera may be unknown and may vary depending on the circumstances.^{[8]}^{[9]}
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Transcription
Contents
Definition
The notion of scale space applies to signals of arbitrary numbers of variables. The most common case in the literature applies to twodimensional images, which is what is presented here. For a given image , its linear (Gaussian) scalespace representation is a family of derived signals defined by the convolution of with the twodimensional Gaussian kernel
such that
where the semicolon in the argument of implies that the convolution is performed only over the variables , while the scale parameter after the semicolon just indicates which scale level is being defined. This definition of works for a continuum of scales , but typically only a finite discrete set of levels in the scalespace representation would be actually considered.
The scale parameter is the variance of the Gaussian filter and as a limit for the filter becomes an impulse function such that that is, the scalespace representation at scale level is the image itself. As increases, is the result of smoothing with a larger and larger filter, thereby removing more and more of the details which the image contains. Since the standard deviation of the filter is , details which are significantly smaller than this value are to a large extent removed from the image at scale parameter , see the following figure and^{[10]} for graphical illustrations.
Why a Gaussian filter?
When faced with the task of generating a multiscale representation one may ask: could any filter g of lowpass type and with a parameter t which determines its width be used to generate a scale space? The answer is no, as it is of crucial importance that the smoothing filter does not introduce new spurious structures at coarse scales that do not correspond to simplifications of corresponding structures at finer scales. In the scalespace literature, a number of different ways have been expressed to formulate this criterion in precise mathematical terms.
The conclusion from several different axiomatic derivations that have been presented is that the Gaussian scale space constitutes the canonical way to generate a linear scale space, based on the essential requirement that new structures must not be created when going from a fine scale to any coarser scale.^{[2]}^{[3]}^{[5]}^{[8]}^{[11]}^{[12]}^{[13]}^{[14]}^{[15]}^{[16]}^{[17]}^{[18]} Conditions, referred to as scalespace axioms, that have been used for deriving the uniqueness of the Gaussian kernel include linearity, shift invariance, semigroup structure, nonenhancement of local extrema, scale invariance and rotational invariance. In the works,^{[14]}^{[19]}^{[20]} the uniqueness claimed in the arguments based on scale invariance originally due to Iijima (1962) has been criticized, and alternative selfsimilar scalespace kernels have been proposed. The Gaussian kernel is, however, a unique choice according to the scalespace axiomatics based on causality^{[2]} or nonenhancement of local extrema.^{[15]}^{[17]}
Alternative definition
Equivalently, the scalespace family can be defined as the solution of the diffusion equation (for example in terms of the heat equation),
with initial condition . This formulation of the scalespace representation L means that it is possible to interpret the intensity values of the image f as a "temperature distribution" in the image plane and that the process which generates the scalespace representation as a function of t corresponds to heat diffusion in the image plane over time t (assuming the thermal conductivity of the material equal to the arbitrarily chosen constant ½). Although this connection may appear superficial for a reader not familiar with differential equations, it is indeed the case that the main scalespace formulation in terms of nonenhancement of local extrema is expressed in terms of a sign condition on partial derivatives in the 2+1D volume generated by the scale space, thus within the framework of partial differential equations. Furthermore, a detailed analysis of the discrete case shows that the diffusion equation provides a unifying link between continuous and discrete scale spaces, which also generalizes to nonlinear scale spaces, for example, using anisotropic diffusion. Hence, one may say that the primary way to generate a scale space is by the diffusion equation, and that the Gaussian kernel arises as the Green's function of this specific partial differential equation.
Motivations
The motivation for generating a scalespace representation of a given data set originates from the basic observation that realworld objects are composed of different structures at different scales. This implies that realworld objects, in contrast to idealized mathematical entities such as points or lines, may appear in different ways depending on the scale of observation. For example, the concept of a "tree" is appropriate at the scale of meters, while concepts such as leaves and molecules are more appropriate at finer scales. For a computer vision system analysing an unknown scene, there is no way to know a priori what scales are appropriate for describing the interesting structures in the image data. Hence, the only reasonable approach is to consider descriptions at multiple scales in order to be able to capture the unknown scale variations that may occur. Taken to the limit, a scalespace representation considers representations at all scales.^{[8]}
Another motivation to the scalespace concept originates from the process of performing a physical measurement on realworld data. In order to extract any information from a measurement process, one has to apply operators of noninfinitesimal size to the data. In many branches of computer science and applied mathematics, the size of the measurement operator is disregarded in the theoretical modelling of a problem. The scalespace theory on the other hand explicitly incorporates the need for a noninfinitesimal size of the image operators as an integral part of any measurement as well as any other operation that depends on a realworld measurement.^{[4]}
There is a close link between scalespace theory and biological vision. Many scalespace operations show a high degree of similarity with receptive field profiles recorded from the mammalian retina and the first stages in the visual cortex. In these respects, the scalespace framework can be seen as a theoretically wellfounded paradigm for early vision, which in addition has been thoroughly tested by algorithms and experiments.^{[3]}^{[8]}
Gaussian derivatives
At any scale in scale space, we can apply local derivative operators to the scalespace representation:
Due to the commutative property between the derivative operator and the Gaussian smoothing operator, such scalespace derivatives can equivalently be computed by convolving the original image with Gaussian derivative operators. For this reason they are often also referred to as Gaussian derivatives:
The uniqueness of the Gaussian derivative operators as local operations derived from a scalespace representation can be obtained by similar axiomatic derivations as are used for deriving the uniqueness of the Gaussian kernel for scalespace smoothing.^{[3]}^{[21]}
Visual front end
These Gaussian derivative operators can in turn be combined by linear or nonlinear operators into a larger variety of different types of feature detectors, which in many cases can be well modelled by differential geometry. Specifically, invariance (or more appropriately covariance) to local geometric transformations, such as rotations or local affine transformations, can be obtained by considering differential invariants under the appropriate class of transformations or alternatively by normalizing the Gaussian derivative operators to a locally determined coordinate frame determined from e.g. a preferred orientation in the image domain or by applying a preferred local affine transformation to a local image patch (see the article on affine shape adaptation for further details).
When Gaussian derivative operators and differential invariants are used in this way as basic feature detectors at multiple scales, the uncommitted first stages of visual processing are often referred to as a visual frontend. This overall framework has been applied to a large variety of problems in computer vision, including feature detection, feature classification, image segmentation, image matching, motion estimation, computation of shape cues and object recognition. The set of Gaussian derivative operators up to a certain order is often referred to as the Njet and constitutes a basic type of feature within the scalespace framework.
Detector examples
Following the idea of expressing visual operation in terms of differential invariants computed at multiple scales using Gaussian derivative operators, we can express an edge detector from the set of points that satisfy the requirement that the gradient magnitude
should assume a local maximum in the gradient direction
By working out the differential geometry, it can be shown ^{[3]} that this differential edge detector can equivalently be expressed from the zerocrossings of the secondorder differential invariant
that satisfy the following sign condition on a thirdorder differential invariant:
Similarly, multiscale blob detectors at any given fixed scale^{[8]} can be obtained from local maxima and local minima of either the Laplacian operator (also referred to as the Laplacian of Gaussian)
or the determinant of the Hessian matrix
In an analogous fashion, corner detectors and ridge and valley detectors can be expressed as local maxima, minima or zerocrossings of multiscale differential invariants defined from Gaussian derivatives. The algebraic expressions for the corner and ridge detection operators are, however, somewhat more complex and the reader is referred to the articles on corner detection and ridge detection for further details.
Scalespace operations have also been frequently used for expressing coarsetofine methods, in particular for tasks such as image matching and for multiscale image segmentation.
Scale selection
The theory presented so far describes a wellfounded framework for representing image structures at multiple scales. In many cases it is, however, also necessary to select locally appropriate scales for further analysis. This need for scale selection originates from two major reasons; (i) realworld objects may have different size, and this size may be unknown to the vision system, and (ii) the distance between the object and the camera can vary, and this distance information may also be unknown a priori. A highly useful property of scalespace representation is that image representations can be made invariant to scales, by performing automatic local scale selection^{[8]}^{[9]}^{[22]}^{[23]}^{[24]}^{[25]}^{[26]}^{[27]} based on local maxima (or minima) over scales of normalized derivatives
where is a parameter that is related to the dimensionality of the image feature. This algebraic expression for scale normalized Gaussian derivative operators originates from the introduction of normalized derivatives according to
 and
It can be theoretically shown that a scale selection module working according to this principle will satisfy the following scale covariance property: if for a certain type of image feature a local maximum is assumed in a certain image at a certain scale , then under a rescaling of the image by a scale factor the local maximum over scales in the rescaled image will be transformed to the scale level .
Scale invariant feature detection
Following this approach of gammanormalized derivatives, it can be shown that different types of scale adaptive and scale invariant feature detectors^{[8]}^{[9]}^{[22]}^{[23]}^{[24]}^{[28]}^{[29]}^{[26]} can be expressed for tasks such as blob detection, corner detection, ridge detection, edge detection and spatiotemporal interest points (see the specific articles on these topics for indepth descriptions of how these scaleinvariant feature detectors are formulated). Furthermore, the scale levels obtained from automatic scale selection can be used for determining regions of interest for subsequent affine shape adaptation^{[30]} to obtain affine invariant interest points^{[31]}^{[32]} or for determining scale levels for computing associated image descriptors, such as locally scale adapted Njets.
Recent work has shown that also more complex operations, such as scaleinvariant object recognition can be performed in this way, by computing local image descriptors (Njets or local histograms of gradient directions) at scaleadapted interest points obtained from scalespace extrema of the normalized Laplacian operator (see also scaleinvariant feature transform^{[33]}) or the determinant of the Hessian (see also SURF);^{[34]} see also the Scholarpedia article on the scaleinvariant feature transform^{[35]} for a more general outlook of object recognition approaches based on receptive field responses^{[18]}^{[36]}^{[37]}^{[38]} in terms Gaussian derivative operators or approximations thereof.
Related multiscale representations
An image pyramid is a discrete representation in which a scale space is sampled in both space and scale. For scale invariance, the scale factors should be sampled exponentially, for example as integer powers of 2 or root 2. When properly constructed, the ratio of the sample rates in space and scale are held constant so that the impulse response is identical in all levels of the pyramid.^{[39]}^{[40]}^{[41]} Fast, O(N), algorithms exist for computing a scale invariant image pyramid in which the image or signal is repeatedly smoothed then subsampled. Values for scale space between pyramid samples can easily be estimated using interpolation within and between scales and allowing for scale and position estimates with sub resolution accuracy.^{[41]}
In a scalespace representation, the existence of a continuous scale parameter makes it possible to track zero crossings over scales leading to socalled deep structure. For features defined as zerocrossings of differential invariants, the implicit function theorem directly defines trajectories across scales,^{[3]}^{[42]} and at those scales where bifurcations occur, the local behaviour can be modelled by singularity theory.^{[3]}^{[42]}^{[43]}^{[44]}
Extensions of linear scalespace theory concern the formulation of nonlinear scalespace concepts more committed to specific purposes.^{[45]}^{[46]} These nonlinear scalespaces often start from the equivalent diffusion formulation of the scalespace concept, which is subsequently extended in a nonlinear fashion. A large number of evolution equations have been formulated in this way, motivated by different specific requirements (see the abovementioned book references for further information). It should be noted, however, that not all of these nonlinear scalespaces satisfy similar "nice" theoretical requirements as the linear Gaussian scalespace concept. Hence, unexpected artifacts may sometimes occur and one should be very careful of not using the term "scalespace" for just any type of oneparameter family of images.
A firstorder extension of the isotropic Gaussian scale space is provided by the affine (Gaussian) scale space.^{[3]} One motivation for this extension originates from the common need for computing image descriptors subject for realworld objects that are viewed under a perspective camera model. To handle such nonlinear deformations locally, partial invariance (or more correctly covariance) to local affine deformations can be achieved by considering affine Gaussian kernels with their shapes determined by the local image structure,^{[30]} see the article on affine shape adaptation for theory and algorithms. Indeed, this affine scale space can also be expressed from a nonisotropic extension of the linear (isotropic) diffusion equation, while still being within the class of linear partial differential equations.
There exists a more general extension of the Gaussian scalespace model to affine and spatiotemporal scalespaces.^{[17]}^{[18]}^{[47]} In addition to variabilities over scale, which original scalespace theory was designed to handle, this generalized scalespace theory also comprises other types of variabilities caused by geometric transformations in the image formation process, including variations in viewing direction approximated by local affine transformations, and relative motions between objects in the world and the observer, approximated by local Galilean transformations. This generalized scalespace theory leads to predictions about receptive field profiles in good qualitative agreement with receptive field profiles measured by cell recordings in biological vision.^{[47]}^{[48]}^{[49]}
There are strong relations between scalespace theory and wavelet theory, although these two notions of multiscale representation have been developed from somewhat different premises. There has also been work on other multiscale approaches, such as pyramids and a variety of other kernels, that do not exploit or require the same requirements as true scalespace descriptions do.
Relations to biological vision and hearing
There are interesting relations between scalespace representation and biological vision and hearing. Neurophysiological studies of biological vision have shown that there are receptive field profiles in the mammalian retina and visual cortex, which can be well modelled by linear Gaussian derivative operators, in some cases also complemented by a nonisotropic affine scalespace model, a spatiotemporal scalespace model and/or nonlinear combinations of such linear operators.^{[17]}^{[48]}^{[49]}^{[47]}^{[50]}^{[51]} Regarding biological hearing there are receptive field profiles in the inferior colliculus and the primary auditory cortex that can be well modelled by spectratemporal receptive fields that can be well modelled by Gaussian derivates over logarithmic frequencies and windowed Fourier transforms over time with the window functions being temporal scalespace kernels.^{[52]}^{[53]}
Normative theories for visual and auditory receptive fields founded on the scalespace framework are described in the article on axiomatic theory of receptive fields.
Implementation issues
When implementing scalespace smoothing in practice there are a number of different approaches that can be taken in terms of continuous or discrete Gaussian smoothing, implementation in the Fourier domain, in terms of pyramids based on binomial filters that approximate the Gaussian or using recursive filters. More details about this are given in a separate article on scale space implementation.
See also
References
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External links
 Lindeberg, Tony (2008). "Scalespace". Encyclopedia of Computer Science and Engineering (Benjamin Wah, Ed), John Wiley and Sons. IV: 2495–2504. doi:10.1002/9780470050118.ecse609. ISBN 9780470050118.
 Lindeberg, Tony, "Scalespace: A framework for handling image structures at multiple scales", In: Proc. CERN School of Computing, Egmond aan Zee, The Netherlands, 821 September, 1996 (online web tutorial)
 Lindeberg, Tony: Scalespace theory: A basic tool for analysing structures at different scales, in J. of Applied Statistics, 21(2), pp. 224–270, 1994 (longer pdf tutorial on scalespace)
 Lindeberg, Tony, "Principles for automatic scale selection", In: B. Jähne (et al., eds.), Handbook on Computer Vision and Applications, volume 2, pp 239—274, Academic Press, Boston, USA, 1999. (tutorial on approaches to automatic scale selection)
 Lindeberg, Tony: "Scalespace theory" In: Encyclopedia of Mathematics, (Michiel Hazewinkel, ed) Kluwer, 1997
 Powers of ten interactive Java tutorial at Molecular Expressions website
 Online resource with spacetime receptive fields of visual neurons provided by Izumi Ohzawa at Osaka University
 Web archive backup: Lecture on scalespace at the University of Massachusetts (pdf)
 Multiscale analysis for optimized vessel segmentation of fundus retina images PhD Thesis
 Peak detection in 1D data using a scalespace approach BSDlicensed MATLAB code