Scale space
Scale-space theory is a framework for multi-scale 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 one-parameter family of smoothed images, the scale-space representation, parametrized by the size of the smoothing kernel used for suppressing fine-scale structures. 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 scale-space level at scale.
The main type of scale space is the linear scale space, which has wide applicability as well as the attractive property of being possible to derive from a small set of scale-space axioms. The corresponding scale-space 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 real-world 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.
Definition
The notion of scale space applies to signals of arbitrary numbers of variables. The most common case in the literature applies to two-dimensional images, which is what is presented here. Consider a given image where is the greyscale value of the pixel at position . The linear scale-space representation of is a family of derived signals defined by the convolution of with the two-dimensional Gaussian kernelsuch 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 scale-space 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 scale-space 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 that the image contains. Since the standard deviation of the filter is, details that are significantly smaller than this value are to a large extent removed from the image at scale parameter, see the following figures and for graphical illustrations.
Why a Gaussian filter?
When faced with the task of generating a multi-scale representation one may ask: could any filter g of low-pass 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 scale-space 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.
Conditions, referred to as scale-space axioms, that have been used for deriving the uniqueness of the Gaussian kernel include linearity, shift invariance, semi-group structure, non-enhancement of local extrema, scale invariance and rotational invariance.
In the works, the uniqueness claimed in the arguments based on scale invariance has been criticized, and alternative self-similar scale-space kernels have been proposed. The Gaussian kernel is, however, a unique choice according to the scale-space axiomatics based on causality or non-enhancement of local extrema.
Alternative definition
Equivalently, the scale-space family can be defined as the solution of the diffusion equation,with initial condition. This formulation of the scale-space 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 that generates the scale-space representation as a function of t corresponds to heat diffusion in the image plane over time t. Although this connection may appear superficial for a reader not familiar with differential equations, it is indeed the case that the main scale-space formulation in terms of non-enhancement of local extrema is expressed in terms of a sign condition on partial derivatives in the 2+1-D 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 scale-space representation of a given data set originates from the basic observation that real-world objects are composed of different structures at different scales. This implies that real-world 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 scale-space representation considers representations at all scales.
Another motivation to the scale-space concept originates from the process of performing a physical measurement on real-world data. In order to extract any information from a measurement process, one has to apply operators of non-infinitesimal 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 scale-space theory on the other hand explicitly incorporates the need for a non-infinitesimal size of the image operators as an integral part of any measurement as well as any other operation that depends on a real-world measurement.
There is a close link between scale-space theory and biological vision. Many scale-space 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 scale-space framework can be seen as a theoretically well-founded paradigm for early vision, which in addition has been thoroughly tested by algorithms and experiments.
Gaussian derivatives
At any scale in scale space, we can apply local derivative operators to the scale-space representation:Due to the commutative property between the derivative operator and the Gaussian smoothing operator, such scale-space 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 scale-space representation can be obtained by similar axiomatic derivations as are used for deriving the uniqueness of the Gaussian kernel for scale-space smoothing.
Visual front end
These Gaussian derivative operators can in turn be combined by linear or non-linear operators into a larger variety of different types of feature detectors, which in many cases can be well modelled by differential geometry. Specifically, invariance 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.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 front-end. 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 N-jet and constitutes a basic type of feature within the scale-space framework.
Detector examples
Following the idea of expressing visual operations 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 magnitudeshould assume a local maximum in the gradient direction
By working out the differential geometry, it can be shown that this differential edge detector can equivalently be expressed from the zero-crossings of the second-order differential invariant
that satisfy the following sign condition on a third-order differential invariant:
Similarly, multi-scale blob detectors at any given fixed scale can be obtained from local maxima and local minima of either the Laplacian operator
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 zero-crossings of multi-scale 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.
Scale-space operations have also been frequently used for expressing coarse-to-fine methods, in particular for tasks such as image matching and for multi-scale image segmentation.