scale space.docx

1、scale spaceScale spaceScale-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

2、 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.1234567 The parameter t in this family is referred to as the scale parameter, with the

3、 interpretation that image structures of spatial size smaller than about have largely been smoothed away in the scale-space level at scale t.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

4、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

5、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 circums

6、tances.8DefinitionThe 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. For a given image f(x,y), its linear (Gaussian) scale-space representation is a family of deriv

7、ed signals L(x,y;t) defined by the convolution of f(x,y) with the Gaussian kernelsuch thatwhere the semicolon in the argument of L implies that the convolution is performed only over the variables x,y, while the scale parameter t after the semicolon just indicates which scale level is being defined.

8、 This definition of L works for a continuum of scales , but typically only a finite discrete set of levels in the scale-space representation would be actually considered.t is the variance of the Gaussian filter and as a limit for t = 0 the filter g becomes an impulse function such that L(x,y;0) = f(

9、x,y), that is, the scale-space representation at scale level t = 0 is the image f itself. As t increases, L is the result of smoothing f 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 wh

10、ich are significantly smaller than this value are to a large extent removed from the image at scale parameter t, see the following figure and 9 for graphical illustrations. Scale-space representation L(x,y;t) at scale t = 0, corresponding to the original image f Scale-space representation L(x,y;t) a

11、t scale t = 1 Scale-space representation L(x,y;t) at scale t = 4 Scale-space representation L(x,y;t) at scale t = 16 Scale-space representation L(x,y;t) at scale t = 64 Scale-space representation L(x,y;t) at scale t = 256edit Why a Gaussian filter?When faced with the task of generating a multi-scale

12、 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 cor

13、respond 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

14、 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.2358101112131415 Conditions, referred to as scale-space axioms, that have been used f

15、or deriving the uniqueness of the Gaussian kernel include linearity, shift invariance, semi-group structure, non-enhancement of local extrema, scale invariance and rotational invariance.Equivalently, the scale-space family can be defined as the solution of the diffusion equation (for example in term

16、s of the heat equation),with initial condition L(x,y;0) = f(x,y). 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 which generates the scale-space repr

17、esentation 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 ). Although this connection may appear superficial for a reader not familiar with differential equations, it is indee

18、d 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 analys

19、is of the discrete case shows that the diffusion equation provides a unifying link between continuous and discrete scale-spaces, which also generalizes to non-linear scale-spaces, for example, using anisotropic diffusion. Hence, one may say that the primary way to generate a scale-space is by the di

20、ffusion equation, and that the Gaussian kernel arises as the Greens function of this specific partial differential equation.MotivationsThe motivation for generating a scale-space representation of a given data set originates from the basic observation that real-world objects are composed of differen

21、t 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

22、 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 de

23、scriptions 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.8Another motivation to the scale-space concept originates from the process of performing a physical measu

24、rement 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

25、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.4There is a close link between scale-space theory

26、 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

27、 vision, which in addition has been thoroughly tested by algorithms and experiments.38edit Gaussian derivatives and the notion of a visual front-endAt any scale in scale-space, we can apply local derivative operators to the scale-space representation:Due to the commutative property between the deriv

28、ative 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:Interestingly, the uniqueness of the Gaussian der

29、ivative 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.316These Gaussian derivative operators can in turn be combined by linear or non-li

30、near 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 o

31、btained 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 a

32、ffine 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 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