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Multi scale approaches

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Joel saltz discusses integrative multi scale approaches to biomedical informatics research


The scale space representation of a signal obtained by Gaussian smoothing satisfies a number of special properties, scale-space axioms, which make it into a special form of multi-scale representation. There are, however, also other types of "multi-scale approaches" in the areas of computer vision, image processing and signal processing, in particular the notion of wavelets. The purpose of this article is to describe a few of these approaches:

Contents

Scale-space theory for one-dimensional signals

For one-dimensional signals, there exists quite a well-developed theory for continuous and discrete kernels that guarantee that new local extrema or zero-crossings cannot be created by a convolution operation. For continuous signals, it holds that all scale-space kernels can be decomposed into the following sets of primitive smoothing kernels:

  • the Gaussian kernel : g ( x , t ) = 1 2 π t exp ⁡ ( − x 2 / 2 t ) where t > 0 ,
  • truncated exponential kernels (filters with one real pole in the s-plane):
  • translations,
  • rescalings.

    • For discrete signals, we can, up to trivial translations and rescalings, decompose any discrete scale-space kernel into the following primitive operations:

  • the discrete Gaussian kernel
  • generalized binomial kernels corresponding to linear smoothing of the form
    • f o u t ( x ) = p f i n ( x ) + q f i n ( x − 1 ) where p , q > 0 f o u t ( x ) = p f i n ( x ) + q f i n ( x + 1 ) where p , q > 0 ,
  • first-order recursive filters corresponding to linear smoothing of the form
    • f o u t ( x ) = f i n ( x ) + α f o u t ( x − 1 ) where α > 0 f o u t ( x ) = f i n ( x ) + β f o u t ( x + 1 ) where β > 0 ,
  • the one-sided Poisson kernel
    • p ( n , t ) = e − t t n n ! for n ≥ 0 where t ≥ 0 p ( n , t ) = e − t t − n ( − n ) ! for n ≤ 0 where t ≥ 0 .

    From this classification, it is apparent that it we require a continuous semi-group structure, there are only three classes of scale-space kernels with a continuous scale parameter; the Gaussian kernel which forms the scale-space of continuous signals, the discrete Gaussian kernel which forms the scale-space of discrete signals and the time-causal Poisson kernel that forms a temporal scale-space over discrete time. If we on the other hand sacrifice the continuous semi-group structure, there are more options:

    For discrete signals, the use of generalized binomial kernels provides a formal basis for defining the smoothing operation in a pyramid. For temporal data, the one-sided truncated exponential kernels and the first-order recursive filters provide a way to define time-causal scale-spaces that allow for efficient numerical implementation and respect causality over time without access to the future. The first-order recursive filters also provide a framework for defining recursive approximations to the Gaussian kernel that in a weaker sense preserve some of the scale-space properties.

    References

    Multi-scale approaches Wikipedia
    (Text) CC BY-SA