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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.¹ 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 ) {\displaystyle g(x,t)={\frac {1}{\sqrt {2\pi t}}}\exp({-x^{2}/2t})} where t > 0 {\displaystyle t>0} ,
• truncated exponential kernels (filters with one real pole in the s-plane):

h ( x ) = exp ⁡ ( − a x ) {\displaystyle h(x)=\exp({-ax})} if x ≥ 0 {\displaystyle x\geq 0} and 0 otherwise where a > 0 {\displaystyle a>0} h ( x ) = exp ⁡ ( b x ) {\displaystyle h(x)=\exp({bx})} if x ≤ 0 {\displaystyle x\leq 0} and 0 otherwise where b > 0 {\displaystyle b>0} ,

• 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

T ( n , t ) = I n ( α t ) {\displaystyle T(n,t)=I_{n}(\alpha t)} where α , t > 0 {\displaystyle \alpha ,t>0} where I n {\displaystyle I_{n}} are the modified Bessel functions of integer order,

• 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 ) {\displaystyle f_{out}(x)=pf_{in}(x)+qf_{in}(x-1)} where p , q > 0 {\displaystyle p,q>0} f o u t ( x ) = p f i n ( x ) + q f i n ( x + 1 ) {\displaystyle f_{out}(x)=pf_{in}(x)+qf_{in}(x+1)} where p , q > 0 {\displaystyle 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 ) {\displaystyle f_{out}(x)=f_{in}(x)+\alpha f_{out}(x-1)} where α > 0 {\displaystyle \alpha >0} f o u t ( x ) = f i n ( x ) + β f o u t ( x + 1 ) {\displaystyle f_{out}(x)=f_{in}(x)+\beta f_{out}(x+1)} where β > 0 {\displaystyle \beta >0} ,

• the one-sided Poisson kernel

p ( n , t ) = e − t t n n ! {\displaystyle p(n,t)=e^{-t}{\frac {t^{n}}{n!}}} for n ≥ 0 {\displaystyle n\geq 0} where t ≥ 0 {\displaystyle t\geq 0} p ( n , t ) = e − t t − n ( − n ) ! {\displaystyle p(n,t)=e^{-t}{\frac {t^{-n}}{(-n)!}}} for n ≤ 0 {\displaystyle n\leq 0} where t ≥ 0 {\displaystyle t\geq 0} .

From this classification, it is apparent that 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 ²³ 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.⁴⁵

See also

• Scale space
• Scale space implementation
• Scale-space segmentation

References

1. Lindeberg, T., "Scale-space for discrete signals," PAMI(12), No. 3, March 1990, pp. 234-254. http://www.nada.kth.se/~tony/abstracts/Lin90-PAMI.html ↩

2. Richard F. Lyon. "Speech recognition in scale space," Proc. of 1987 ICASSP. San Diego, March, pp. 29.3.14, 1987. http://www.dicklyon.com/tech/Scans/ICASSP87_ScaleSpace-Lyon.pdf ↩

3. Lindeberg, T. and Fagerstrom, F.: Scale-space with causal time direction, Proc. 4th European Conference on Computer Vision, Cambridge, England, April 1996. Springer-Verlag LNCS Vol 1064, pages 229--240. http://www.nada.kth.se/cvap/abstracts/cvap189.html ↩

4. Young, I.I., van Vliet, L.J.: Recursive implementation of the Gaussian filter, Signal Processing, vol. 44, no. 2, 1995, 139-151. http://citeseer.ist.psu.edu/young95recursive.html ↩

5. Deriche, R: Recursively implementing the Gaussian and its derivatives, INRIA Research Report 1893, 1993. http://citeseer.ist.psu.edu/deriche93recursively.html ↩