Scale-Space Diagram
A Scale-Space Diagram is a Multiscale Signal Representation that can handle image structure at different scales.
- AKA: Scale Space Representation.
- See: Scale Invariant, Multiscale Mathematics, Signal (Information Theory), Knowledge Representation, Computer Vision, Image Processing, Signal Processing, Physics, Biological Vision, Scale (Ratio), Smoothing, Convolution Kernel.
References
2020a
- (Wikipedia, 2020) ⇒ https://en.wikipedia.org/wiki/Scale_space Retrieved:2020-1-21.
- 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[1][2][3][4][5][6][7] The parameter [math]\displaystyle{ t }[/math] in this family is referred to as the scale parameter, with the interpretation that image structures of spatial size smaller than about [math]\displaystyle{ \sqrt{t} }[/math] have largely been smoothed away in the scale-space level at scale [math]\displaystyle{ t }[/math] .
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 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.[8][9]
- 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[1][2][3][4][5][6][7] The parameter [math]\displaystyle{ t }[/math] in this family is referred to as the scale parameter, with the interpretation that image structures of spatial size smaller than about [math]\displaystyle{ \sqrt{t} }[/math] have largely been smoothed away in the scale-space level at scale [math]\displaystyle{ t }[/math] .
- ↑ Witkin, A. P. “Scale-space filtering", Proc. 8th Int. Joint Conf. Art. Intell., Karlsruhe, Germany,1019–1022, 1983.
- ↑ Koenderink, Jan "The structure of images", Biological Cybernetics, 50:363–370, 1984
- ↑ Lindeberg, T., Scale-Space Theory in Computer Vision, Kluwer Academic Publishers, 1994, ISBN 0-7923-9418-6
- ↑ T. Lindeberg (1994). “Scale-space theory: A basic tool for analysing structures at different scales". Journal of Applied Statistics (Supplement on Advances in Applied Statistics: Statistics and Images: 2). 21 (2). pp. 224–270. doi:10.1080/757582976.
- ↑ Florack, Luc, Image Structure, Kluwer Academic Publishers, 1997.
- ↑ Sporring, Jon et al. (Eds), Gaussian Scale-Space Theory, Kluwer Academic Publishers, 1997.
- ↑ ter Haar Romeny, Bart M. (2008). Front-End Vision and Multi-Scale Image Analysis: Multi-scale Computer Vision Theory and Applications, written in Mathematica. Springer Science & Business Media. ISBN 978-1-4020-8840-7.
- ↑ Lindeberg, Tony (2008). “Scale-space". Encyclopedia of Computer Science and Engineering (Benjamin Wah, Ed), John Wiley and Sons. IV: 2495–2504. doi:10.1002/9780470050118.ecse609. ISBN 978-0470050118.
- ↑ T. Lindeberg (2014) "Scale selection", Computer Vision: A Reference Guide, (K. Ikeuchi, Editor), Springer, pages 701–713.
2020b
- (CS UMD, 2020) ⇒ https://www.cs.umd.edu/hcil/pad++/papers/jvlc-96-pad/zoom-6.html Retrieved:2020-1-21.
- QUOTE: In an effort to understand multiscale spaces better, we have developed an analytical tool for describing them which we call space-scale diagrams. By representing the spatial structure of an information world at all its different magnifications simultaneously, these diagrams allow us to visualize various aspects of zoomable interfaces and analyze their properties(...)
The basic one-dimensional diagram concept is illustrated in Figure 9. This diagram shows six points that are copied over and over at all possible magnifications. These copies are stacked up systematically to create a two dimensional diagram whose horizontal axes represents the original spatial dimension and whose vertical axis represents the degree of magnification (or scale). Because the diagram shows an infinite number of magnifications, each point is represented by a line emanating from the origin. We call these lines great rays. In the 2-D analog, whole 2D pictures would be stacked up at all magnifications, forming a 3D space-scale diagram, with points still becoming great rays and 2D regions becoming cones.
- QUOTE: In an effort to understand multiscale spaces better, we have developed an analytical tool for describing them which we call space-scale diagrams. By representing the spatial structure of an information world at all its different magnifications simultaneously, these diagrams allow us to visualize various aspects of zoomable interfaces and analyze their properties(...)
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2013
- (Marvel et al., 2013) ⇒ K. Marvel, D. Ivanova, and K. E. Taylor (2013). "Scale Space Methods For Climate Model Analysis". Journal of Geophysical Research: Atmospheres, 118(11), 5082-5097.
- QUOTE: Evolving data forward in time using the diffusion equation allows us to construct a continuous hierarchy of representations of the data at different spatial scales parameterized by the length scale $\sigma$. This “scale space” representation is widely used in image processing because the features of the diffusion equation give rise to several useful properties unique to the Gaussian kernel (Babaud et al., 1986).
1995
- (Furnas & Bederson, 1995) ⇒ George W. Furnas, and Benjamin B. Bederson (1995, May). "Space-Scale Diagrams: Understanding Multiscale Interfaces". In: Proceedings of the SIGCHI Conference on Human Factors in Computing Systems.
- QUOTE: The basic idea of a space-scale diagram is quite simple. Consider, for example, a square 2D picture (Figure 1a). The space-scale diagram for this picture would be obtained by creating many copies of the original 2-D picture, one at each possible magnification, and stacking them up to form an inverted pyramid (Figure 1b). While the horizontal axes represent the original spatial dimensions, the vertical axis represents scale, i.e. the magnification of the picture at that level. In theory, this representation is continuous and infinite: all magnifications appear from 0 to infinity, and the “picture” may be a whole 2D plane if needed.
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1986
- (Babaud et al., 1986) ⇒ Jean Babaud, Andrew P. Witkin, Michel Baudin, and Richard O. Duda (1986). “Uniqueness of the Gaussian Kernel for Scale-Space Filtering". IEEE transactions on pattern analysis and machine intelligence, (1), 26-33. DOI: 10.1109/TPAMI.1986.4767749