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| {{For|other features called ridges see|Ridge (disambiguation)}}
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| {{Technical|date=September 2012}}
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| The attempt (via software) to locate ridges (or edges) in an image.
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| In [[mathematic]]s and [[computer vision]], the '''ridges''' (or the '''ridge set''') of a [[smooth function]] of two variables are a set of curves whose points are, in one or more ways to be made precise below, [[local maxima]] of the function in at least one dimension. This notion captures the intuition of geographical [[ridge]]s. For a function of <math>N</math> variables, its ridges are a set of curves whose points are local maxima in <math>N-1</math> dimensions. In this respect, the notion of ridge points extends the concept of a [[maxima and minima|local maximum]]. Correspondingly, the notion of '''valleys''' for a function can be defined by replacing the condition of a local maximum with the condition of a [[local minimum]]. The union of ridge sets and valley sets, together with a related set of points called the '''connector set''' form a connected set of curves that partition, intersect, or meet at the critical points of the function. This union of sets together is called the function's '''relative critical set'''.<ref name=Damon99>{{cite journal |author=Damon, J. |title=Properties of Ridges and Cores in Two-Dimensional Images |journal=J Math Imaging Vis |volume=10 |issue=2 |doi=10.1023/A:1008379107611 |url=http://www.springerlink.com/content/l6564431145n1880/ |pages=163–174 |date=March 1999 }}</ref><ref name=Miller98>Miller, J. ''Relative Critical Sets in <math>\mathbb{R}^n</math> and Applications to Image Analysis.'' Ph.D. Dissertation. University of North Carolina. 1998.</ref>
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| Ridge sets, valley sets, and relative critical sets represent important geometric information intrinsic to a function. In a way, they provide a compact representation of important features of the function, but the extent to which they can be used to determine global features of the function is an open question. The primary motivation for the creation of '''ridge detection''' and '''valley detection''' procedures has come from [[image analysis]] and [[computer vision]] and is to capture the interior of elongated objects in the image domain. Ridge-related representations in terms of [[Watershed (algorithm)|watershed]]s have been used for [[image segmentation]]. There have also been attempts to capture the shapes of objects by graph-based representations that reflect ridges, valleys and critical points in the image domain. Such representations may, however, be highly noise sensitive if computed at a single scale only. Because scale-space theoretic computations involve convolution with the Gaussian (smoothing) kernel, it has been hoped that use of multi-scale ridges, valleys and critical points in the context of [[scale space]] theory should allow for more a robust representation of objects (or shapes) in the image.
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| In this respect, ridges and valleys can be seen as a complement to natural [[interest point detection|interest points]] or local extremal points. With appropriately defined concepts, ridges and valleys in the [[intensity landscape]] (or in some other representation derived from the intensity landscape) may form a [[scale invariant]] [[topological skeleton|skeleton]] for organizing spatial constraints on local appearance, with a number of qualitative similarities to the way the Blum's [[medial axis|medial axis transform]] provides a [[shape skeleton]] for [[binary image]]s. In typical applications, ridge and valley descriptors are often used for detecting roads in [[aerial image]]s and for detecting [[blood vessel]]s in [[retinal image]]s or three-dimensional [[magnetic resonance imaging|magnetic resonance images]].
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| == Differential geometric definition of ridges and valleys at a fixed scale in a two-dimensional image ==
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| Let <math>f(x, y)</math> denote a two-dimensional function, and let <math>L</math> be the [[scale space|scale-space representation]] of <math>f(x, y)</math> obtained by convolving <math>f(x, y)</math> with a Gaussian function
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| :<math>g(x, y, t) = \frac{1}{2 \pi t} e^{-(x^2+y^2)/2t}</math>.
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| Furthermore, let <math>L_{pp}</math> and <math>L_{qq}</math> denote the [[eigenvalue]]s of the [[Hessian matrix]]
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| :<math>H = \begin{bmatrix}
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| L_{xx} & L_{xy} \\
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| L_{xy} & L_{yy}
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| \end{bmatrix}</math>
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| of the [[scale space|scale-space representation]] <math>L</math>. With a coordinate transformation (a rotation) applied to local directional derivative operators,
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| :<math>\partial_p = \sin \beta \partial_x - \cos \beta \partial_y, \partial_q = \cos \beta \partial_x + \sin \beta \partial_y </math>
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| where p and q are coordinates of the rotated coordinate system.
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| It can be shown that the mixed derivative <math>L_{pq}</math> in the transformed coordinate system is zero if we choose
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| :<math>\cos \beta = \sqrt{\frac{1}{2} \left( 1 + \frac{L_{xx}-L_{yy}}{\sqrt{(L_{xx}-L_{yy})^2 + 4 L_{xy}^2}} \right)}</math>,<math> \sin \beta = \sgn(Lxy) \sqrt{\frac{1}{2} \left( 1 - \frac{L_{xx}-L_{yy}}{\sqrt{(L_{xx}-L_{yy})^2 + 4 L_{xy}^2}} \right)} </math>.
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| Then, a formal differential geometric definition of the ridges of <math>f(x, y)</math> at a fixed scale <math>t</math> can be expressed as the set of points that satisfy
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| <ref>{{cite journal
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| | author=T. Lindeberg
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| | title=Scale-space
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| | journal=Encyclopedia of Computer Science and Engineering (Benjamin Wah, ed), John Wiley and Sons
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| | volume = IV
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| | pages = 2495–2504
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| | year = 2008/2009
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| | doi=10.1002/9780470050118.ecse609
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| | url = http://www.nada.kth.se/~tony/abstracts/Lin08-EncCompSci.html
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| }}</ref>
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| :<math>L_{p} = 0, L_{pp} \leq 0, |L_{pp}| \geq |L_{qq}|.</math>
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| Correspondingly, the valleys of <math>f(x, y)</math> at scale <math>t</math> are the set of points
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| :<math>L_{q} = 0, L_{qq} \geq 0, |L_{qq}| \geq |L_{pp}|.</math>
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| In terms of a <math>(u, v)</math> coordinate system with the <math>v</math> direction parallel to the image gradient
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| :<math>\partial_u = \sin \alpha \partial_x - \cos \alpha \partial_y, \partial_v = \cos \alpha \partial_x + \sin \alpha \partial_y </math>
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| where
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| :<math>\cos \alpha = \frac{L_x}{\sqrt{L_x^2 + L_y^2}}, \sin \alpha = \frac{L_y}{\sqrt{L_x^2 + L_y^2}} </math>
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| it can be shown that this ridge and valley definition can instead be equivalently<ref>{{cite journal |author=Lindeberg, T |title=Scale-space theory: A basic tool for analysing structures at different scales |journal=J. of Applied Statistics |volume=21 |issue=2 |pages=224–270 |year=1994 |url=http://www.nada.kth.se/~tony/abstracts/Lin94-SI-abstract.html}}</ref> be written as
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| :<math> L_{uv} = 0, L_{uu}^2 - L_{vv}^2 \geq 0 </math>
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| where
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| :<math>L_v^2 L_{uu} = L_x^2 L_{yy} - 2 L_x L_y L_{xy} + L_y^2 L_{xx},</math>
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| :<math>L_v^2 L_{uv} = L_x L_y (L_{xx} - L_{yy}) - (L_x^2 - L_y^2) L_{xy}, </math>
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| :<math>L_v^2 L_{vv} = L_x^2 L_{xx} + 2 L_x L_y L_{xy} + L_y^2 L_{yy} </math>
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| and the sign of <math>L_{uu}</math> determines the polarity; <math>L_{uu}<0</math> for ridges and <math>L_{uu}>0</math> for valleys.
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| == Computation of variable scale ridges from two-dimensional images ==
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| A main problem with the fixed scale ridge definition presented above is that it can be very sensitive to the choice of the scale level. Experiments show that the scale parameter of the Gaussian pre-smoothing kernel must be carefully tuned to the width of the ridge structure in the image domain, in order for the ridge detector to produce a connected curve reflecting the underlying image structures. To handle this problem in the absence of prior information, the notion of ''scale-space ridges'' has been introduced, which treats the scale parameter as an inherent property of the ridge definition and allows the scale levels to vary along a scale-space ridge. Moreover, the concept of a scale-space ridge also allows the scale parameter to be automatically tuned to the width of the ridge structures in the image domain, in fact as a consequence of a well-stated definition. In the literature, a number of different approaches have been proposed based on this idea.
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| Let <math>R(x, y, t)</math> denote a measure of ridge strength (to be specified below). Then, for a two-dimensional image, a scale-space ridge is the set of points that satisfy
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| :<math>L_{p} = 0, L_{pp} \leq 0, \partial_t(R) = 0, \partial_{tt}(R) \leq 0,</math>
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| where <math>t</math> is the scale parameter in the [[scale space|scale-space representation]]. Similarly, a ''scale-space valley'' is the set of points that satisfy
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| :<math>L_{q} = 0, L_{qq} \geq 0, \partial_t(R) = 0, \partial_{tt}(R) \leq 0.</math>
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| An immediate consequence of this definition is that for a two-dimensional image the concept of scale-space ridges sweeps out a set of one-dimensional curves in the three-dimensional scale-space, where the scale parameter is allowed to vary along the scale-space ridge (or the scale-space valley). The ridge descriptor in the image domain will then be a projection of this three-dimensional curve into the two-dimensional image plane, where the attribute scale information at every ridge point can be used as a natural estimate of the width of the ridge structure in the image domain in a neighbourhood of that point.
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| In the literature, various measures of ridge strength have been proposed. When Lindeberg (1996, 1998)<ref>{{cite journal |doi=10.1023/A:1008097225773 |author=Lindeberg, T. |title=Edge detection and ridge detection with automatic scale selection |journal=International Journal of Computer Vision |volume=30 |issue=2 |pages=117–154 |year=1998 |url=http://www.nada.kth.se/cvap/abstracts/cvap191.html}} Earlier version presented at IEEE Conference on Pattern Recognition and Computer Vision, CVPR'96, San Francisco, California, pages 465–470, june 1996</ref> coined the term scale-space ridge, he considered three measures of ridge strength:
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| <ul>
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| <li>
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| The main principal curvature
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| :<math>L_{pp, \gamma-norm} = \frac{t^{\gamma}}{2} \left( L_{xx}+L_{yy} - \sqrt{(L_{xx}-L_{yy})^2 + 4 L_{xy}^2} \right)</math>
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| expressed in terms of ''<math>\gamma</math>-normalized derivatives'' with
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| :<math>\partial_{\xi} = t^{\gamma/2} \partial_x, \partial_{\eta} = t^{\gamma/2} \partial_y</math>.
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| <li>
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| The square of the <math>\gamma</math>-normalized square eigenvalue difference
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| :<math>N_{\gamma-norm} = \left( L_{pp, \gamma-norm}^2 - L_{qq, \gamma-norm}^2 \right)^2 = t^{4 \gamma} (L_{xx}+L_{yy})^2 \left( (L_{xx}-L_{yy})^2 + 4 L_{xy}^2 \right). </math>
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| <li>
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| The square of the <math>\gamma</math>-normalized eigenvalue difference
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| :<math>A_{\gamma-norm} = \left( L_{pp, \gamma-norm} - L_{qq, \gamma-norm} \right)^2 = t^{2 \gamma} \left( (L_{xx}-L_{yy})^2 + 4 L_{xy}^2 \right). </math>
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| </ul>
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| The notion of <math>\gamma</math>-normalized derivatives is essential here, since it allows the ridge and valley detector algorithms to be calibrated properly. By requiring that for a one-dimensional Gaussian ridge embedded in two (or three dimensions) the detection scale should be equal to the width of the ridge structure when measured in units of length (a requirement of a match between the size of the detection filter and the image structure it responds to), it follows that one should choose <math>\gamma = 3/4</math>. Out of these three measures of ridge strength, the first entity <math>L_{pp, \gamma-norm}</math> is a general purpose ridge strength measure with many applications such as blood vessel detection and road extraction. Nevertheless, the entity <math>A_{\gamma-norm}</math> has been used in applications such as fingerprint enhancement,<ref>{{cite journal |author=Almansa, A., Lindeberg, T. |title=Fingerprint Enhancement by Shape Adaptation of Scale-Space Operators with Automatic Scale-Selection |journal=IEEE Transactions on Image Processing |volume=9 |issue=12 |pages=2027–42 |year=2000 |url=http://www.nada.kth.se/cvap/abstracts/cvap226.html |pmid=18262941 |doi=10.1109/83.887971}}</ref> real-time hand tracking and gesture recognition<ref>[http://www.nada.kth.se/cvap/abstracts/BreLapLin-FG02.html L. Bretzner, I. Laptev and T. Lindeberg: Hand Gesture Recognition using Multi-Scale Colour Features, Hierarchical Models and Particle Filtering, Proc. IEEE Conference on Face and Gesture 2002, Washington DC, 423–428.]</ref> as well as for modelling local image statistics for detecting and tracking humans in images and video.<ref>{{cite journal |author=Sidenbladh, H., Black, M. |title=Learning the statistics of people in images and video |journal=International Journal of Computer Vision |volume=54 |issue=1–2 |pages=183–209 |year=2003 |url=http://www.csc.kth.se/~hedvig/publications/ijcv_03.pdf |format=PDF}}</ref>
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| There are also other closely related ridge definitions that make use of normalized derivatives with the implicit assumption of <math>\gamma = 1</math>.<ref>J. Furst and J. Miller, "The Maximal Scale Ridge: Incorporating Scale in the Ridge Definition", ''Scale Space Theory in Computer Vision: Proceedings of the First International Conference on, Scale Space '97,'' pp. 93–104. Springer Lecture Notes in Computer Science, vol. 1682.</ref> ''Develop these approaches in further detail.'' When detecting ridges with <math>\gamma = 1</math>, however, the detection scale will be twice as large as for <math>\gamma = 3/4</math>, resulting in more shape distortions and a lower ability to capture ridges and valleys with nearby interfering image structures in the image domain.
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| == History ==
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| The notion of ridges and valleys in digital images was introduced by [[Haralick]] in 1983<ref>{{cite journal |doi=10.1016/0734-189X(83)90094-4 |author=Haralick, R. |title=Ridges and Valleys on Digital Images |journal=Computer Vision, Graphics, and Image Processing |volume=22 |issue=10 |pages=28–38 |date=April 1983 }}</ref> and by Crowley concerning [[difference of Gaussians]] [[pyramid (image processing)|pyramid]]s in 1984.<ref>{{cite journal |doi=10.1109/TPAMI.1984.4767500 |author=Crowley, J.L., Parker, A.C. |title=A Representation for Shape Based on Peaks and Ridges in the Difference of Low Pass Transform |journal=IEEE Trans Pattern Anal Mach Intell |volume=6 |issue=2 |pages=156–170 |date=March 1984 |url=http://www-prima.inrialpes.fr/Prima/Homepages/jlc/papers/Crowley-Parker84.pdf |format=PDF |pmid=21869180}}</ref><ref>{{cite journal |doi=10.1109/TPAMI.1987.4767876 |author=Crowley, J.L., Sanderson, A. |title=Multiple Resolution Representation and Probabilistic Matching of 2-D Gray-Scale Shape |journal=IEEE Trans Pattern Anal Mach Intell |volume=9 |issue=1 |pages=113–121 |date=January 1987 |url=http://www-prima.inrialpes.fr/Prima/Homepages/jlc/papers/Crowley-Sanderson-PAMI87.pdf |format=PDF}}</ref> The application of ridge descriptors to medical image analysis has been extensively studied by Pizer and his co-workers<ref>{{cite journal |doi=10.1109/34.216734 |author=Gauch, J.M., Pizer, S.M. |title=Multiresolution Analysis of Ridges and Valleys in Grey-Scale Images |journal=IEEE Trans Pattern Anal Mach Intell |volume=15 |issue=6 |pages=635–646 |date=June 1993 }}</ref><ref>{{cite journal |doi=10.1007/BF01262402 |author=Eberly D., Gardner R., Morse B., Pizer S., Scharlach C. |title=Ridges for image analysis |journal=Journal of Mathematical Imaging and Vision |volume=4 |issue=4 |pages=353–373 |date=December 1994 }}</ref><ref>{{cite journal |doi=10.1006/cviu.1997.0563 |author=Pizer, Stephen M., Eberly, David, Fritsch, Daniel S. |title=Zoom-invariant vision of figural shape: the mathematics of cores |journal=Computer Vision and Image Understanding |volume=69 |issue=1 |pages=55–71 |date=January 1998 }}</ref> resulting in their notion of M-reps.<ref>[http://citeseer.ist.psu.edu/pizer01segmentation.html S. Pizer, S. Joshi, T. Fletcher, M. Styner, G. Tracton, J. Chen (2001) "Segmentation of Single-Figure Objects by Deformable M-reps", Proceedings of the 4th International Conference on Medical Image Computing and Computer-Assisted Intervention, Springer Lecture Notes In Computer Science; Vol. 2208, pp. 862–871]</ref> Ridge detection has also been furthered by Lindeberg with the introduction of <math>\gamma</math>-normalized derivatives and scale-space ridges defined from local maximization of the appropriately normalized main principal curvature of the Hessian matrix (or other measures of ridge strength) over space and over scale. These notions have later been developed with application to road extraction by Steger et al.<ref>{{cite journal |author=Steger C. |title=An unbiased detector of curvilinear structures |journal=IEEE Trans Pattern Anal Mach Intell |volume=20 |issue=2 |pages=113–125 |year=1998 |doi=10.1109/34.659930 |url=http://csdl2.computer.org/persagen/DLAbsToc.jsp?resourcePath=/dl/trans/tp/&toc=comp/trans/tp/1998/02/i2toc.xml&DOI=10.1109/34.659930}}</ref><ref>{{cite journal |doi=10.1007/s001380050121 |author=Laptev I., Mayer H., Lindeberg T., Eckstein W., Steger C., Baumgartner A. |title=Automatic extraction of roads from aerial images based on scale-space and snakes |journal=Machine Vision and Applications |volume=12 |issue=1 |pages=23 |year=2000 |url=http://www.irisa.fr/vista/Papers/2000_mva_laptev.pdf |format=PDF}}</ref> and to blood vessel segmentation by Frangi et al.<ref>{{cite journal |author=Frangi AF, Niessen WJ, Hoogeveen RM, van Walsum T, Viergever MA |title=Model-based quantitation of 3-D magnetic resonance angiographic images |journal=IEEE Trans Med Imaging |volume=18 |issue=10 |pages=946–56 |date=October 1999 |pmid=10628954 |doi=10.1109/42.811279 }}</ref> as well as to the detection of curvilinear and tubular structures by Sato et al.<ref>{{cite journal |author=Sato Y, Nakajima S, Shiraga N, Atsumi H, Yoshida S, ''et al'' |title=Three-dimensional multi-scale line filter for segmentation and visualization of curvilinear structures in medical images |journal=Medical Image Analysis |volume=2 |issue=2 |pages=143–168 |year=1998 |url=http://www.image.med.osaka-u.ac.jp/member/yoshi/paper/linefilter.pdf}}</ref> and Krissian et al.<ref>{{cite journal |doi=10.1006/cviu.2000.0866 |author=Krissian K., Malandain G., Ayache N., Vaillan R., Trousset Y. |title=Model-based detection of tubular structures in 3D images |journal=Computer Vision and Image Understanding |volume=80 |issue=2 |pages=130–171 |year=2000 |url=http://portal.acm.org/citation.cfm?id=363419.363423}}</ref> A review of several of the classical ridge definitions at a fixed scale including relations between them has been given by Koenderink and van Doorn.<ref>{{cite journal |doi=10.1016/0167-8655(94)90134-1 |author=Koenderink, Jan J., van Doorn, Andrea J. |title=2+1-D differential geometry |journal=Pattern. Recognition Letters |volume=15 |pages=439–443 |date=May 1994 }}</ref> A review of vessel extraction techniques has been presented by Kirbas and Quek.<ref>{{cite journal |doi=10.1145/1031120.1031121 |author=Kirbas C, Quek F |title=A review of vessel extraction techniques and algorithms |journal=ACM Computing Surveys |volume=36 |issue=2 |pages=81–121 |year=2004 |url=ftp://cmp.felk.cvut.cz/pub/cmp/articles/matas/ps/Curvilinear/blood-vessel_review.pdf |format=PDF}}</ref>
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| ==Definition of ridges and valleys in N dimensions==
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| In its broadest sense, the notion of ridge generalizes the idea of a local maximum of a real-valued function. A point <math>\mathbf{x}_0</math> in the domain of a function <math>f:\mathbb{R}^n \rightarrow \mathbb{R}</math>is a local maximum of the function if there is a distance <math>\delta>0</math> with the property that if <math>\mathbf{x}</math> is within <math>\delta</math> units of <math>\mathbf{x}_0</math>, then <math>f(\mathbf{x}) < f(\mathbf{x}_0)</math>. It is well known that critical points, of which local maxima are just one type, are isolated points in a function's domain in all but the most unusual situations (''i.e.'', the nongeneric cases).
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| Consider relaxing the condition that <math>f(\mathbf{x}) < f(\mathbf{x}_0)</math> for <math>\mathbf{x}</math> in an entire neighborhood of <math>\mathbf{x}_0</math> slightly to require only that this hold on an <math>n-1</math> dimensional subset. Presumably this relaxation allows the set of points which satisfy the criteria, which we will call the ridge, to have a single degree of freedom, at least in the generic case. This means that the set of ridge points will form a 1-dimensional locus, or a ridge curve. Notice that the above can be modified to generalize the idea to local minima and result in what might call 1-dimensional valley curves.
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| This following ridge definition follows the book by Eberly<ref>{{cite book |author=Eberly, D. |title=Ridges in Image and Data Analysis |publisher=Kluwer |year=1996 |isbn=0-7923-4268-2 }}</ref> and can be seen as a generalization of some of the abovementioned ridge definitions. Let <math>U \subset \mathbb{R}^n</math> be an open set, and <math>f:U \rightarrow \mathbb{R}</math> be smooth. Let <math>\mathbf{x}_0 \in U</math>. Let <math>\nabla_{\mathbf{x}_0}f</math> be the gradient of <math>f</math> at <math>\mathbf{x}_0</math>, and let <math>H_{\mathbf{x}_0}(f)</math> be the <math>n \times n</math> Hessian matrix of <math>f</math>at <math>\mathbf{x}_0</math>. Let <math>\lambda_1 \leq \lambda_2 \leq \cdots \leq \lambda_n</math> be the <math>n</math> ordered eigenvalues of <math>H_{\mathbf{x}_0}(f)</math> and let <math>\mathbf{e}_i</math> be a unit eigenvector in the eigenspace for <math>\lambda_i</math>. (For this, one should assume that all the eigenvalues are distinct.)
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| The point <math>\mathbf{x}_0</math> is a point on the 1-dimensional ridge of <math>f</math> if the following conditions hold:
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| <ol>
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| <li><math>\lambda_{n-1}<0</math>, and
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| <li><math>\nabla_{\mathbf{x}_0} f \cdot \mathbf{e}_i=0</math> for <math>i=1, 2, \ldots, n-1</math>.
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| </ol>
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| This makes precise the concept that <math>f</math> restricted to ''this particular'' <math>n-1</math>-dimensional subspace has a local maxima at <math>\mathbf{x}_0</math>.
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| This definition naturally generalizes to the ''k''-dimensional ridge as follows: the point <math>\mathbf{x}_0</math> is a point on the ''k''-dimensional ridge of <math>f</math> if the following conditions hold:
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| <ol>
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| <li><math>\lambda_{n-k}<0</math>, and
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| <li><math>\nabla_{\mathbf{x}_0} f \cdot \mathbf{e}_i=0</math> for <math>i=1, 2, \ldots, n-k</math>.
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| </ol>
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| In many ways, these definitions naturally generalize that of a local maximum of a function. Properties of maximal convexity ridges are put on a solid mathematical footing by Damon<ref name=Damon99/> and Miller.<ref name=Miller98/> Their properties in one-parameter families was established by Keller.<ref>Kerrel, R. ''Generic Transitions of Relative Critical Sets in Parameterized Families with Applications to Image Analysis.'' University of North Carolina. 1999.</ref>
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| == Maximal Scale Ridge ==
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| The following definition can be traced to Fritsch<ref>Fritsch, DS, Eberly,D., Pizer, SM, and McAuliffe, MJ. "Stimulated cores and
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| their applications in medical imaging." Information Processing in Medical Imaging, Y. Bizais, C Barillot, R DiPaola, eds., Kluwer Series in Computational Imaging and Vision, pp. 365–368.</ref> who was interested in extracting geometric information about figures in two dimensional greyscale images. Fritsch filtered his image with a "medialness" filter that gave him information analogous to "distant to the boundary" data in scale-space. Ridges of this image, once projected to the original image, were to be analogous to a shape skeleton (''e.g.'', the Blum Medial Axis) of the original image.
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| What follows is a definition for the maximal scale ridge of a function of three variables, one of which is a "scale" parameter. One thing that we want to be true in this definition is, if <math>(\mathbf{x},\sigma)</math> is a point on this ridge, then the value of the function at the point is maximal in the scale dimension. Let <math>f(\mathbf{x},\sigma)</math> be a smooth differentiable function on <math>U \subset \mathbb{R}^2 \times \mathbb{R}_{+}</math>. The <math>(\mathbf{x},\sigma)</math> is a point on the maximal scale ridge if and only if
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| <ol>
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| <li> <math>\frac{\partial f}{\partial \sigma}=0</math> and <math>\frac{\partial^2 f}{\partial \sigma^2}<0</math>, and
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| <li><math>\nabla f \cdot \mathbf{e}_1=0</math> and <math>\mathbf{v}_1^t H(f) \mathbf{v}_1 <0</math>.
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| </ol>
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| == Relations between edge detection and ridge detection ==
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| The purpose of ridge detection is usually to capture the major axis of symmetry of an elongated object,{{Citation needed|date=July 2008}} whereas the purpose of [[edge detection]] is usually to capture the boundary of the object. However, some literature on edge detection erroneously{{Citation needed|date=July 2008}} includes the notion of ridges into the concept of edges, which confuses the situation.
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| In terms of definitions, there is a close connection between edge detectors and ridge detectors. With the formulation of non-maximum as given by Canny,<ref>{{cite journal |doi=10.1109/TPAMI.1986.4767851 |author=Canny J. |title=A computational approach to edge detection |journal=IEEE Trans Pattern Anal Mach Intell |volume=8 |issue=6 |pages=679–698 |year=1986 |url=http://portal.acm.org/citation.cfm?id=11275}}</ref> it holds that edges are defined as the points where the gradient magnitude assumes a local maximum in the gradient direction. Following a differential geometric way of expressing this definition,<ref>{{cite journal |doi=10.1007/BF01664794 |author=Lindeberg T. |title=Discrete Derivative Approximations with Scale-Space Properties: A Basis for Low-Level Feature Extraction |journal=J. of Mathematical Imaging and Vision |volume=3 |issue=4 |pages=349–376 |year=1993 |url=http://www.nada.kth.se/~tony/abstracts/Lin93-JMIV.html}}</ref> we can in the above-mentioned <math>(u, v)</math>-coordinate system state that the gradient magnitude of the scale-space representation, which is equal to the first-order directional derivative in the <math>v</math>-direction <math>L_v</math>, should have its first order directional derivative in the <math>v</math>-direction equal to zero
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| :<math>\partial_v(L_v) = 0</math>
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| while the second-order directional derivative in the <math>v</math>-direction of <math>L_v</math> should be negative, i.e.,
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| :<math>\partial_{vv}(L_v) \leq 0</math>.
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| Written out as an explicit expression in terms of local partial derivatives <math>L_x</math>, <math>L_y</math> ... <math>L_{yyy}</math>, this edge definition can be expressed as the zero-crossing curves of the differential invariant
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| :<math>L_v^2 L_{vv} = L_x^2 \, L_{xx} + 2 \, L_x \, L_y \, L_{xy} + L_y^2 \, L_{yy} = 0,</math>
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| that satisfy a sign-condition on the following differential invariant
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| :<math>L_v^3 L_{vvv} = L_x^3 \, L_{xxx} + 3 \, L_x^2 \, L_y \, L_{xxy} + 3 \, L_x \, L_y^2 \, L_{xyy} + L_y^3 \, L_{yyy} \leq 0</math>
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| (see the article on [[edge detection]] for more information). Notably, the edges obtained in this way are the ridges of the gradient magnitude.
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| == See also ==
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| *[[Scale space]]
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| *[[Edge detection]]
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| *[[Interest point detection]]
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| *[[Blob detection]]
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| *[[Computer vision]]
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| ==References==
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| <references/>
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| ==External links==
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| *[http://www.letteraturaoperaomnia.org/thesis/thesis_italy_XX_ciclo_andrea_anzalone.html Multiscale analysis for optimized vessel segmentation of fundus retina images] Ph.D Thesis
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| [[Category:Feature detection]]
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| [[Category:Multivariable calculus]]
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| [[Category:Smooth functions]]
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| [[Category:Singularity theory]]
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