Ridge detection

 ( Feature Detection ) 

In image processing, ridge detection is the attempt, via software, to locate ridges in an image, defined as curves whose points are local maxima of the function, akin to geographical .

For a function of N variables, its ridges are a set of curves whose points are local maxima in N − 1 dimensions. In this respect, the notion of ridge points extends the concept of a 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.Miller, J. Relative Critical Sets in $\backslash mathbb\{R\}^n$ and Applications to Image Analysis. Ph.D. Dissertation. University of North Carolina. 1998.

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 watersheds 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.

In this respect, ridges and valleys can be seen as a complement to natural 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 skeleton for organizing spatial constraints on local appearance, with a number of qualitative similarities to the way the Blum's medial axis provides a shape skeleton for . In typical applications, ridge and valley descriptors are often used for detecting roads in aerial images and for detecting in retinal images or three-dimensional magnetic resonance images.

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Differential geometric definition of ridges and valleys at a fixed scale in a two-dimensional image Let $f(x,\; y)$ denote a two-dimensional function, and let $L$ be the scale space of $f(x,\; y)$ obtained by convolving $f(x,\; y)$ with a Gaussian function

$g(x,\; y,\; t)\; =\; \backslash frac\{1\}\{2\; \backslash pi\; t\}\; e^\{-(x^2+y^2)/2t\}$.

Furthermore, let $L\_\{pp\}$ and $L\_\{qq\}$ denote the of the Hessian matrix

$H\; =\; \backslash begin\{bmatrix\}$

L_{xx} & L_{xy} \\ L_{xy} & L_{yy} \end{bmatrix} of the scale space $L$ with a coordinate transformation (a rotation) applied to local directional derivative operators,

$\backslash partial\_p\; =\; \backslash sin\; \backslash beta\; \backslash partial\_x\; -\; \backslash cos\; \backslash beta\; \backslash partial\_y,\; \backslash partial\_q\; =\; \backslash cos\; \backslash beta\; \backslash partial\_x\; +\; \backslash sin\; \backslash beta\; \backslash partial\_y$

where p and q are coordinates of the rotated coordinate system.

It can be shown that the mixed derivative $L\_\{pq\}$ in the transformed coordinate system is zero if we choose

$\backslash cos\; \backslash beta\; =\; \backslash sqrt\{\backslash frac\{1\}\{2\}\; \backslash left(\; 1\; +\; \backslash frac\{L\_\{xx\}-L\_\{yy\}\}\{\backslash sqrt\{(L\_\{xx\}-L\_\{yy\})^2\; +\; 4\; L\_\{xy\}^2\}\}\; \backslash right)\}$,$\backslash sin\; \backslash beta\; =\; \backslash sgn(L\_\{xy\})\; \backslash sqrt\{\backslash frac\{1\}\{2\}\; \backslash left(\; 1\; -\; \backslash frac\{L\_\{xx\}-L\_\{yy\}\}\{\backslash sqrt\{(L\_\{xx\}-L\_\{yy\})^2\; +\; 4\; L\_\{xy\}^2\}\}\; \backslash right)\}$.

Then, a formal differential geometric definition of the ridges of $f(x,\; y)$ at a fixed scale $t$ can be expressed as the set of points that satisfy

$L\_\{p\}\; =\; 0,\; L\_\{pp\}\; \backslash leq\; 0,\; |L\_\{pp\}|\; \backslash geq\; |L\_\{qq\}|.$

Correspondingly, the valleys of $f(x,\; y)$ at scale $t$ are the set of points

$L\_\{q\}\; =\; 0,\; L\_\{qq\}\; \backslash geq\; 0,\; |L\_\{qq\}|\; \backslash geq\; |L\_\{pp\}|.$

In terms of a $(u,\; v)$ coordinate system with the $v$ direction parallel to the image gradient

$\backslash partial\_u\; =\; \backslash sin\; \backslash alpha\; \backslash partial\_x\; -\; \backslash cos\; \backslash alpha\; \backslash partial\_y,\; \backslash partial\_v\; =\; \backslash cos\; \backslash alpha\; \backslash partial\_x\; +\; \backslash sin\; \backslash alpha\; \backslash partial\_y$

where

$\backslash cos\; \backslash alpha\; =\; \backslash frac\{L\_x\}\{\backslash sqrt\{L\_x^2\; +\; L\_y^2\}\},\; \backslash sin\; \backslash alpha\; =\; \backslash frac\{L\_y\}\{\backslash sqrt\{L\_x^2\; +\; L\_y^2\}\}$

it can be shown that this ridge and valley definition can instead be equivalently written as

$L\_\{uv\}\; =\; 0,\; L\_\{uu\}^2\; -\; L\_\{vv\}^2\; \backslash geq\; 0$

where

$L\_v^2\; L\_\{uu\}\; =\; L\_x^2\; L\_\{yy\}\; -\; 2\; L\_x\; L\_y\; L\_\{xy\}\; +\; L\_y^2\; L\_\{xx\},$

$L\_v^2\; L\_\{uv\}\; =\; L\_x\; L\_y\; (L\_\{xx\}\; -\; L\_\{yy\})\; -\; (L\_x^2\; -\; L\_y^2)\; L\_\{xy\},$

$L\_v^2\; L\_\{vv\}\; =\; L\_x^2\; L\_\{xx\}\; +\; 2\; L\_x\; L\_y\; L\_\{xy\}\; +\; L\_y^2\; L\_\{yy\}$

and the sign of $L\_\{uu\}$ determines the polarity; for ridges and $L\_\{uu\}>0$ for valleys.

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Computation of variable scale ridges from two-dimensional images 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.

Let $R(x,\; y,\; t)$ 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

$L\_\{p\}\; =\; 0,\; L\_\{pp\}\; \backslash leq\; 0,\; \backslash partial\_t(R)\; =\; 0,\; \backslash partial\_\{tt\}(R)\; \backslash leq\; 0,$

where $t$ is the scale parameter in the scale space. Similarly, a scale-space valley is the set of points that satisfy

$L\_\{q\}\; =\; 0,\; L\_\{qq\}\; \backslash geq\; 0,\; \backslash partial\_t(R)\; =\; 0,\; \backslash partial\_\{tt\}(R)\; \backslash leq\; 0.$

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.

In the literature, various measures of ridge strength have been proposed. When Lindeberg (1996, 1998) Earlier version presented at IEEE Conference on Pattern Recognition and Computer Vision, CVPR'96, San Francisco, California, pages 465–470, June 1996 coined the term scale-space ridge, he considered three measures of ridge strength:

• The main principal curvature

:$L\_\{pp,\; \backslash gamma-norm\}\; =\; \backslash frac\{t^\{\backslash gamma\}\}\{2\}\; \backslash left(\; L\_\{xx\}+L\_\{yy\}\; -\; \backslash sqrt\{(L\_\{xx\}-L\_\{yy\})^2\; +\; 4\; L\_\{xy\}^2\}\; \backslash right)$

expressed in terms of $\backslash gamma$-normalized derivatives with

:$\backslash partial\_\{\backslash xi\}\; =\; t^\{\backslash gamma/2\}\; \backslash partial\_x,\; \backslash partial\_\{\backslash eta\}\; =\; t^\{\backslash gamma/2\}\; \backslash partial\_y$.

• The square of the $\backslash gamma$-normalized square eigenvalue difference

:$N\_\{\backslash gamma-norm\}\; =\; \backslash left(\; L\_\{pp,\; \backslash gamma-norm\}^2\; -\; L\_\{qq,\; \backslash gamma-norm\}^2\; \backslash right)^2\; =\; t^\{4\; \backslash gamma\}\; (L\_\{xx\}+L\_\{yy\})^2\; \backslash left(\; (L\_\{xx\}-L\_\{yy\})^2\; +\; 4\; L\_\{xy\}^2\; \backslash right).$

• The square of the $\backslash gamma$-normalized eigenvalue difference

:$A\_\{\backslash gamma-norm\}\; =\; \backslash left(\; L\_\{pp,\; \backslash gamma-norm\}\; -\; L\_\{qq,\; \backslash gamma-norm\}\; \backslash right)^2\; =\; t^\{2\; \backslash gamma\}\; \backslash left(\; (L\_\{xx\}-L\_\{yy\})^2\; +\; 4\; L\_\{xy\}^2\; \backslash right).$

The notion of $\backslash gamma$-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 $\backslash gamma\; =\; 3/4$. Out of these three measures of ridge strength, the first entity $L\_\{pp,\; \backslash gamma-norm\}$ is a general purpose ridge strength measure with many applications such as blood vessel detection and road extraction. Nevertheless, the entity $A\_\{\backslash gamma-norm\}$ has been used in applications such as fingerprint enhancement, real-time hand tracking and gesture recognition 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. as well as for modelling local image statistics for detecting and tracking humans in images and video.

There are also other closely related ridge definitions that make use of normalized derivatives with the implicit assumption of $\backslash gamma\; =\; 1$.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. Develop these approaches in further detail. When detecting ridges with $\backslash gamma\; =\; 1$, however, the detection scale will be twice as large as for $\backslash gamma\; =\; 3/4$, 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 The notion of ridges and valleys in digital images was introduced by Haralick in 1983 and by Crowley concerning difference of Gaussians pyramids in 1984. The application of ridge descriptors to medical image analysis has been extensively studied by Pizer and his co-workers resulting in their notion of M-reps. 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 Ridge detection has also been furthered by Lindeberg with the introduction of $\backslash gamma$-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. and to blood vessel segmentation by Frangi et al. as well as to the detection of curvilinear and tubular structures by Sato et al. and Krissian et al. 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. A review of vessel extraction techniques has been presented by Kirbas and Quek.

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Definition of ridges and valleys in N dimensions In its broadest sense, the notion of ridge generalizes the idea of a local maximum of a real-valued function. A point $\backslash mathbf\{x\}\_0$ in the domain of a function $f:\backslash mathbb\{R\}^n\; \backslash rightarrow\; \backslash mathbb\{R\}$ is a local maximum of the function if there is a distance $\backslash delta>0$ with the property that if $\backslash mathbf\{x\}$ is within $\backslash delta$ units of $\backslash mathbf\{x\}\_0$, then . 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).

Consider relaxing the condition that for $\backslash mathbf\{x\}$ in an entire neighborhood of $\backslash mathbf\{x\}\_0$ slightly to require only that this hold on an $n-1$ 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.

This following ridge definition follows the book by Eberly

and can be seen as a generalization of some of the abovementioned ridge definitions. Let $U\; \backslash subset\; \backslash mathbb\{R\}^n$ be an open set, and $f:U\; \backslash rightarrow\; \backslash mathbb\{R\}$ be smooth. Let $\backslash mathbf\{x\}\_0\; \backslash in\; U$. Let $\backslash nabla\_\{\backslash mathbf\{x\}\_0\}f$ be the gradient of $f$ at $\backslash mathbf\{x\}\_0$, and let $H\_\{\backslash mathbf\{x\}\_0\}(f)$ be the $n\; \backslash times\; n$ Hessian matrix of $f$ at $\backslash mathbf\{x\}\_0$. Let $\backslash lambda\_1\; \backslash leq\; \backslash lambda\_2\; \backslash leq\; \backslash cdots\; \backslash leq\; \backslash lambda\_n$ be the $n$ ordered eigenvalues of $H\_\{\backslash mathbf\{x\}\_0\}(f)$ and let $\backslash mathbf\{e\}\_i$ be a unit eigenvector in the eigenspace for $\backslash lambda\_i$. (For this, one should assume that all the eigenvalues are distinct.)

The point $\backslash mathbf\{x\}\_0$ is a point on the 1-dimensional ridge of $f$ if the following conditions hold:

1. , and
2. $\backslash nabla\_\{\backslash mathbf\{x\}\_0\}\; f\; \backslash cdot\; \backslash mathbf\{e\}\_i=0$ for $i=1,\; 2,\; \backslash ldots,\; n-1$.

This makes precise the concept that $f$ restricted to this particular $n-1$-dimensional subspace has a local maximum at $\backslash mathbf\{x\}\_0$.

This definition naturally generalizes to the k-dimensional ridge as follows: the point $\backslash mathbf\{x\}\_0$ is a point on the k-dimensional ridge of $f$ if the following conditions hold:

1. , and
2. $\backslash nabla\_\{\backslash mathbf\{x\}\_0\}\; f\; \backslash cdot\; \backslash mathbf\{e\}\_i=0$ for $i=1,\; 2,\; \backslash ldots,\; n-k$.

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 and Miller. Their properties in one-parameter families was established by Keller.Kerrel, R. Generic Transitions of Relative Critical Sets in Parameterized Families with Applications to Image Analysis. University of North Carolina. 1999.

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Maximal scale ridge The following definition can be traced to FritschFritsch, DS, Eberly, D., Pizer, SM, and McAuliffe, MJ. "Stimulated cores and 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. 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 Medial axis) of the original image.

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 $(\backslash mathbf\{x\},\backslash sigma)$ is a point on this ridge, then the value of the function at the point is maximal in the scale dimension. Let $f(\backslash mathbf\{x\},\backslash sigma)$ be a smooth differentiable function on $U\; \backslash subset\; \backslash mathbb\{R\}^2\; \backslash times\; \backslash mathbb\{R\}\_\{+\}$. The $(\backslash mathbf\{x\},\backslash sigma)$ is a point on the maximal scale ridge if and only if

1. $\backslash frac\{\backslash partial\; f\}\{\backslash partial\; \backslash sigma\}=0$ and , and
2. $\backslash nabla\; f\; \backslash cdot\; \backslash mathbf\{e\}\_1=0$ and .

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Relations between edge detection and ridge detection The purpose of ridge detection is usually to capture the major axis of symmetry of an elongated object, whereas the purpose of edge detection is usually to capture the boundary of the object. However, some literature on edge detection erroneously includes the notion of ridges into the concept of edges, which confuses the situation.

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, 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, we can in the above-mentioned $(u,\; v)$-coordinate system state that the gradient magnitude of the scale-space representation, which is equal to the first-order directional derivative in the $v$-direction $L\_v$, should have its first order directional derivative in the $v$-direction equal to zero

$\backslash partial\_v(L\_v)\; =\; 0$

while the second-order directional derivative in the $v$-direction of $L\_v$ should be negative, i.e.,

$\backslash partial\_\{vv\}(L\_v)\; \backslash leq\; 0$.

Written out as an explicit expression in terms of local partial derivatives $L\_x$, $L\_y$ ... $L\_\{yyy\}$, this edge definition can be expressed as the zero-crossing curves of the differential invariant

$L\_v^2\; L\_\{vv\}\; =\; L\_x^2\; \backslash ,\; L\_\{xx\}\; +\; 2\; \backslash ,\; L\_x\; \backslash ,\; L\_y\; \backslash ,\; L\_\{xy\}\; +\; L\_y^2\; \backslash ,\; L\_\{yy\}\; =\; 0,$

that satisfy a sign-condition on the following differential invariant

$L\_v^3\; L\_\{vvv\}\; =\; L\_x^3\; \backslash ,\; L\_\{xxx\}\; +\; 3\; \backslash ,\; L\_x^2\; \backslash ,\; L\_y\; \backslash ,\; L\_\{xxy\}\; +\; 3\; \backslash ,\; L\_x\; \backslash ,\; L\_y^2\; \backslash ,\; L\_\{xyy\}\; +\; L\_y^3\; \backslash ,\; L\_\{yyy\}\; \backslash leq\; 0$

(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

• Blob detection
• Computer vision
• Edge detection
• Feature detection (computer vision)
• Interest point detection
• Scale space

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