Let ${\displaystyle f(x,y)}$  denote a two-dimensional function, and let ${\displaystyle L}$  be the scale-space representation of ${\displaystyle f(x,y)}$  obtained by convolving ${\displaystyle f(x,y)}$  with a Gaussian function

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

Furthermore, let ${\displaystyle L_{pp}}$  and ${\displaystyle L_{qq}}$  denote the eigenvalues of the Hessian matrix

${\displaystyle H={\begin{bmatrix}L_{xx}&L_{xy}\\L_{xy}&L_{yy}\end{bmatrix}}}$ 

of the scale-space representation ${\displaystyle L}$  with a coordinate transformation (a rotation) applied to local directional derivative operators,

${\displaystyle \partial _{p}=\sin \beta \partial _{x}-\cos \beta \partial _{y},\partial _{q}=\cos \beta \partial _{x}+\sin \beta \partial _{y}}$ 

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

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

${\displaystyle \cos \beta ={\sqrt {{\frac {1}{2}}\left(1+{\frac {L_{xx}-L_{yy}}{\sqrt {(L_{xx}-L_{yy})^{2}+4L_{xy}^{2}}}}\right)}}}$ ,${\displaystyle \sin \beta =\operatorname {sgn}(L_{xy}){\sqrt {{\frac {1}{2}}\left(1-{\frac {L_{xx}-L_{yy}}{\sqrt {(L_{xx}-L_{yy})^{2}+4L_{xy}^{2}}}}\right)}}}$ .

Then, a formal differential geometric definition of the ridges of ${\displaystyle f(x,y)}$  at a fixed scale ${\displaystyle t}$  can be expressed as the set of points that satisfy ^[3]

${\displaystyle L_{p}=0,L_{pp}\leq 0,|L_{pp}|\geq |L_{qq}|.}$ 

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

${\displaystyle L_{q}=0,L_{qq}\geq 0,|L_{qq}|\geq |L_{pp}|.}$ 

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

${\displaystyle \partial _{u}=\sin \alpha \partial _{x}-\cos \alpha \partial _{y},\partial _{v}=\cos \alpha \partial _{x}+\sin \alpha \partial _{y}}$ 

where

${\displaystyle \cos \alpha ={\frac {L_{x}}{\sqrt {L_{x}^{2}+L_{y}^{2}}}},\sin \alpha ={\frac {L_{y}}{\sqrt {L_{x}^{2}+L_{y}^{2}}}}}$ 

it can be shown that this ridge and valley definition can instead be equivalently^[4] be written as

${\displaystyle L_{uv}=0,L_{uu}^{2}-L_{vv}^{2}\geq 0}$ 

where

${\displaystyle L_{v}^{2}L_{uu}=L_{x}^{2}L_{yy}-2L_{x}L_{y}L_{xy}+L_{y}^{2}L_{xx},}$ 

${\displaystyle L_{v}^{2}L_{uv}=L_{x}L_{y}(L_{xx}-L_{yy})-(L_{x}^{2}-L_{y}^{2})L_{xy},}$ 

${\displaystyle L_{v}^{2}L_{vv}=L_{x}^{2}L_{xx}+2L_{x}L_{y}L_{xy}+L_{y}^{2}L_{yy}}$ 

and the sign of ${\displaystyle L_{uu}}$  determines the polarity; ${\displaystyle L_{uu}<0}$  for ridges and ${\displaystyle L_{uu}>0}$  for valleys.

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 ${\displaystyle R(x,y,t)}$  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

${\displaystyle L_{p}=0,L_{pp}\leq 0,\partial _{t}(R)=0,\partial _{tt}(R)\leq 0,}$ 

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

${\displaystyle L_{q}=0,L_{qq}\geq 0,\partial _{t}(R)=0,\partial _{tt}(R)\leq 0.}$ 

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)^[5] coined the term scale-space ridge, he considered three measures of ridge strength:

• The main principal curvature

${\displaystyle L_{pp,\gamma -norm}={\frac {t^{\gamma }}{2}}\left(L_{xx}+L_{yy}-{\sqrt {(L_{xx}-L_{yy})^{2}+4L_{xy}^{2}}}\right)}$ 

expressed in terms of ${\displaystyle \gamma }$ -normalized derivatives with

${\displaystyle \partial _{\xi }=t^{\gamma /2}\partial _{x},\partial _{\eta }=t^{\gamma /2}\partial _{y}}$ .

• The square of the ${\displaystyle \gamma }$ -normalized square eigenvalue difference

${\displaystyle N_{\gamma -norm}=\left(L_{pp,\gamma -norm}^{2}-L_{qq,\gamma -norm}^{2}\right)^{2}=t^{4\gamma }(L_{xx}+L_{yy})^{2}\left((L_{xx}-L_{yy})^{2}+4L_{xy}^{2}\right).}$ 

• The square of the ${\displaystyle \gamma }$ -normalized eigenvalue difference

${\displaystyle A_{\gamma -norm}=\left(L_{pp,\gamma -norm}-L_{qq,\gamma -norm}\right)^{2}=t^{2\gamma }\left((L_{xx}-L_{yy})^{2}+4L_{xy}^{2}\right).}$ 

The notion of ${\displaystyle \gamma }$ -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 ${\displaystyle \gamma =3/4}$ . Out of these three measures of ridge strength, the first entity ${\displaystyle L_{pp,\gamma -norm}}$  is a general purpose ridge strength measure with many applications such as blood vessel detection and road extraction. Nevertheless, the entity ${\displaystyle A_{\gamma -norm}}$  has been used in applications such as fingerprint enhancement,^[6] real-time hand tracking and gesture recognition^[7] as well as for modelling local image statistics for detecting and tracking humans in images and video.^[8]

There are also other closely related ridge definitions that make use of normalized derivatives with the implicit assumption of ${\displaystyle \gamma =1}$ .^[9] Develop these approaches in further detail. When detecting ridges with ${\displaystyle \gamma =1}$ , however, the detection scale will be twice as large as for ${\displaystyle \gamma =3/4}$ , resulting in more shape distortions and a lower ability to capture ridges and valleys with nearby interfering image structures in the image domain.

The notion of ridges and valleys in digital images was introduced by Haralick in 1983^[10] and by Crowley concerning difference of Gaussians pyramids in 1984.^[11][12] The application of ridge descriptors to medical image analysis has been extensively studied by Pizer and his co-workers^[13][14][15] resulting in their notion of M-reps.^[16] Ridge detection has also been furthered by Lindeberg with the introduction of ${\displaystyle \gamma }$ -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.^[17][18] and to blood vessel segmentation by Frangi et al.^[19] as well as to the detection of curvilinear and tubular structures by Sato et al.^[20] and Krissian et al.^[21] 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.^[22] A review of vessel extraction techniques has been presented by Kirbas and Quek.^[23]

In its broadest sense, the notion of ridge generalizes the idea of a local maximum of a real-valued function. A point ${\displaystyle \mathbf {x} _{0}}$  in the domain of a function ${\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }$  is a local maximum of the function if there is a distance ${\displaystyle \delta >0}$  with the property that if ${\displaystyle \mathbf {x} }$  is within ${\displaystyle \delta }$  units of ${\displaystyle \mathbf {x} _{0}}$ , then ${\displaystyle f(\mathbf {x} )<f(\mathbf {x} _{0})}$ . 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 ${\displaystyle f(\mathbf {x} )<f(\mathbf {x} _{0})}$  for ${\displaystyle \mathbf {x} }$  in an entire neighborhood of ${\displaystyle \mathbf {x} _{0}}$  slightly to require only that this hold on an ${\displaystyle n-1}$  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^[24] and can be seen as a generalization of some of the abovementioned ridge definitions. Let ${\displaystyle U\subset \mathbb {R} ^{n}}$  be an open set, and ${\displaystyle f:U\rightarrow \mathbb {R} }$  be smooth. Let ${\displaystyle \mathbf {x} _{0}\in U}$ . Let ${\displaystyle \nabla _{\mathbf {x} _{0}}f}$  be the gradient of ${\displaystyle f}$  at ${\displaystyle \mathbf {x} _{0}}$ , and let ${\displaystyle H_{\mathbf {x} _{0}}(f)}$  be the ${\displaystyle n\times n}$  Hessian matrix of ${\displaystyle f}$  at ${\displaystyle \mathbf {x} _{0}}$ . Let ${\displaystyle \lambda _{1}\leq \lambda _{2}\leq \cdots \leq \lambda _{n}}$  be the ${\displaystyle n}$  ordered eigenvalues of ${\displaystyle H_{\mathbf {x} _{0}}(f)}$  and let ${\displaystyle \mathbf {e} _{i}}$  be a unit eigenvector in the eigenspace for ${\displaystyle \lambda _{i}}$ . (For this, one should assume that all the eigenvalues are distinct.)

The point ${\displaystyle \mathbf {x} _{0}}$  is a point on the 1-dimensional ridge of ${\displaystyle f}$  if the following conditions hold:

1. ${\displaystyle \lambda _{n-1}<0}$ , and
2. ${\displaystyle \nabla _{\mathbf {x} _{0}}f\cdot \mathbf {e} _{i}=0}$  for ${\displaystyle i=1,2,\ldots ,n-1}$ .

This makes precise the concept that ${\displaystyle f}$  restricted to this particular ${\displaystyle n-1}$ -dimensional subspace has a local maximum at ${\displaystyle \mathbf {x} _{0}}$ .

This definition naturally generalizes to the k-dimensional ridge as follows: the point ${\displaystyle \mathbf {x} _{0}}$  is a point on the k-dimensional ridge of ${\displaystyle f}$  if the following conditions hold:

1. ${\displaystyle \lambda _{n-k}<0}$ , and
2. ${\displaystyle \nabla _{\mathbf {x} _{0}}f\cdot \mathbf {e} _{i}=0}$  for ${\displaystyle i=1,2,\ldots ,n-k}$ .

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^[1] and Miller.^[2] Their properties in one-parameter families was established by Keller.^[25]

The following definition can be traced to Fritsch^[26] 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.

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

1. ${\displaystyle {\frac {\partial f}{\partial \sigma }}=0}$  and ${\displaystyle {\frac {\partial ^{2}f}{\partial \sigma ^{2}}}<0}$ , and
2. ${\displaystyle \nabla f\cdot \mathbf {e} _{1}=0}$  and ${\displaystyle \mathbf {e} _{1}^{t}H(f)\mathbf {e} _{1}<0}$ .

The purpose of ridge detection is usually to capture the major axis of symmetry of an elongated object,^{[citation needed]} whereas the purpose of edge detection is usually to capture the boundary of the object. However, some literature on edge detection erroneously^{[citation needed]} 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,^[27] 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,^[28] we can in the above-mentioned ${\displaystyle (u,v)}$ -coordinate system state that the gradient magnitude of the scale-space representation, which is equal to the first-order directional derivative in the ${\displaystyle v}$ -direction ${\displaystyle L_{v}}$ , should have its first order directional derivative in the ${\displaystyle v}$ -direction equal to zero

${\displaystyle \partial _{v}(L_{v})=0}$ 

while the second-order directional derivative in the ${\displaystyle v}$ -direction of ${\displaystyle L_{v}}$  should be negative, i.e.,

${\displaystyle \partial _{vv}(L_{v})\leq 0}$ .

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

${\displaystyle L_{v}^{2}L_{vv}=L_{x}^{2}\,L_{xx}+2\,L_{x}\,L_{y}\,L_{xy}+L_{y}^{2}\,L_{yy}=0,}$ 

that satisfy a sign-condition on the following differential invariant

${\displaystyle L_{v}^{3}L_{vvv}=L_{x}^{3}\,L_{xxx}+3\,L_{x}^{2}\,L_{y}\,L_{xxy}+3\,L_{x}\,L_{y}^{2}\,L_{xyy}+L_{y}^{3}\,L_{yyy}\leq 0}$ 

(see the article on edge detection for more information). Notably, the edges obtained in this way are the ridges of the gradient magnitude.

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