4343:-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.
4075:-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
464:. 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
332:
3788:
3150:
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.
4714:
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
3149:
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
1482:
7581:
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,
2828:
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
439:
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
5725:
The following definition can be traced to
Fritsch 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.
2833:
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
6121:
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
4048:
2011:
1094:
6860:
2159:
1295:
3344:
5716:
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.
7939:
Fritsch, 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.
3475:
4322:
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. Ridge detection has also been furthered by
Lindeberg with the introduction of
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1300:
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2996:
5087:
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7182:
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,
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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.
6039:
5737:
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
313:
6376:
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3153:
In the literature, various measures of ridge strength have been proposed. When
Lindeberg (1996, 1998) coined the term scale-space ridge, he considered three measures of ridge strength:
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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.
5710:
5414:
2829:
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
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5532:
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5240:
5181:
5028:
4908:
4686:
4518:
4378:
4200:
3783:{\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).}
3378:
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4657:
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4447:
4301:
4107:
2818:
2782:
4979:
4267:
4237:
105:
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6467:
4575:. 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 (
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432:, 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
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1477:{\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)}}}
476:
2410:
7672:
Frangi AF, Niessen WJ, Hoogeveen RM, van Walsum T, Viergever MA (October 1999). "Model-based quantitation of 3-D magnetic resonance angiographic images".
843:
299:
6154:-coordinate system state that the gradient magnitude of the scale-space representation, which is equal to the first-order directional derivative in the
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2572:
4303:, resulting in more shape distortions and a lower ability to capture ridges and valleys with nearby interfering image structures in the image domain.
3035:
2884:
95:
90:
7111:
Earlier version presented at IEEE Conference on
Pattern Recognition and Computer Vision, CVPR'96, San Francisco, California, pages 465–470, June 1996
2878:
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
4718:
This following ridge definition follows the book by Eberly and can be seen as a generalization of some of the abovementioned ridge definitions. Let
7833:
7567:
7473:
7432:
7358:
7236:
7168:
4043:{\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).}
2006:{\displaystyle \partial _{u}=\sin \alpha \partial _{x}-\cos \alpha \partial _{y},\partial _{v}=\cos \alpha \partial _{x}+\sin \alpha \partial _{y}}
491:
350:
5033:
4160:
is a general purpose ridge strength measure with many applications such as blood vessel detection and road extraction. Nevertheless, the entity
1089:{\displaystyle \partial _{p}=\sin \beta \partial _{x}-\cos \beta \partial _{y},\partial _{q}=\cos \beta \partial _{x}+\sin \beta \partial _{y}}
7848:
8041:
7522:
Pizer, Stephen M., Eberly, David, Fritsch, Daniel S. (January 1998). "Zoom-invariant vision of figural shape: the mathematics of cores".
5941:
5605:
5309:
657:
6855:{\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}
5811:
153:
6044:
2154:{\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}}}}}
1290:{\displaystyle \cos \beta ={\sqrt {{\frac {1}{2}}\left(1+{\frac {L_{xx}-L_{yy}}{\sqrt {(L_{xx}-L_{yy})^{2}+4L_{xy}^{2}}}}\right)}}}
1720:
1549:
4383:
3339:{\displaystyle L_{pp,\gamma -norm}={\frac {t^{\gamma }}{2}}\left(L_{xx}+L_{yy}-{\sqrt {(L_{xx}-L_{yy})^{2}+4L_{xy}^{2}}}\right)}
6114:
The purpose of ridge detection is usually to capture the major axis of symmetry of an elongated object, whereas the purpose of
4213:
There are also other closely related ridge definitions that make use of normalized derivatives with the implicit assumption of
5898:
7908:
4715:
the above can be modified to generalize the idea to local minima and result in what might call 1-dimensional valley curves.
4585:
4523:
7727:"Three-dimensional multi-scale line filter for segmentation and visualization of curvilinear structures in medical images"
2170:
6893:
7373:
486:
for organizing spatial constraints on local appearance, with a number of qualitative similarities to the way the Blum's
7026:
4351:
In its broadest sense, the notion of ridge generalizes the idea of a local maximum of a real-valued function. A point
6000:
517:
100:
368:
244:
148:
456:
and is to capture the interior of elongated objects in the image domain. Ridge-related representations in terms of
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259:
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7181:
7122:
7043:
7006:
7998:
4721:
228:
115:
5771:
is a point on this ridge, then the value of the function at the point is maximal in the scale dimension. Let
143:
7446:
Gauch, J.M., Pizer, S.M. (June 1993). "Multiresolution
Analysis of Ridges and Valleys in Grey-Scale Images".
7255:
Scale Space Theory in
Computer Vision: Proceedings of the First International Conference on, Scale Space '97,
5774:
5112:
4913:
223:
110:
6227:
5861:
5740:
7926:
Generic Transitions of Relative Critical Sets in Parameterized Families with Applications to Image Analysis
4790:
202:
7999:"Discrete Derivative Approximations with Scale-Space Properties: A Basis for Low-Level Feature Extraction"
8056:
6869:
for more information). Notably, the edges obtained in this way are the ridges of the gradient magnitude.
5564:
5268:
4112:
181:
133:
3470:{\displaystyle \partial _{\xi }=t^{\gamma /2}\partial _{x},\partial _{\eta }=t^{\gamma /2}\partial _{y}}
7726:
5665:
5369:
4319:
4210:
as well as for modelling local image statistics for detecting and tracking humans in images and video.
511:
287:
282:
249:
7487:
Eberly D.; Gardner R.; Morse B.; Pizer S.; Scharlach C. (December 1994). "Ridges for image analysis".
6967:
5726:
Ridges of this image, once projected to the original image, were to be analogous to a shape skeleton (
5508:
5472:
5216:
5157:
5004:
4884:
4662:
4494:
4354:
4163:
8051:
7393:
7123:"Fingerprint Enhancement by Shape Adaptation of Scale-Space Operators with Automatic Scale-Selection"
26:
7868:
7686:
7319:
424:
for a function can be defined by replacing the condition of a local maximum with the condition of a
7607:
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6898:
518:
Differential geometric definition of ridges and valleys at a fixed scale in a two-dimensional image
472:
6469:, this edge definition can be expressed as the zero-crossing curves of the differential invariant
5186:
4640:
4452:
4426:
4316:
4272:
4078:
2787:
2751:
965:
with a coordinate transformation (a rotation) applied to local directional derivative operators,
218:
138:
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7681:
7602:
7531:
7388:
7314:
4958:
4246:
4216:
498:. In typical applications, ridge and valley descriptors are often used for detecting roads in
17:
7300:"A Representation for Shape Based on Peaks and Ridges in the Difference of Low Pass Transform"
6625:{\displaystyle L_{v}^{2}L_{vv}=L_{x}^{2}\,L_{xx}+2\,L_{x}\,L_{y}\,L_{xy}+L_{y}^{2}\,L_{yy}=0,}
2840:
7827:
7561:
7467:
7426:
7352:
7250:
7230:
7162:
6439:
4474:
4326:
4058:
3799:
3487:
3354:
2164:
it can be shown that this ridge and valley definition can instead be equivalently written as
457:
428:. The union of ridge sets and valley sets, together with a related set of points called the
67:
47:
2721:
1662:
1491:
1105:
803:
773:
619:
584:
525:
52:
7134:
7055:
6412:
6385:
6299:
6177:
6125:
1833:
483:
468:
theory should allow for more a robust representation of objects (or shapes) in the image.
8:
7195:
5446:
4691:
4207:
475:
or local extremal points. With appropriately defined concepts, ridges and valleys in the
346:
42:
7138:
7059:
6122:
differential geometric way of expressing this definition, we can in the above-mentioned
8018:
7979:
7881:
7788:
7707:
7654:
7631:
7580:
7549:
7504:
7414:
7374:"Multiple Resolution Representation and Probabilistic Matching of 2-D Gray-Scale Shape"
7340:
7218:
7102:
6942:
6279:
6204:
6157:
5541:
5423:
5245:
5092:
4984:
4864:
3004:
1865:
1697:
1526:
948:
560:
507:
499:
461:
277:
7768:
7745:
2561:{\displaystyle L_{v}^{2}L_{uv}=L_{x}L_{y}(L_{xx}-L_{yy})-(L_{x}^{2}-L_{y}^{2})L_{xy},}
416:− 1 dimensions. In this respect, the notion of ridge points extends the concept of a
7971:
7904:
7819:
7749:
7699:
7406:
7332:
7280:
7150:
7022:
417:
8022:
7983:
7806:
Koenderink, Jan J., van Doorn, Andrea J. (May 1994). "2+1-D differential geometry".
7521:
7418:
7344:
7106:
6946:
6925:
Damon, J. (March 1999). "Properties of Ridges and Cores in Two-Dimensional Images".
932:{\displaystyle H={\begin{bmatrix}L_{xx}&L_{xy}\\L_{xy}&L_{yy}\end{bmatrix}}}
8010:
7963:
7873:
7815:
7792:
7780:
7741:
7711:
7691:
7658:
7646:
7612:
7541:
7508:
7496:
7455:
7398:
7324:
7276:
7222:
7210:
7142:
7094:
7063:
7014:
6934:
2823:
382:
197:
81:
62:
7885:
7632:"Automatic extraction of roads from aerial images based on scale-space and snakes"
7630:
Laptev I.; Mayer H.; Lindeberg T.; Eckstein W.; Steger C.; Baumgartner A. (2000).
7553:
7018:
2399:{\displaystyle L_{v}^{2}L_{uu}=L_{x}^{2}L_{yy}-2L_{x}L_{y}L_{xy}+L_{y}^{2}L_{xx},}
479:(or in some other representation derived from the intensity landscape) may form a
7295:
6883:
2708:{\displaystyle L_{v}^{2}L_{vv}=L_{x}^{2}L_{xx}+2L_{x}L_{y}L_{xy}+L_{y}^{2}L_{yy}}
480:
453:
176:
3139:{\displaystyle L_{q}=0,L_{qq}\geq 0,\partial _{t}(R)=0,\partial _{tt}(R)\leq 0.}
2991:{\displaystyle L_{p}=0,L_{pp}\leq 0,\partial _{t}(R)=0,\partial _{tt}(R)\leq 0,}
7967:
7402:
7328:
7044:"Scale-space theory: A basic tool for analysing structures at different scales"
6888:
6878:
6866:
6115:
835:
449:
124:
57:
33:
7951:
7214:
7098:
6938:
8035:
4203:
425:
72:
7925:
7877:
6382:
Written out as an explicit expression in terms of local partial derivatives
7975:
7784:
7753:
7703:
7545:
7410:
7336:
7154:
503:
495:
471:
In this respect, ridges and valleys can be seen as a complement to natural
412:
variables, its ridges are a set of curves whose points are local maxima in
398:
7766:
7650:
5082:{\displaystyle \lambda _{1}\leq \lambda _{2}\leq \cdots \leq \lambda _{n}}
6903:
5731:
4202:
has been used in applications such as fingerprint enhancement, real-time
3022:
943:
578:
487:
465:
268:
7805:
7725:
Sato Y, Nakajima S, Shiraga N, Atsumi H, Yoshida S, et al. (1998).
7629:
8014:
7500:
6109:
5210:. (For this, one should assume that all the eigenvalues are distinct.)
831:
7767:
Krissian K.; Malandain G.; Ayache N.; Vaillan R.; Trousset Y. (2000).
7695:
7616:
7459:
7146:
7067:
6635:
that satisfy a sign-condition on the following differential invariant
7251:
The Maximal Scale Ridge: Incorporating Scale in the Ridge Definition
4311:
The notion of ridges and valleys in digital images was introduced by
7593:
Steger C. (1998). "An unbiased detector of curvilinear structures".
7486:
7267:
Haralick, R. (April 1983). "Ridges and Valleys on Digital Images".
7257:
pp. 93–104. Springer Lecture Notes in Computer Science, vol. 1682.
7083:"Edge detection and ridge detection with automatic scale selection"
5988:{\displaystyle {\frac {\partial ^{2}f}{\partial \sigma ^{2}}}<0}
5655:{\displaystyle \nabla _{\mathbf {x} _{0}}f\cdot \mathbf {e} _{i}=0}
5359:{\displaystyle \nabla _{\mathbf {x} _{0}}f\cdot \mathbf {e} _{i}=0}
4346:
4312:
4109:. Out of these three measures of ridge strength, the first entity
1488:
Then, a formal differential geometric definition of the ridges of
759:{\displaystyle g(x,y,t)={\frac {1}{2\pi t}}e^{-(x^{2}+y^{2})/2t}}
7671:
5851:{\displaystyle U\subset \mathbb {R} ^{2}\times \mathbb {R} _{+}}
2824:
Computation of variable scale ridges from two-dimensional images
1099:
where p and q are coordinates of the rotated coordinate system.
6098:{\displaystyle \mathbf {e} _{1}^{t}H(f)\mathbf {e} _{1}<0}
4269:, however, the detection scale will be twice as large as for
402:
394:
7371:
6201:, should have its first order directional derivative in the
440:
open question. The primary motivation for the creation of
7293:
1820:{\displaystyle L_{q}=0,L_{qq}\geq 0,|L_{qq}|\geq |L_{pp}|.}
1649:{\displaystyle L_{p}=0,L_{pp}\leq 0,|L_{pp}|\geq |L_{qq}|.}
1132:
in the transformed coordinate system is zero if we choose
254:
7769:"Model-based detection of tubular structures in 3D images"
7120:
4423:
is a local maximum of the function if there is a distance
4416:{\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} }
7849:"A review of vessel extraction techniques and algorithms"
7724:
7193:
7013:. Vol. IV. John Wiley and Sons. pp. 2495–2504.
6995:
Ph.D. Dissertation. University of North Carolina. 1998.
7445:
5931:{\displaystyle {\frac {\partial f}{\partial \sigma }}=0}
7196:"Learning the statistics of people in images and video"
4630:{\displaystyle f(\mathbf {x} )<f(\mathbf {x} _{0})}
4568:{\displaystyle f(\mathbf {x} )<f(\mathbf {x} _{0})}
858:
6970:
6644:
6478:
6442:
6415:
6388:
6332:
6302:
6282:
6276:
while the second-order directional derivative in the
6230:
6207:
6180:
6160:
6128:
6047:
6003:
5944:
5901:
5892:
is a point on the maximal scale ridge if and only if
5864:
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6110:
Relations between edge detection and ridge detection
2244:{\displaystyle L_{uv}=0,L_{uu}^{2}-L_{vv}^{2}\geq 0}
1543:can be expressed as the set of points that satisfy
341:
may be too technical for most readers to understand
6985:
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6624:
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2005:
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8033:
7011:Encyclopedia of Computer Science and Engineering
6034:{\displaystyle \nabla f\cdot \mathbf {e} _{1}=0}
4347:Definition of ridges and valleys in N dimensions
7269:Computer Vision, Graphics, and Image Processing
4688:slightly to require only that this hold on an
7372:Crowley, J.L., Sanderson, A. (January 1987).
7080:
5501:This definition naturally generalizes to the
5469:-dimensional subspace has a local maximum at
307:
7996:
7952:"A computational approach to edge detection"
7832:: CS1 maint: multiple names: authors list (
7566:: CS1 maint: multiple names: authors list (
7472:: CS1 maint: multiple names: authors list (
7431:: CS1 maint: multiple names: authors list (
7357:: CS1 maint: multiple names: authors list (
7266:
7235:: CS1 maint: multiple names: authors list (
7167:: CS1 maint: multiple names: authors list (
7041:
7004:
5183:be a unit eigenvector in the eigenspace for
6371:{\displaystyle \partial _{vv}(L_{v})\leq 0}
4854:{\displaystyle \nabla _{\mathbf {x} _{0}}f}
4780:{\displaystyle f:U\rightarrow \mathbb {R} }
4241:Develop these approaches in further detail.
557:denote a two-dimensional function, and let
8003:Journal of Mathematical Imaging and Vision
7898:
7846:
7489:Journal of Mathematical Imaging and Vision
6958:
6956:
5505:-dimensional ridge as follows: the point
1102:It can be shown that the mixed derivative
314:
300:
7867:
7685:
7606:
7592:
7535:
7392:
7318:
6973:
6924:
6920:
6918:
6829:
6794:
6778:
6767:
6744:
6733:
6717:
6694:
6599:
6567:
6556:
6545:
6525:
5838:
5823:
5242:is a point on the 1-dimensional ridge of
4773:
4746:{\displaystyle U\subset \mathbb {R} ^{n}}
4733:
4409:
4395:
1882:direction parallel to the image gradient
369:Learn how and when to remove this message
353:, without removing the technical details.
7949:
7203:International Journal of Computer Vision
7087:International Journal of Computer Vision
3504:-normalized square eigenvalue difference
389:is the attempt, via software, to locate
7773:Computer Vision and Image Understanding
7524:Computer Vision and Image Understanding
6953:
5808:be a smooth differentiable function on
5801:{\displaystyle f(\mathbf {x} ,\sigma )}
5147:{\displaystyle H_{\mathbf {x} _{0}}(f)}
4948:{\displaystyle H_{\mathbf {x} _{0}}(f)}
8034:
6915:
6266:{\displaystyle \partial _{v}(L_{v})=0}
5885:{\displaystyle (\mathbf {x} ,\sigma )}
5764:{\displaystyle (\mathbf {x} ,\sigma )}
5720:
401:of the function, akin to geographical
16:For other features called ridges, see
7127:IEEE Transactions on Image Processing
4815:{\displaystyle \mathbf {x} _{0}\in U}
4582:Consider relaxing the condition that
397:, defined as curves whose points are
351:make it understandable to non-experts
7930:University of North Carolina. 1999.
5420:This makes precise the concept that
325:
8042:Feature detection (computer vision)
7956:IEEE Trans Pattern Anal Mach Intell
7595:IEEE Trans Pattern Anal Mach Intell
7448:IEEE Trans Pattern Anal Mach Intell
7381:IEEE Trans Pattern Anal Mach Intell
7307:IEEE Trans Pattern Anal Mach Intell
7121:Almansa, A., Lindeberg, T. (2000).
6993:and Applications to Image Analysis.
6894:Feature detection (computer vision)
5593:{\displaystyle \lambda _{n-k}<0}
5297:{\displaystyle \lambda _{n-1}<0}
4153:{\displaystyle L_{pp,\gamma -norm}}
13:
7194:Sidenbladh, H., Black, M. (2003).
6334:
6232:
6004:
5963:
5949:
5913:
5905:
5610:
5558:if the following conditions hold:
5314:
5262:if the following conditions hold:
4830:
4315:in 1983 and by Crowley concerning
3458:
3427:
3414:
3383:
3109:
3081:
3029:is the set of points that satisfy
2958:
2930:
1994:
1972:
1950:
1937:
1915:
1893:
1077:
1055:
1033:
1020:
998:
976:
211:Affine invariant feature detection
14:
8068:
7901:Ridges in Image and Data Analysis
5705:{\displaystyle i=1,2,\ldots ,n-k}
5409:{\displaystyle i=1,2,\ldots ,n-1}
3816:-normalized eigenvalue difference
420:. Correspondingly, the notion of
149:Maximally stable extremal regions
106:Hessian feature strength measures
6986:{\displaystyle \mathbb {R} ^{n}}
6079:
6050:
6015:
5869:
5785:
5748:
5636:
5616:
5527:{\displaystyle \mathbf {x} _{0}}
5514:
5491:{\displaystyle \mathbf {x} _{0}}
5478:
5340:
5320:
5235:{\displaystyle \mathbf {x} _{0}}
5222:
5176:{\displaystyle \mathbf {e} _{i}}
5163:
5123:
5023:{\displaystyle \mathbf {x} _{0}}
5010:
4924:
4903:{\displaystyle \mathbf {x} _{0}}
4890:
4836:
4796:
4681:{\displaystyle \mathbf {x} _{0}}
4668:
4645:
4614:
4596:
4552:
4534:
4513:{\displaystyle \mathbf {x} _{0}}
4500:
4457:
4373:{\displaystyle \mathbf {x} _{0}}
4360:
4195:{\displaystyle A_{\gamma -norm}}
1659:Correspondingly, the valleys of
330:
7990:
7943:
7933:
7917:
7892:
7840:
7799:
7760:
7718:
7665:
7639:Machine Vision and Applications
7623:
7586:
7574:
7515:
7480:
7439:
7365:
7287:
7260:
7243:
7187:
7175:
7114:
7074:
7035:
6998:
6359:
6346:
6254:
6241:
6141:
6129:
6074:
6068:
5879:
5865:
5795:
5781:
5758:
5744:
5141:
5135:
4942:
4936:
4769:
4624:
4609:
4600:
4592:
4562:
4547:
4538:
4530:
4405:
3999:
3966:
3739:
3706:
3692:
3659:
3296:
3263:
3127:
3121:
3096:
3090:
3021:is the scale parameter in the
2976:
2970:
2945:
2939:
2865:
2847:
2539:
2503:
2497:
2465:
1849:
1837:
1810:
1792:
1784:
1766:
1681:
1669:
1639:
1621:
1613:
1595:
1510:
1498:
1431:
1398:
1338:
1322:
1244:
1211:
740:
714:
682:
664:
638:
626:
603:
591:
544:
532:
1:
7746:10.1016/s1361-8415(98)80009-1
7048:Journal of Applied Statistics
7019:10.1002/9780470050118.ecse609
6909:
4659:in an entire neighborhood of
144:Determinant of Hessian (DoH)
139:Difference of Gaussians (DoG)
7820:10.1016/0167-8655(94)90134-1
7281:10.1016/0734-189X(83)90094-4
5203:{\displaystyle \lambda _{i}}
4652:{\displaystyle \mathbf {x} }
4464:{\displaystyle \mathbf {x} }
4442:{\displaystyle \delta >0}
4380:in the domain of a function
3157:The main principal curvature
203:Generalized structure tensor
7:
7808:Pattern Recognition Letters
6872:
6323:should be negative, i.e.,
4296:{\displaystyle \gamma =3/4}
4243:When detecting ridges with
4102:{\displaystyle \gamma =3/4}
2813:{\displaystyle L_{uu}>0}
2777:{\displaystyle L_{uu}<0}
1862:coordinate system with the
182:Generalized Hough transform
134:Laplacian of Gaussian (LoG)
10:
8073:
7968:10.1109/TPAMI.1986.4767851
7403:10.1109/TPAMI.1987.4767876
7329:10.1109/TPAMI.1984.4767500
7249:J. Furst and J. Miller, "
6964:Relative Critical Sets in
6221:-direction equal to zero
4449:with the property that if
4306:
3023:scale-space representation
944:scale-space representation
651:with a Gaussian function
579:scale-space representation
448:procedures has come from
15:
7847:Kirbas C, Quek F (2004).
7009:. In Benjamin Wah (ed.).
5734:) of the original image.
4974:{\displaystyle n\times n}
4579:, the nongeneric cases).
4262:{\displaystyle \gamma =1}
4232:{\displaystyle \gamma =1}
2748:determines the polarity;
512:magnetic resonance images
6899:Interest point detection
2871:{\displaystyle R(x,y,t)}
7878:10.1145/1031120.1031121
7215:10.1023/a:1023765619733
7099:10.1023/A:1008097225773
6939:10.1023/A:1008379107611
6462:{\displaystyle L_{yyy}}
5109:ordered eigenvalues of
4484:{\displaystyle \delta }
4336:{\displaystyle \gamma }
4317:difference of Gaussians
4068:{\displaystyle \gamma }
3809:{\displaystyle \gamma }
3497:{\displaystyle \gamma }
3371:-normalized derivatives
3364:{\displaystyle \gamma }
616:obtained by convolving
219:Affine shape adaptation
8047:Multivariable calculus
7785:10.1006/cviu.2000.0866
7734:Medical Image Analysis
7674:IEEE Trans Med Imaging
7546:10.1006/cviu.1997.0563
7081:Lindeberg, T. (1998).
6987:
6856:
6626:
6463:
6430:
6403:
6372:
6317:
6290:
6267:
6215:
6195:
6168:
6148:
6099:
6035:
5989:
5932:
5886:
5852:
5802:
5765:
5706:
5656:
5594:
5552:
5538:-dimensional ridge of
5528:
5492:
5463:
5434:
5410:
5360:
5298:
5256:
5236:
5204:
5177:
5148:
5103:
5083:
5024:
4995:
4975:
4949:
4904:
4875:
4855:
4816:
4781:
4747:
4708:
4682:
4653:
4631:
4569:
4514:
4485:
4465:
4443:
4417:
4374:
4337:
4297:
4263:
4233:
4196:
4154:
4103:
4069:
4044:
3810:
3784:
3498:
3471:
3365:
3350:expressed in terms of
3340:
3140:
3015:
2992:
2872:
2814:
2778:
2742:
2741:{\displaystyle L_{uu}}
2709:
2562:
2400:
2245:
2155:
2007:
1876:
1856:
1821:
1714:are the set of points
1708:
1688:
1687:{\displaystyle f(x,y)}
1650:
1537:
1517:
1516:{\displaystyle f(x,y)}
1478:
1291:
1126:
1125:{\displaystyle L_{pq}}
1090:
959:
933:
824:
823:{\displaystyle L_{qq}}
794:
793:{\displaystyle L_{pp}}
760:
645:
644:{\displaystyle f(x,y)}
610:
609:{\displaystyle f(x,y)}
571:
551:
550:{\displaystyle f(x,y)}
283:Implementation details
18:Ridge (disambiguation)
7997:Lindeberg T. (1993).
7856:ACM Computing Surveys
7651:10.1007/s001380050121
7042:Lindeberg, T (1994).
7005:T. Lindeberg (2009).
6988:
6857:
6627:
6464:
6431:
6429:{\displaystyle L_{y}}
6404:
6402:{\displaystyle L_{x}}
6373:
6318:
6316:{\displaystyle L_{v}}
6291:
6268:
6216:
6196:
6194:{\displaystyle L_{v}}
6169:
6149:
6147:{\displaystyle (u,v)}
6100:
6036:
5990:
5933:
5887:
5853:
5803:
5766:
5707:
5657:
5595:
5553:
5529:
5493:
5464:
5435:
5411:
5361:
5299:
5257:
5237:
5205:
5178:
5149:
5104:
5084:
5025:
4996:
4976:
4950:
4905:
4876:
4856:
4817:
4782:
4748:
4709:
4683:
4654:
4632:
4570:
4515:
4486:
4466:
4444:
4418:
4375:
4338:
4298:
4264:
4234:
4197:
4155:
4104:
4070:
4045:
3811:
3785:
3499:
3472:
3366:
3341:
3141:
3016:
2993:
2873:
2815:
2779:
2743:
2710:
2563:
2401:
2246:
2156:
2008:
1877:
1857:
1855:{\displaystyle (u,v)}
1822:
1709:
1689:
1651:
1538:
1518:
1479:
1292:
1127:
1091:
960:
934:
825:
795:
761:
646:
611:
572:
552:
510:or three-dimensional
488:medial axis transform
434:relative critical set
101:Level curve curvature
6968:
6865:(see the article on
6642:
6476:
6440:
6413:
6386:
6330:
6300:
6280:
6228:
6205:
6178:
6158:
6126:
6045:
6001:
5942:
5899:
5862:
5812:
5775:
5741:
5666:
5606:
5565:
5542:
5509:
5473:
5447:
5424:
5370:
5310:
5269:
5246:
5217:
5187:
5158:
5113:
5093:
5034:
5005:
4985:
4959:
4914:
4885:
4865:
4826:
4791:
4757:
4753:be an open set, and
4722:
4692:
4663:
4641:
4586:
4524:
4495:
4475:
4453:
4427:
4384:
4355:
4327:
4273:
4247:
4217:
4164:
4113:
4079:
4059:
3826:
3800:
3514:
3488:
3379:
3355:
3167:
3036:
3005:
2885:
2841:
2788:
2752:
2722:
2573:
2411:
2261:
2171:
2023:
1889:
1866:
1834:
1721:
1698:
1663:
1550:
1527:
1492:
1301:
1139:
1106:
972:
949:
844:
804:
774:
658:
620:
585:
561:
526:
7899:Eberly, D. (1996).
7139:2000ITIP....9.2027L
7060:1994JApSt..21..225L
6828:
6793:
6732:
6693:
6659:
6598:
6524:
6493:
6064:
5721:Maximal scale ridge
5462:{\displaystyle n-1}
4861:be the gradient of
4707:{\displaystyle n-1}
4208:gesture recognition
4031:
3771:
3631:
3589:
3328:
2691:
2621:
2590:
2538:
2520:
2428:
2379:
2309:
2278:
2234:
2213:
2147:
2129:
2083:
2065:
1463:
1276:
477:intensity landscape
460:have been used for
237:Feature description
8057:Singularity theory
8015:10.1007/BF01664794
7501:10.1007/BF01262402
6983:
6927:J Math Imaging Vis
6852:
6814:
6779:
6718:
6679:
6645:
6622:
6584:
6510:
6479:
6459:
6426:
6399:
6368:
6313:
6286:
6263:
6211:
6191:
6164:
6144:
6095:
6048:
6031:
5985:
5928:
5882:
5848:
5798:
5761:
5702:
5652:
5590:
5548:
5534:is a point on the
5524:
5488:
5459:
5430:
5406:
5356:
5294:
5252:
5232:
5200:
5173:
5144:
5099:
5079:
5020:
4991:
4981:Hessian matrix of
4971:
4945:
4900:
4871:
4851:
4812:
4777:
4743:
4704:
4678:
4649:
4627:
4565:
4510:
4481:
4461:
4439:
4413:
4370:
4333:
4293:
4259:
4229:
4192:
4150:
4099:
4065:
4040:
4014:
3806:
3796:The square of the
3780:
3754:
3593:
3551:
3494:
3484:The square of the
3467:
3361:
3336:
3311:
3136:
3027:scale-space valley
3011:
2988:
2868:
2831:scale-space ridges
2810:
2774:
2738:
2705:
2677:
2607:
2576:
2558:
2524:
2506:
2414:
2396:
2365:
2295:
2264:
2241:
2217:
2196:
2151:
2133:
2115:
2069:
2051:
2003:
1872:
1852:
1817:
1704:
1684:
1646:
1533:
1513:
1474:
1446:
1287:
1259:
1122:
1086:
955:
929:
923:
820:
790:
770:Furthermore, let
756:
641:
606:
567:
547:
502:and for detecting
462:image segmentation
408:For a function of
278:Scale-space axioms
7950:Canny J. (1986).
7910:978-0-7923-4268-7
7696:10.1109/42.811279
7617:10.1109/34.659930
7460:10.1109/34.216734
7147:10.1109/83.887971
7068:10.1080/757582976
6289:{\displaystyle v}
6214:{\displaystyle v}
6167:{\displaystyle v}
5977:
5920:
5551:{\displaystyle f}
5433:{\displaystyle f}
5255:{\displaystyle f}
5102:{\displaystyle n}
4994:{\displaystyle f}
4874:{\displaystyle f}
3329:
3222:
3014:{\displaystyle t}
2149:
2148:
2085:
2084:
1875:{\displaystyle v}
1707:{\displaystyle t}
1536:{\displaystyle t}
1523:at a fixed scale
1472:
1465:
1464:
1351:
1285:
1278:
1277:
1164:
958:{\displaystyle L}
704:
570:{\displaystyle L}
379:
378:
371:
324:
323:
27:Feature detection
8064:
8052:Smooth functions
8027:
8026:
7994:
7988:
7987:
7947:
7941:
7937:
7931:
7921:
7915:
7914:
7896:
7890:
7889:
7871:
7853:
7844:
7838:
7837:
7831:
7823:
7803:
7797:
7796:
7764:
7758:
7757:
7731:
7722:
7716:
7715:
7689:
7669:
7663:
7662:
7636:
7627:
7621:
7620:
7610:
7590:
7584:
7578:
7572:
7571:
7565:
7557:
7539:
7519:
7513:
7512:
7484:
7478:
7477:
7471:
7463:
7443:
7437:
7436:
7430:
7422:
7396:
7394:10.1.1.1015.9294
7378:
7369:
7363:
7362:
7356:
7348:
7322:
7304:
7291:
7285:
7284:
7264:
7258:
7247:
7241:
7240:
7234:
7226:
7209:(1–2): 183–209.
7200:
7191:
7185:
7179:
7173:
7172:
7166:
7158:
7118:
7112:
7110:
7078:
7072:
7071:
7039:
7033:
7032:
7002:
6996:
6992:
6990:
6989:
6984:
6982:
6981:
6976:
6960:
6951:
6950:
6922:
6861:
6859:
6858:
6853:
6845:
6844:
6827:
6822:
6810:
6809:
6792:
6787:
6777:
6776:
6760:
6759:
6743:
6742:
6731:
6726:
6710:
6709:
6692:
6687:
6675:
6674:
6658:
6653:
6631:
6629:
6628:
6623:
6612:
6611:
6597:
6592:
6580:
6579:
6566:
6565:
6555:
6554:
6538:
6537:
6523:
6518:
6506:
6505:
6492:
6487:
6468:
6466:
6465:
6460:
6458:
6457:
6435:
6433:
6432:
6427:
6425:
6424:
6408:
6406:
6405:
6400:
6398:
6397:
6377:
6375:
6374:
6369:
6358:
6357:
6345:
6344:
6322:
6320:
6319:
6314:
6312:
6311:
6295:
6293:
6292:
6287:
6272:
6270:
6269:
6264:
6253:
6252:
6240:
6239:
6220:
6218:
6217:
6212:
6200:
6198:
6197:
6192:
6190:
6189:
6173:
6171:
6170:
6165:
6153:
6151:
6150:
6145:
6104:
6102:
6101:
6096:
6088:
6087:
6082:
6063:
6058:
6053:
6040:
6038:
6037:
6032:
6024:
6023:
6018:
5994:
5992:
5991:
5986:
5978:
5976:
5975:
5974:
5961:
5957:
5956:
5946:
5937:
5935:
5934:
5929:
5921:
5919:
5911:
5903:
5891:
5889:
5888:
5883:
5872:
5857:
5855:
5854:
5849:
5847:
5846:
5841:
5832:
5831:
5826:
5807:
5805:
5804:
5799:
5788:
5770:
5768:
5767:
5762:
5751:
5732:Blum medial axis
5711:
5709:
5708:
5703:
5661:
5659:
5658:
5653:
5645:
5644:
5639:
5627:
5626:
5625:
5624:
5619:
5599:
5597:
5596:
5591:
5583:
5582:
5557:
5555:
5554:
5549:
5533:
5531:
5530:
5525:
5523:
5522:
5517:
5497:
5495:
5494:
5489:
5487:
5486:
5481:
5468:
5466:
5465:
5460:
5439:
5437:
5436:
5431:
5415:
5413:
5412:
5407:
5365:
5363:
5362:
5357:
5349:
5348:
5343:
5331:
5330:
5329:
5328:
5323:
5303:
5301:
5300:
5295:
5287:
5286:
5261:
5259:
5258:
5253:
5241:
5239:
5238:
5233:
5231:
5230:
5225:
5209:
5207:
5206:
5201:
5199:
5198:
5182:
5180:
5179:
5174:
5172:
5171:
5166:
5153:
5151:
5150:
5145:
5134:
5133:
5132:
5131:
5126:
5108:
5106:
5105:
5100:
5088:
5086:
5085:
5080:
5078:
5077:
5059:
5058:
5046:
5045:
5029:
5027:
5026:
5021:
5019:
5018:
5013:
5000:
4998:
4997:
4992:
4980:
4978:
4977:
4972:
4954:
4952:
4951:
4946:
4935:
4934:
4933:
4932:
4927:
4909:
4907:
4906:
4901:
4899:
4898:
4893:
4880:
4878:
4877:
4872:
4860:
4858:
4857:
4852:
4847:
4846:
4845:
4844:
4839:
4821:
4819:
4818:
4813:
4805:
4804:
4799:
4787:be smooth. Let
4786:
4784:
4783:
4778:
4776:
4752:
4750:
4749:
4744:
4742:
4741:
4736:
4713:
4711:
4710:
4705:
4687:
4685:
4684:
4679:
4677:
4676:
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4658:
4656:
4655:
4650:
4648:
4636:
4634:
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4628:
4623:
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4599:
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4509:
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4422:
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4414:
4412:
4404:
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4398:
4379:
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4376:
4371:
4369:
4368:
4363:
4342:
4340:
4339:
4334:
4302:
4300:
4299:
4294:
4289:
4268:
4266:
4265:
4260:
4238:
4236:
4235:
4230:
4201:
4199:
4198:
4193:
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4190:
4159:
4157:
4156:
4151:
4149:
4148:
4108:
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4105:
4100:
4095:
4074:
4072:
4071:
4066:
4049:
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4036:
4032:
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4025:
4007:
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3997:
3996:
3981:
3980:
3960:
3959:
3944:
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3938:
3934:
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3932:
3896:
3895:
3853:
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3815:
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3807:
3789:
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3786:
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3776:
3772:
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3765:
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3746:
3737:
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3721:
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3700:
3699:
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3689:
3674:
3673:
3658:
3657:
3642:
3641:
3636:
3632:
3630:
3625:
3588:
3583:
3541:
3540:
3503:
3501:
3500:
3495:
3476:
3474:
3473:
3468:
3466:
3465:
3456:
3455:
3451:
3435:
3434:
3422:
3421:
3412:
3411:
3407:
3391:
3390:
3370:
3368:
3367:
3362:
3345:
3343:
3342:
3337:
3335:
3331:
3330:
3327:
3322:
3304:
3303:
3294:
3293:
3278:
3277:
3262:
3257:
3256:
3241:
3240:
3223:
3218:
3217:
3208:
3203:
3202:
3145:
3143:
3142:
3137:
3120:
3119:
3089:
3088:
3070:
3069:
3048:
3047:
3020:
3018:
3017:
3012:
2997:
2995:
2994:
2989:
2969:
2968:
2938:
2937:
2919:
2918:
2897:
2896:
2877:
2875:
2874:
2869:
2819:
2817:
2816:
2811:
2803:
2802:
2783:
2781:
2780:
2775:
2767:
2766:
2747:
2745:
2744:
2739:
2737:
2736:
2718:and the sign of
2714:
2712:
2711:
2706:
2704:
2703:
2690:
2685:
2673:
2672:
2660:
2659:
2650:
2649:
2634:
2633:
2620:
2615:
2603:
2602:
2589:
2584:
2567:
2565:
2564:
2559:
2554:
2553:
2537:
2532:
2519:
2514:
2496:
2495:
2480:
2479:
2464:
2463:
2454:
2453:
2441:
2440:
2427:
2422:
2405:
2403:
2402:
2397:
2392:
2391:
2378:
2373:
2361:
2360:
2348:
2347:
2338:
2337:
2322:
2321:
2308:
2303:
2291:
2290:
2277:
2272:
2250:
2248:
2247:
2242:
2233:
2228:
2212:
2207:
2186:
2185:
2160:
2158:
2157:
2152:
2150:
2146:
2141:
2128:
2123:
2114:
2113:
2112:
2103:
2086:
2082:
2077:
2064:
2059:
2050:
2049:
2048:
2039:
2012:
2010:
2009:
2004:
2002:
2001:
1980:
1979:
1958:
1957:
1945:
1944:
1923:
1922:
1901:
1900:
1881:
1879:
1878:
1873:
1861:
1859:
1858:
1853:
1826:
1824:
1823:
1818:
1813:
1808:
1807:
1795:
1787:
1782:
1781:
1769:
1755:
1754:
1733:
1732:
1713:
1711:
1710:
1705:
1693:
1691:
1690:
1685:
1655:
1653:
1652:
1647:
1642:
1637:
1636:
1624:
1616:
1611:
1610:
1598:
1584:
1583:
1562:
1561:
1542:
1540:
1539:
1534:
1522:
1520:
1519:
1514:
1483:
1481:
1480:
1475:
1473:
1471:
1467:
1466:
1462:
1457:
1439:
1438:
1429:
1428:
1413:
1412:
1397:
1396:
1395:
1394:
1379:
1378:
1365:
1352:
1344:
1342:
1337:
1336:
1296:
1294:
1293:
1288:
1286:
1284:
1280:
1279:
1275:
1270:
1252:
1251:
1242:
1241:
1226:
1225:
1210:
1209:
1208:
1207:
1192:
1191:
1178:
1165:
1157:
1155:
1131:
1129:
1128:
1123:
1121:
1120:
1095:
1093:
1092:
1087:
1085:
1084:
1063:
1062:
1041:
1040:
1028:
1027:
1006:
1005:
984:
983:
964:
962:
961:
956:
938:
936:
935:
930:
928:
927:
920:
919:
905:
904:
888:
887:
873:
872:
829:
827:
826:
821:
819:
818:
799:
797:
796:
791:
789:
788:
765:
763:
762:
757:
755:
754:
747:
739:
738:
726:
725:
705:
703:
689:
650:
648:
647:
642:
615:
613:
612:
607:
576:
574:
573:
568:
556:
554:
553:
548:
446:valley detection
383:image processing
374:
367:
363:
360:
354:
334:
333:
326:
316:
309:
302:
198:Structure tensor
190:Structure tensor
82:Corner detection
23:
22:
8072:
8071:
8067:
8066:
8065:
8063:
8062:
8061:
8032:
8031:
8030:
7995:
7991:
7948:
7944:
7938:
7934:
7922:
7918:
7911:
7897:
7893:
7869:10.1.1.460.8544
7851:
7845:
7841:
7825:
7824:
7804:
7800:
7765:
7761:
7729:
7723:
7719:
7687:10.1.1.502.5994
7670:
7666:
7634:
7628:
7624:
7591:
7587:
7579:
7575:
7559:
7558:
7520:
7516:
7485:
7481:
7465:
7464:
7444:
7440:
7424:
7423:
7376:
7370:
7366:
7350:
7349:
7320:10.1.1.161.3102
7302:
7294:Crowley, J.L.,
7292:
7288:
7265:
7261:
7248:
7244:
7228:
7227:
7198:
7192:
7188:
7180:
7176:
7160:
7159:
7133:(12): 2027–42.
7119:
7115:
7079:
7075:
7040:
7036:
7029:
7003:
6999:
6977:
6972:
6971:
6969:
6966:
6965:
6961:
6954:
6923:
6916:
6912:
6884:Computer vision
6875:
6834:
6830:
6823:
6818:
6799:
6795:
6788:
6783:
6772:
6768:
6749:
6745:
6738:
6734:
6727:
6722:
6699:
6695:
6688:
6683:
6664:
6660:
6654:
6649:
6643:
6640:
6639:
6604:
6600:
6593:
6588:
6572:
6568:
6561:
6557:
6550:
6546:
6530:
6526:
6519:
6514:
6498:
6494:
6488:
6483:
6477:
6474:
6473:
6447:
6443:
6441:
6438:
6437:
6420:
6416:
6414:
6411:
6410:
6393:
6389:
6387:
6384:
6383:
6353:
6349:
6337:
6333:
6331:
6328:
6327:
6307:
6303:
6301:
6298:
6297:
6281:
6278:
6277:
6248:
6244:
6235:
6231:
6229:
6226:
6225:
6206:
6203:
6202:
6185:
6181:
6179:
6176:
6175:
6159:
6156:
6155:
6127:
6124:
6123:
6112:
6083:
6078:
6077:
6059:
6054:
6049:
6046:
6043:
6042:
6019:
6014:
6013:
6002:
5999:
5998:
5970:
5966:
5962:
5952:
5948:
5947:
5945:
5943:
5940:
5939:
5912:
5904:
5902:
5900:
5897:
5896:
5868:
5863:
5860:
5859:
5842:
5837:
5836:
5827:
5822:
5821:
5813:
5810:
5809:
5784:
5776:
5773:
5772:
5747:
5742:
5739:
5738:
5723:
5667:
5664:
5663:
5640:
5635:
5634:
5620:
5615:
5614:
5613:
5609:
5607:
5604:
5603:
5572:
5568:
5566:
5563:
5562:
5543:
5540:
5539:
5518:
5513:
5512:
5510:
5507:
5506:
5482:
5477:
5476:
5474:
5471:
5470:
5448:
5445:
5444:
5442:this particular
5425:
5422:
5421:
5371:
5368:
5367:
5344:
5339:
5338:
5324:
5319:
5318:
5317:
5313:
5311:
5308:
5307:
5276:
5272:
5270:
5267:
5266:
5247:
5244:
5243:
5226:
5221:
5220:
5218:
5215:
5214:
5194:
5190:
5188:
5185:
5184:
5167:
5162:
5161:
5159:
5156:
5155:
5127:
5122:
5121:
5120:
5116:
5114:
5111:
5110:
5094:
5091:
5090:
5073:
5069:
5054:
5050:
5041:
5037:
5035:
5032:
5031:
5014:
5009:
5008:
5006:
5003:
5002:
4986:
4983:
4982:
4960:
4957:
4956:
4928:
4923:
4922:
4921:
4917:
4915:
4912:
4911:
4894:
4889:
4888:
4886:
4883:
4882:
4866:
4863:
4862:
4840:
4835:
4834:
4833:
4829:
4827:
4824:
4823:
4800:
4795:
4794:
4792:
4789:
4788:
4772:
4758:
4755:
4754:
4737:
4732:
4731:
4723:
4720:
4719:
4693:
4690:
4689:
4672:
4667:
4666:
4664:
4661:
4660:
4644:
4642:
4639:
4638:
4618:
4613:
4612:
4595:
4587:
4584:
4583:
4556:
4551:
4550:
4533:
4525:
4522:
4521:
4504:
4499:
4498:
4496:
4493:
4492:
4476:
4473:
4472:
4456:
4454:
4451:
4450:
4428:
4425:
4424:
4408:
4399:
4394:
4393:
4385:
4382:
4381:
4364:
4359:
4358:
4356:
4353:
4352:
4349:
4328:
4325:
4324:
4309:
4285:
4274:
4271:
4270:
4248:
4245:
4244:
4218:
4215:
4214:
4171:
4167:
4165:
4162:
4161:
4120:
4116:
4114:
4111:
4110:
4091:
4080:
4077:
4076:
4060:
4057:
4056:
4026:
4018:
4002:
3998:
3989:
3985:
3973:
3969:
3965:
3961:
3952:
3948:
3939:
3904:
3900:
3867:
3863:
3862:
3858:
3857:
3833:
3829:
3827:
3824:
3823:
3801:
3798:
3797:
3766:
3758:
3742:
3738:
3729:
3725:
3713:
3709:
3705:
3701:
3695:
3691:
3682:
3678:
3666:
3662:
3650:
3646:
3637:
3626:
3597:
3584:
3555:
3550:
3546:
3545:
3521:
3517:
3515:
3512:
3511:
3489:
3486:
3485:
3461:
3457:
3447:
3443:
3439:
3430:
3426:
3417:
3413:
3403:
3399:
3395:
3386:
3382:
3380:
3377:
3376:
3356:
3353:
3352:
3323:
3315:
3299:
3295:
3286:
3282:
3270:
3266:
3261:
3249:
3245:
3233:
3229:
3228:
3224:
3213:
3209:
3207:
3174:
3170:
3168:
3165:
3164:
3112:
3108:
3084:
3080:
3062:
3058:
3043:
3039:
3037:
3034:
3033:
3025:. Similarly, a
3006:
3003:
3002:
2961:
2957:
2933:
2929:
2911:
2907:
2892:
2888:
2886:
2883:
2882:
2842:
2839:
2838:
2826:
2795:
2791:
2789:
2786:
2785:
2784:for ridges and
2759:
2755:
2753:
2750:
2749:
2729:
2725:
2723:
2720:
2719:
2696:
2692:
2686:
2681:
2665:
2661:
2655:
2651:
2645:
2641:
2626:
2622:
2616:
2611:
2595:
2591:
2585:
2580:
2574:
2571:
2570:
2546:
2542:
2533:
2528:
2515:
2510:
2488:
2484:
2472:
2468:
2459:
2455:
2449:
2445:
2433:
2429:
2423:
2418:
2412:
2409:
2408:
2384:
2380:
2374:
2369:
2353:
2349:
2343:
2339:
2333:
2329:
2314:
2310:
2304:
2299:
2283:
2279:
2273:
2268:
2262:
2259:
2258:
2229:
2221:
2208:
2200:
2178:
2174:
2172:
2169:
2168:
2142:
2137:
2124:
2119:
2108:
2104:
2102:
2078:
2073:
2060:
2055:
2044:
2040:
2038:
2024:
2021:
2020:
1997:
1993:
1975:
1971:
1953:
1949:
1940:
1936:
1918:
1914:
1896:
1892:
1890:
1887:
1886:
1867:
1864:
1863:
1835:
1832:
1831:
1809:
1800:
1796:
1791:
1783:
1774:
1770:
1765:
1747:
1743:
1728:
1724:
1722:
1719:
1718:
1699:
1696:
1695:
1664:
1661:
1660:
1638:
1629:
1625:
1620:
1612:
1603:
1599:
1594:
1576:
1572:
1557:
1553:
1551:
1548:
1547:
1528:
1525:
1524:
1493:
1490:
1489:
1458:
1450:
1434:
1430:
1421:
1417:
1405:
1401:
1387:
1383:
1371:
1367:
1366:
1364:
1357:
1353:
1343:
1341:
1329:
1325:
1302:
1299:
1298:
1271:
1263:
1247:
1243:
1234:
1230:
1218:
1214:
1200:
1196:
1184:
1180:
1179:
1177:
1170:
1166:
1156:
1154:
1140:
1137:
1136:
1113:
1109:
1107:
1104:
1103:
1080:
1076:
1058:
1054:
1036:
1032:
1023:
1019:
1001:
997:
979:
975:
973:
970:
969:
950:
947:
946:
922:
921:
912:
908:
906:
897:
893:
890:
889:
880:
876:
874:
865:
861:
854:
853:
845:
842:
841:
811:
807:
805:
802:
801:
781:
777:
775:
772:
771:
743:
734:
730:
721:
717:
710:
706:
693:
688:
659:
656:
655:
621:
618:
617:
586:
583:
582:
562:
559:
558:
527:
524:
523:
520:
481:scale invariant
473:interest points
454:computer vision
442:ridge detection
387:ridge detection
375:
364:
358:
355:
347:help improve it
344:
335:
331:
320:
177:Hough transform
169:Hough transform
163:Ridge detection
91:Harris operator
21:
12:
11:
5:
8070:
8060:
8059:
8054:
8049:
8044:
8029:
8028:
8009:(4): 349–376.
7989:
7962:(6): 679–698.
7942:
7932:
7916:
7909:
7891:
7839:
7814:(5): 439–443.
7798:
7779:(2): 130–171.
7759:
7740:(2): 143–168.
7717:
7680:(10): 946–56.
7664:
7622:
7608:10.1.1.42.2266
7601:(2): 113–125.
7585:
7573:
7537:10.1.1.38.3116
7514:
7495:(4): 353–373.
7479:
7454:(6): 635–646.
7438:
7387:(1): 113–121.
7364:
7313:(2): 156–170.
7298:(March 1984).
7286:
7259:
7242:
7186:
7174:
7113:
7093:(2): 117–154.
7073:
7054:(2): 224–270.
7034:
7028:978-0470050118
7027:
6997:
6980:
6975:
6952:
6933:(2): 163–174.
6913:
6911:
6908:
6907:
6906:
6901:
6896:
6891:
6889:Edge detection
6886:
6881:
6879:Blob detection
6874:
6871:
6867:edge detection
6863:
6862:
6851:
6848:
6843:
6840:
6837:
6833:
6826:
6821:
6817:
6813:
6808:
6805:
6802:
6798:
6791:
6786:
6782:
6775:
6771:
6766:
6763:
6758:
6755:
6752:
6748:
6741:
6737:
6730:
6725:
6721:
6716:
6713:
6708:
6705:
6702:
6698:
6691:
6686:
6682:
6678:
6673:
6670:
6667:
6663:
6657:
6652:
6648:
6633:
6632:
6621:
6618:
6615:
6610:
6607:
6603:
6596:
6591:
6587:
6583:
6578:
6575:
6571:
6564:
6560:
6553:
6549:
6544:
6541:
6536:
6533:
6529:
6522:
6517:
6513:
6509:
6504:
6501:
6497:
6491:
6486:
6482:
6456:
6453:
6450:
6446:
6423:
6419:
6396:
6392:
6380:
6379:
6367:
6364:
6361:
6356:
6352:
6348:
6343:
6340:
6336:
6310:
6306:
6296:-direction of
6285:
6274:
6273:
6262:
6259:
6256:
6251:
6247:
6243:
6238:
6234:
6210:
6188:
6184:
6163:
6143:
6140:
6137:
6134:
6131:
6116:edge detection
6111:
6108:
6107:
6106:
6094:
6091:
6086:
6081:
6076:
6073:
6070:
6067:
6062:
6057:
6052:
6030:
6027:
6022:
6017:
6012:
6009:
6006:
5996:
5984:
5981:
5973:
5969:
5965:
5960:
5955:
5951:
5927:
5924:
5918:
5915:
5910:
5907:
5881:
5878:
5875:
5871:
5867:
5845:
5840:
5835:
5830:
5825:
5820:
5817:
5797:
5794:
5791:
5787:
5783:
5780:
5760:
5757:
5754:
5750:
5746:
5722:
5719:
5714:
5713:
5701:
5698:
5695:
5692:
5689:
5686:
5683:
5680:
5677:
5674:
5671:
5651:
5648:
5643:
5638:
5633:
5630:
5623:
5618:
5612:
5601:
5589:
5586:
5581:
5578:
5575:
5571:
5547:
5521:
5516:
5485:
5480:
5458:
5455:
5452:
5440:restricted to
5429:
5418:
5417:
5405:
5402:
5399:
5396:
5393:
5390:
5387:
5384:
5381:
5378:
5375:
5355:
5352:
5347:
5342:
5337:
5334:
5327:
5322:
5316:
5305:
5293:
5290:
5285:
5282:
5279:
5275:
5251:
5229:
5224:
5197:
5193:
5170:
5165:
5143:
5140:
5137:
5130:
5125:
5119:
5098:
5076:
5072:
5068:
5065:
5062:
5057:
5053:
5049:
5044:
5040:
5017:
5012:
4990:
4970:
4967:
4964:
4944:
4941:
4938:
4931:
4926:
4920:
4897:
4892:
4870:
4850:
4843:
4838:
4832:
4811:
4808:
4803:
4798:
4775:
4771:
4768:
4765:
4762:
4740:
4735:
4730:
4727:
4703:
4700:
4697:
4675:
4670:
4647:
4626:
4621:
4616:
4611:
4608:
4605:
4602:
4598:
4594:
4591:
4564:
4559:
4554:
4549:
4546:
4543:
4540:
4536:
4532:
4529:
4507:
4502:
4480:
4459:
4438:
4435:
4432:
4411:
4407:
4402:
4397:
4392:
4389:
4367:
4362:
4348:
4345:
4332:
4308:
4305:
4292:
4288:
4284:
4281:
4278:
4258:
4255:
4252:
4228:
4225:
4222:
4189:
4186:
4183:
4180:
4177:
4174:
4170:
4147:
4144:
4141:
4138:
4135:
4132:
4129:
4126:
4123:
4119:
4098:
4094:
4090:
4087:
4084:
4064:
4055:The notion of
4053:
4052:
4051:
4050:
4039:
4035:
4029:
4024:
4021:
4017:
4013:
4010:
4005:
4001:
3995:
3992:
3988:
3984:
3979:
3976:
3972:
3968:
3964:
3958:
3955:
3951:
3947:
3942:
3937:
3931:
3928:
3925:
3922:
3919:
3916:
3913:
3910:
3907:
3903:
3899:
3894:
3891:
3888:
3885:
3882:
3879:
3876:
3873:
3870:
3866:
3861:
3856:
3851:
3848:
3845:
3842:
3839:
3836:
3832:
3818:
3817:
3805:
3793:
3792:
3791:
3790:
3779:
3775:
3769:
3764:
3761:
3757:
3753:
3750:
3745:
3741:
3735:
3732:
3728:
3724:
3719:
3716:
3712:
3708:
3704:
3698:
3694:
3688:
3685:
3681:
3677:
3672:
3669:
3665:
3661:
3656:
3653:
3649:
3645:
3640:
3635:
3629:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3596:
3592:
3587:
3582:
3579:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3554:
3549:
3544:
3539:
3536:
3533:
3530:
3527:
3524:
3520:
3506:
3505:
3493:
3481:
3480:
3479:
3478:
3464:
3460:
3454:
3450:
3446:
3442:
3438:
3433:
3429:
3425:
3420:
3416:
3410:
3406:
3402:
3398:
3394:
3389:
3385:
3360:
3348:
3347:
3346:
3334:
3326:
3321:
3318:
3314:
3310:
3307:
3302:
3298:
3292:
3289:
3285:
3281:
3276:
3273:
3269:
3265:
3260:
3255:
3252:
3248:
3244:
3239:
3236:
3232:
3227:
3221:
3216:
3212:
3206:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3173:
3159:
3158:
3147:
3146:
3135:
3132:
3129:
3126:
3123:
3118:
3115:
3111:
3107:
3104:
3101:
3098:
3095:
3092:
3087:
3083:
3079:
3076:
3073:
3068:
3065:
3061:
3057:
3054:
3051:
3046:
3042:
3010:
2999:
2998:
2987:
2984:
2981:
2978:
2975:
2972:
2967:
2964:
2960:
2956:
2953:
2950:
2947:
2944:
2941:
2936:
2932:
2928:
2925:
2922:
2917:
2914:
2910:
2906:
2903:
2900:
2895:
2891:
2867:
2864:
2861:
2858:
2855:
2852:
2849:
2846:
2825:
2822:
2820:for valleys.
2809:
2806:
2801:
2798:
2794:
2773:
2770:
2765:
2762:
2758:
2735:
2732:
2728:
2716:
2715:
2702:
2699:
2695:
2689:
2684:
2680:
2676:
2671:
2668:
2664:
2658:
2654:
2648:
2644:
2640:
2637:
2632:
2629:
2625:
2619:
2614:
2610:
2606:
2601:
2598:
2594:
2588:
2583:
2579:
2568:
2557:
2552:
2549:
2545:
2541:
2536:
2531:
2527:
2523:
2518:
2513:
2509:
2505:
2502:
2499:
2494:
2491:
2487:
2483:
2478:
2475:
2471:
2467:
2462:
2458:
2452:
2448:
2444:
2439:
2436:
2432:
2426:
2421:
2417:
2406:
2395:
2390:
2387:
2383:
2377:
2372:
2368:
2364:
2359:
2356:
2352:
2346:
2342:
2336:
2332:
2328:
2325:
2320:
2317:
2313:
2307:
2302:
2298:
2294:
2289:
2286:
2282:
2276:
2271:
2267:
2252:
2251:
2240:
2237:
2232:
2227:
2224:
2220:
2216:
2211:
2206:
2203:
2199:
2195:
2192:
2189:
2184:
2181:
2177:
2162:
2161:
2145:
2140:
2136:
2132:
2127:
2122:
2118:
2111:
2107:
2101:
2098:
2095:
2092:
2089:
2081:
2076:
2072:
2068:
2063:
2058:
2054:
2047:
2043:
2037:
2034:
2031:
2028:
2014:
2013:
2000:
1996:
1992:
1989:
1986:
1983:
1978:
1974:
1970:
1967:
1964:
1961:
1956:
1952:
1948:
1943:
1939:
1935:
1932:
1929:
1926:
1921:
1917:
1913:
1910:
1907:
1904:
1899:
1895:
1871:
1851:
1848:
1845:
1842:
1839:
1830:In terms of a
1828:
1827:
1816:
1812:
1806:
1803:
1799:
1794:
1790:
1786:
1780:
1777:
1773:
1768:
1764:
1761:
1758:
1753:
1750:
1746:
1742:
1739:
1736:
1731:
1727:
1703:
1683:
1680:
1677:
1674:
1671:
1668:
1657:
1656:
1645:
1641:
1635:
1632:
1628:
1623:
1619:
1615:
1609:
1606:
1602:
1597:
1593:
1590:
1587:
1582:
1579:
1575:
1571:
1568:
1565:
1560:
1556:
1532:
1512:
1509:
1506:
1503:
1500:
1497:
1486:
1485:
1470:
1461:
1456:
1453:
1449:
1445:
1442:
1437:
1433:
1427:
1424:
1420:
1416:
1411:
1408:
1404:
1400:
1393:
1390:
1386:
1382:
1377:
1374:
1370:
1363:
1360:
1356:
1350:
1347:
1340:
1335:
1332:
1328:
1324:
1321:
1318:
1315:
1312:
1309:
1306:
1283:
1274:
1269:
1266:
1262:
1258:
1255:
1250:
1246:
1240:
1237:
1233:
1229:
1224:
1221:
1217:
1213:
1206:
1203:
1199:
1195:
1190:
1187:
1183:
1176:
1173:
1169:
1163:
1160:
1153:
1150:
1147:
1144:
1119:
1116:
1112:
1097:
1096:
1083:
1079:
1075:
1072:
1069:
1066:
1061:
1057:
1053:
1050:
1047:
1044:
1039:
1035:
1031:
1026:
1022:
1018:
1015:
1012:
1009:
1004:
1000:
996:
993:
990:
987:
982:
978:
954:
940:
939:
926:
918:
915:
911:
907:
903:
900:
896:
892:
891:
886:
883:
879:
875:
871:
868:
864:
860:
859:
857:
852:
849:
836:Hessian matrix
817:
814:
810:
787:
784:
780:
768:
767:
753:
750:
746:
742:
737:
733:
729:
724:
720:
716:
713:
709:
702:
699:
696:
692:
687:
684:
681:
678:
675:
672:
669:
666:
663:
640:
637:
634:
631:
628:
625:
605:
602:
599:
596:
593:
590:
566:
546:
543:
540:
537:
534:
531:
519:
516:
508:retinal images
492:shape skeleton
450:image analysis
377:
376:
359:September 2012
338:
336:
329:
322:
321:
319:
318:
311:
304:
296:
293:
292:
291:
290:
285:
280:
272:
271:
265:
264:
263:
262:
257:
252:
247:
239:
238:
234:
233:
232:
231:
229:Hessian affine
226:
221:
213:
212:
208:
207:
206:
205:
200:
192:
191:
187:
186:
185:
184:
179:
171:
170:
166:
165:
159:
158:
157:
156:
151:
146:
141:
136:
128:
127:
125:Blob detection
121:
120:
119:
118:
113:
108:
103:
98:
96:Shi and Tomasi
93:
85:
84:
78:
77:
76:
75:
70:
65:
60:
55:
50:
45:
37:
36:
34:Edge detection
30:
29:
9:
6:
4:
3:
2:
8069:
8058:
8055:
8053:
8050:
8048:
8045:
8043:
8040:
8039:
8037:
8024:
8020:
8016:
8012:
8008:
8004:
8000:
7993:
7985:
7981:
7977:
7973:
7969:
7965:
7961:
7957:
7953:
7946:
7936:
7929:
7927:
7920:
7912:
7906:
7902:
7895:
7887:
7883:
7879:
7875:
7870:
7865:
7862:(2): 81–121.
7861:
7857:
7850:
7843:
7835:
7829:
7821:
7817:
7813:
7809:
7802:
7794:
7790:
7786:
7782:
7778:
7774:
7770:
7763:
7755:
7751:
7747:
7743:
7739:
7735:
7728:
7721:
7713:
7709:
7705:
7701:
7697:
7693:
7688:
7683:
7679:
7675:
7668:
7660:
7656:
7652:
7648:
7644:
7640:
7633:
7626:
7618:
7614:
7609:
7604:
7600:
7596:
7589:
7583:
7577:
7569:
7563:
7555:
7551:
7547:
7543:
7538:
7533:
7529:
7525:
7518:
7510:
7506:
7502:
7498:
7494:
7490:
7483:
7475:
7469:
7461:
7457:
7453:
7449:
7442:
7434:
7428:
7420:
7416:
7412:
7408:
7404:
7400:
7395:
7390:
7386:
7382:
7375:
7368:
7360:
7354:
7346:
7342:
7338:
7334:
7330:
7326:
7321:
7316:
7312:
7308:
7301:
7297:
7290:
7282:
7278:
7275:(10): 28–38.
7274:
7270:
7263:
7256:
7252:
7246:
7238:
7232:
7224:
7220:
7216:
7212:
7208:
7204:
7197:
7190:
7184:
7178:
7170:
7164:
7156:
7152:
7148:
7144:
7140:
7136:
7132:
7128:
7124:
7117:
7108:
7104:
7100:
7096:
7092:
7088:
7084:
7077:
7069:
7065:
7061:
7057:
7053:
7049:
7045:
7038:
7030:
7024:
7020:
7016:
7012:
7008:
7007:"Scale-space"
7001:
6994:
6978:
6959:
6957:
6948:
6944:
6940:
6936:
6932:
6928:
6921:
6919:
6914:
6905:
6902:
6900:
6897:
6895:
6892:
6890:
6887:
6885:
6882:
6880:
6877:
6876:
6870:
6868:
6849:
6846:
6841:
6838:
6835:
6831:
6824:
6819:
6815:
6811:
6806:
6803:
6800:
6796:
6789:
6784:
6780:
6773:
6769:
6764:
6761:
6756:
6753:
6750:
6746:
6739:
6735:
6728:
6723:
6719:
6714:
6711:
6706:
6703:
6700:
6696:
6689:
6684:
6680:
6676:
6671:
6668:
6665:
6661:
6655:
6650:
6646:
6638:
6637:
6636:
6619:
6616:
6613:
6608:
6605:
6601:
6594:
6589:
6585:
6581:
6576:
6573:
6569:
6562:
6558:
6551:
6547:
6542:
6539:
6534:
6531:
6527:
6520:
6515:
6511:
6507:
6502:
6499:
6495:
6489:
6484:
6480:
6472:
6471:
6470:
6454:
6451:
6448:
6444:
6421:
6417:
6394:
6390:
6365:
6362:
6354:
6350:
6341:
6338:
6326:
6325:
6324:
6308:
6304:
6283:
6260:
6257:
6249:
6245:
6236:
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2019:
2018:
2017:
1998:
1990:
1987:
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1968:
1965:
1962:
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1016:
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994:
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985:
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968:
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731:
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629:
623:
600:
597:
594:
588:
580:
564:
541:
538:
535:
529:
515:
513:
509:
505:
504:blood vessels
501:
500:aerial images
497:
496:binary images
493:
489:
485:
482:
478:
474:
469:
467:
463:
459:
455:
451:
447:
443:
437:
435:
431:
430:connector set
427:
426:local minimum
423:
419:
418:local maximum
415:
411:
406:
404:
400:
396:
392:
388:
384:
373:
370:
362:
352:
348:
342:
339:This article
337:
328:
327:
317:
312:
310:
305:
303:
298:
297:
295:
294:
289:
286:
284:
281:
279:
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273:
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267:
266:
261:
258:
256:
253:
251:
248:
246:
243:
242:
241:
240:
236:
235:
230:
227:
225:
224:Harris affine
222:
220:
217:
216:
215:
214:
210:
209:
204:
201:
199:
196:
195:
194:
193:
189:
188:
183:
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174:
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161:
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155:
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137:
135:
132:
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130:
129:
126:
123:
122:
117:
114:
112:
109:
107:
104:
102:
99:
97:
94:
92:
89:
88:
87:
86:
83:
80:
79:
74:
73:Roberts cross
71:
69:
66:
64:
61:
59:
56:
54:
51:
49:
46:
44:
41:
40:
39:
38:
35:
32:
31:
28:
25:
24:
19:
8006:
8002:
7992:
7959:
7955:
7945:
7935:
7924:
7923:Kerrel, R.
7919:
7900:
7894:
7859:
7855:
7842:
7828:cite journal
7811:
7807:
7801:
7776:
7772:
7762:
7737:
7733:
7720:
7677:
7673:
7667:
7645:(1): 23–31.
7642:
7638:
7625:
7598:
7594:
7588:
7576:
7562:cite journal
7530:(1): 55–71.
7527:
7523:
7517:
7492:
7488:
7482:
7468:cite journal
7451:
7447:
7441:
7427:cite journal
7384:
7380:
7367:
7353:cite journal
7310:
7306:
7296:Parker, A.C.
7289:
7272:
7268:
7262:
7254:
7245:
7231:cite journal
7206:
7202:
7189:
7177:
7163:cite journal
7130:
7126:
7116:
7090:
7086:
7076:
7051:
7047:
7037:
7010:
7000:
6963:
6930:
6926:
6864:
6634:
6381:
6275:
6120:
6113:
5736:
5727:
5724:
5715:
5535:
5502:
5500:
5441:
5419:
5212:
4717:
4581:
4576:
4350:
4310:
4240:
4212:
4054:
3351:
3152:
3148:
3026:
3000:
2836:
2830:
2827:
2717:
2253:
2163:
2015:
1829:
1658:
1487:
1101:
1098:
941:
769:
521:
470:
445:
441:
438:
433:
429:
421:
413:
409:
407:
399:local maxima
390:
386:
380:
365:
356:
340:
162:
53:Differential
7582:pp. 862–871
6962:Miller, J.
6904:Scale space
6174:-direction
832:eigenvalues
830:denote the
490:provides a
466:scale space
269:Scale space
8036:Categories
7903:. Kluwer.
6910:References
5213:The point
4910:, and let
4471:is within
458:watersheds
7864:CiteSeerX
7682:CiteSeerX
7603:CiteSeerX
7532:CiteSeerX
7389:CiteSeerX
7315:CiteSeerX
6847:≤
6363:≤
6335:∂
6233:∂
6011:⋅
6005:∇
5968:σ
5964:∂
5950:∂
5917:σ
5914:∂
5906:∂
5877:σ
5834:×
5819:⊂
5793:σ
5756:σ
5697:−
5688:…
5632:⋅
5611:∇
5577:−
5570:λ
5454:−
5401:−
5392:…
5336:⋅
5315:∇
5281:−
5274:λ
5192:λ
5071:λ
5067:≤
5064:⋯
5061:≤
5052:λ
5048:≤
5039:λ
5030:. Let
4966:×
4831:∇
4807:∈
4770:→
4729:⊂
4699:−
4491:units of
4479:δ
4431:δ
4406:→
4331:γ
4277:γ
4251:γ
4221:γ
4176:−
4173:γ
4134:−
4131:γ
4083:γ
4063:γ
3983:−
3957:γ
3918:−
3915:γ
3898:−
3881:−
3878:γ
3838:−
3835:γ
3804:γ
3723:−
3655:γ
3611:−
3608:γ
3591:−
3569:−
3566:γ
3526:−
3523:γ
3492:γ
3459:∂
3445:γ
3432:η
3428:∂
3415:∂
3401:γ
3388:ξ
3384:∂
3359:γ
3280:−
3259:−
3215:γ
3188:−
3185:γ
3131:≤
3110:∂
3082:∂
3072:≥
2980:≤
2959:∂
2931:∂
2921:≤
2522:−
2501:−
2482:−
2324:−
2236:≥
2215:−
2097:α
2094:
2033:α
2030:
1995:∂
1991:α
1988:
1973:∂
1969:α
1966:
1951:∂
1938:∂
1934:α
1931:
1925:−
1916:∂
1912:α
1909:
1894:∂
1789:≥
1757:≥
1694:at scale
1618:≥
1586:≤
1415:−
1381:−
1362:−
1320:
1311:β
1308:
1228:−
1194:−
1149:β
1146:
1078:∂
1074:β
1071:
1056:∂
1052:β
1049:
1034:∂
1021:∂
1017:β
1014:
1008:−
999:∂
995:β
992:
977:∂
712:−
698:π
8023:16396756
7984:13284142
7976:21869365
7940:365–368.
7754:10646760
7704:10628954
7419:14999508
7411:21869381
7345:14348919
7337:21869180
7183:423–428.
7155:18262941
7107:35328443
6947:10121282
6873:See also
5154:and let
4320:pyramids
4313:Haralick
484:skeleton
288:Pyramids
68:Robinson
7793:3727523
7712:6263198
7659:2561801
7509:9940964
7223:1255196
7135:Bibcode
7056:Bibcode
5858:. The
5089:be the
4955:be the
4822:. Let
4520:, then
4307:History
2254:where
942:of the
834:of the
577:be the
422:valleys
345:Please
63:Prewitt
48:Deriche
8021:
7982:
7974:
7907:
7886:810806
7884:
7866:
7791:
7752:
7710:
7702:
7684:
7657:
7605:
7554:676717
7552:
7534:
7507:
7417:
7409:
7391:
7343:
7335:
7317:
7221:
7153:
7105:
7025:
6945:
5730:, the
3001:where
2016:where
403:ridges
393:in an
391:ridges
8019:S2CID
7980:S2CID
7882:S2CID
7852:(PDF)
7789:S2CID
7730:(PDF)
7708:S2CID
7655:S2CID
7635:(PDF)
7550:S2CID
7505:S2CID
7415:S2CID
7377:(PDF)
7341:S2CID
7303:(PDF)
7219:S2CID
7199:(PDF)
7103:S2CID
6943:S2CID
5995:, and
5600:, and
5304:, and
3373:with
395:image
111:SUSAN
58:Sobel
43:Canny
7972:PMID
7905:ISBN
7834:link
7750:PMID
7700:PMID
7568:link
7474:link
7433:link
7407:PMID
7359:link
7333:PMID
7237:link
7169:link
7151:PMID
7023:ISBN
6436:...
6090:<
6041:and
5980:<
5938:and
5728:e.g.
5662:for
5585:<
5366:for
5289:<
4637:for
4604:<
4577:i.e.
4542:<
4434:>
4206:and
2837:Let
2805:>
2769:<
800:and
522:Let
494:for
452:and
444:and
255:GLOH
250:SURF
245:SIFT
154:PCBR
116:FAST
8011:doi
7964:doi
7874:doi
7816:doi
7781:doi
7742:doi
7692:doi
7647:doi
7613:doi
7542:doi
7497:doi
7456:doi
7399:doi
7325:doi
7277:doi
7253:",
7211:doi
7143:doi
7095:doi
7064:doi
7015:doi
6935:doi
5001:at
4881:at
2091:sin
2027:cos
1985:sin
1963:cos
1928:cos
1906:sin
1317:sgn
1305:sin
1143:cos
1068:sin
1046:cos
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