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43:(a) A simple 3d object. (b) Its medial axis transform. The colors represent the distance from the medial axis to the object's boundary.
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of an object is the set of all points having more than one closest point on the object's boundary. Originally referred to as the
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is a tree whose leaves are the vertices of the polygon, and whose edges are either straight segments or arcs of parabolas.
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for physical models, and for dimensional reduction of complex models. In any dimension, the medial axis of a bounded
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The medial axis together with the associated radius function of the maximally inscribed discs is called the
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Leymarie, Frederic F.; Kimia, Benjamin B. (2008). "From the
Infinitely Large to the Infinitely Small".
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556:(1967). "A transformation for extracting new descriptors of shape". In Wathen-Dunn, Weiant (ed.).
399:{\displaystyle (c-\gamma (s))\cdot {\underline {T}}(s)=(c-\gamma (t))\cdot {\underline {T}}(t)=0,}
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Tagliasacchi, Andrea; Delame, Thomas; Spagnuolo, Michela; Amenta, Nina; Telea, Alexandru (2016).
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35:, a super-set of the medial axis, is the green and yellow curves. One bi-tangent circle is shown.
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is the unit tangent vector at each point. Then there will be a bitangent circle with center
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For most curves, the symmetry set will form a one-dimensional curve and can contain
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and object recognition, while the 3D medial axis has applications in
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is the locus of the centers of circles that are tangent to curve
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in two or more points, where all such circles are contained in
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Lieutier, André (September 2004). "Any open bounded subset of
535:– which can be regarded as a discrete form of the medial axis.
90:. (It follows that the medial axis itself is contained in
149:-dimension hyperspheres. The 2D medial axis is useful for
565:. Cambridge, Massachusetts: MIT Press. pp. 362–380.
208:{\displaystyle \gamma :\mathbf {R} \to \mathbf {R} ^{2}}
145:-dimensional hypersurfaces by replacing 2D circles with
500:. The symmetry set has end points corresponding to the
267:{\displaystyle {\underline {T}}(t)={d\gamma \over dt}}
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Models for the
Perception of Speech and Visual Form
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486:{\displaystyle |c-\gamma (s)|=|c-\gamma (t)|=r.\,}
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113:), meaning that it can be used to reconstruct the
732:– Straight Skeleton builder implemented in java.
605:has the same homotopy type as its medial axis".
16:Points with more than one closest boundary point
134:generally extends to infinity, similar to the
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674:"3D Skeletons: A State-of-the-Art Report"
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31:(blue), and its medial axis (green). The
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726:– a generalization of the medial axis
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641:. Dordrecht: Springer Netherlands.
120:The medial axis is a subset of the
74:In 2D, the medial axis of a subset
67:of the medial axis is known as the
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78:which is bounded by planar curve
639:Computational Imaging and Vision
598:{\displaystyle \mathbb {R} ^{n}}
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63:recognition. In mathematics the
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59:as a tool for biological
724:The Scale Axis Transform
117:of the original domain.
94:.) The medial axis of a
681:Computer Graphics Forum
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155:surface reconstruction
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687:(2). Wiley: 573–597.
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103:medial axis transform
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53:topological skeleton
518:Grassfire transform
163:homotopy equivalent
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523:Local feature size
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165:to the given set.
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693:10.1111/cgf.12865
656:978-1-4020-8657-1
613:(11): 1029–1046.
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33:symmetry set
554:Blum, Harry
278:and radius
49:medial axis
27:(red), its
540:References
57:Harry Blum
701:0167-7055
665:1381-6446
457:γ
454:−
426:γ
423:−
374:_
366:⋅
351:γ
348:−
325:_
317:⋅
302:γ
299:−
251:γ
228:_
191:→
180:γ
151:character
69:cut locus
740:Category
512:See also
502:vertices
159:open set
709:5740454
65:closure
29:evolute
25:ellipse
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215:, and
705:S2CID
677:(PDF)
563:(PDF)
498:cusps
115:shape
61:shape
697:ISSN
661:ISSN
651:ISBN
282:if
47:The
689:doi
643:doi
615:doi
504:of
168:If
161:is
107:MAT
23:An
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187:R
183::
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147:k
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128:S
105:(
92:S
88:S
84:C
80:C
76:S
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