Convolution Surface

A convolution surface is an implicitly represented surface in computer graphics that is described by a control skeleton. This control skeleton can be points , edges or polygons . The surface envelops the skeleton and connects fluently at points where control elements meet.

Convolution Surfaces were introduced in 1991 by Jules Bloomenthal and Ken Shoemake as an extension to Jim Blinn's Blobby Molecules and later adaptations of it, such as Metaballs .

definition

Convolution surfaces are a mixture of potential surfaces and distance surfaces. Distance surfaces describe surfaces with the help of functions that depend on the distance between surface points and control points. Potential surfaces, like metaballs, are formed by the sum of field functions : ${\ displaystyle f_ {i}}$

{\ displaystyle F (\ mathbf {p}) = \ sum _ {i = 1} ^ {N} f_ {i} (\ mathbf {p})}

With the help of the field , an isosurface can be created, a surface that includes all points where a certain threshold value corresponds. ${\ displaystyle F (\ mathbf {p})}$ ${\ displaystyle \ mathbf {p}}$ ${\ displaystyle \ mathbf {p}}$

The field function of a convolution surface consists of the convolution of a geometry function (skeleton) and a core , which corresponds to a volume integral over the entire three-dimensional space.

{\ displaystyle f (\ mathbf {p}) = g (\ mathbf {p}) \ star h (\ mathbf {p}) = \ int _ {R ^ {3}} g (\ mathbf {r}) h (\ mathbf {p} - \ mathbf {r}) d \ mathbf {r}}

The core is a function that corresponds to the field function of a single control point, such as the field function of a metal ball. The kernel originally used corresponds to a Gaussian function , where the distance from the point to the control point is. ${\ displaystyle R ^ {3} \ rightarrow R}$ ${\ displaystyle h (r) = e ^ {(- a ^ {2} r ^ {2})}}$ ${\ displaystyle r}$ ${\ displaystyle \ mathbf {p}}$

A convolution surface, the control skeleton of which consists only of points, is a potential surface.

application

Convolution surfaces are used in the visualization of blood vessels.

In sketch-based modeling , a modeling method in which 2D sketches are automatically converted into 3D objects, convolution surfaces are also used.

literature

J. Bloomenthal, K. Shoemake: Convolution surfaces. In: ACM SIGGRAPH Computer Graphics. Vol. 25, No. 4, 1991, pp. 251-256. ISSN 0097-8930 .
A. Sherstyuk: Convolution surfaces in computer graphics. Doctoral dissertation. Monash University Australia, 1998.
C. Bajaj (Ed.): Introduction to implicit surfaces. Morgan Kaufmann, 1997, ISBN 1-55860-233-X .

Individual evidence

↑ Steffen Oeltze, Bernhard Preim: Visualization of vascular systems with convolution surfaces. In: Image processing for medicine 2004. Springer, Berlin / Heidelberg 2004, ISBN 3-540-21059-8 , pp. 189–193.
↑ Guillaume Pizaine et al: Vessel geometry modeling and segmentation using convolution surfaces and an implicit medial axis. In: Biomedical Imaging: From Nano to Macro. 2011 IEEE International Symposium on. IEEE. 2011, pp. 1421-1424. ISSN 1945-7928
^ A. Alexe, L. Barthe, MP Cani, V. Gaildrat: Shape modeling by sketching using convolution surfaces. In: Sara McMains: ACM SIGGRAPH 2007 courses. ACM, 2007, ISBN 978-1-4503-1823-5 , p. 39.

Web links

Visualized marine animals using convolution surfaces ( Memento from May 29, 2015 in the Internet Archive )

[1] Steffen Oeltze, Bernhard Preim: Visualization of vascular systems with convolution surfaces. In: Image processing for medicine 2004. Springer, Berlin / Heidelberg 2004, ISBN 3-540-21059-8 , pp. 189–193.

[2] Guillaume Pizaine et al: Vessel geometry modeling and segmentation using convolution surfaces and an implicit medial axis. In: Biomedical Imaging: From Nano to Macro. 2011 IEEE International Symposium on. IEEE. 2011, pp. 1421-1424. ISSN 1945-7928

[3] A. Alexe, L. Barthe, MP Cani, V. Gaildrat: Shape modeling by sketching using convolution surfaces. In: Sara McMains: ACM SIGGRAPH 2007 courses. ACM, 2007, ISBN 978-1-4503-1823-5 , p. 39.