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OpenSubdiv/documentation/subdivision_surfaces.rst
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- Hierarchical edits descriptions
- Uniform vs Adaptive feature comparison
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..
Copyright 2013 Pixar
Licensed under the Apache License, Version 2.0 (the "Apache License")
with the following modification; you may not use this file except in
compliance with the Apache License and the following modification to it:
Section 6. Trademarks. is deleted and replaced with:
6. Trademarks. This License does not grant permission to use the trade
names, trademarks, service marks, or product names of the Licensor
and its affiliates, except as required to comply with Section 4(c) of
the License and to reproduce the content of the NOTICE file.
You may obtain a copy of the Apache License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the Apache License with the above modification is
distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
KIND, either express or implied. See the Apache License for the specific
language governing permissions and limitations under the Apache License.
Subdivision Surfaces
--------------------
.. contents::
:local:
:backlinks: none
----
Introduction
============
The most common way to model complex smooth surfaces is by using a patchwork of
bicubic patches such as BSplines or NURBS.
.. image:: images/torus.png
:align: center
:height: 200
However, while they do provide a reliable smooth limit surface definition, bicubic
patch surfaces are limited to 2-dimensional topologies, which only describes a
very small fraction of real-world shapes. This fundamental parametric limitation
requires authoring tools to implementat at least the following functionalities:
- smooth trimming
- seams stitching
Both trimming and stitching need to guarantee the smoothness of the model both
spatially and temporally as the model is animated. Attempting to meet these
requirements introduces a lot of expensive computations and complexity.
Subdivision surfaces on the other hand can represent arbitrary topologies, and
therefore are not constrained by these difficulties.
----
Arbitrary Topology
==================
A subdivision surface, like a parametric surface, is described by its control mesh
of points. The surface itself can approximate or interpolate this control mesh
while being piecewise smooth. But where polygonal surfaces require large numbers
of data points to approximate being smooth, a subdivision surface is smooth -
meaning that polygonal artifacts are never present, no matter how the surface
animates or how closely it is viewed.
Ordinary cubic B-spline surfaces are rectangular grids of tensor-product patches.
Subdivision surfaces generalize these to control grids with arbitrary connectivity.
.. raw:: html
<center>
<p align="center">
<IMG src="images/tetra.0.jpg" style="width: 20%;">
<IMG src="images/tetra.1.jpg" style="width: 20%;">
<IMG src="images/tetra.2.jpg" style="width: 20%;">
<IMG src="images/tetra.3.jpg" style="width: 20%;">
</p>
</center>
----
Manifold Geometry
*****************
Continuous limit surfaces require that the topology be a two-dimensional
manifold. It is therefore possible to model non-manifold geometry that cannot
be represented with a smooth C2 continuous limit. The following examples show
typical cases of non-manifold topological configurations.
----
Non-Manifold Fan
++++++++++++++++
This "fan" configuration shows an edge shared by 3 distinct faces.
.. image:: images/nonmanifold_fan.png
:align: center
:target: images/nonmanifold_fan.png
With this configuration, it is unclear which face should contribute to the
limit surface, as 3 of them share the same edge (which incidentally breaks
half-edge cycles in said data-structures). Fan configurations are not limited
to 3 incident faces: any configuration where an edge is shared by more than
2 faces incurs the same problem.
----
Non-Manifold Disconnected Vertex
++++++++++++++++++++++++++++++++
A vertex is disconnected from any edge and face.
.. image:: images/nonmanifold_vert.png
:align: center
:target: images/nonmanifold_vert.png
This case is fairly trivial: there is no possible way to exact a limit surface here,
so the vertex simply has to be flagged as non-contributing, or discarded gracefully.
----
Boundary Interpolation Rules
============================
Boundary interpolation rules control how boundary face edges and facevarying data
are interpolated.
Vertex Data
***********
The following rule sets can be applied to vertex data interpolation:
+------------------------+----------------------------------------------------------+
| Mode | Behavior |
+========================+==========================================================+
| 0 - **None** | No boundary interpolation behavior should occur |
| | (debug mode - boundaries are undefined) |
+------------------------+----------------------------------------------------------+
| 1 - **EdgeAndCorner** | All the boundary edge-chains are sharp creases and |
| | boundary vertices with exactly two incident edges are |
| | sharp corners |
+------------------------+----------------------------------------------------------+
| 2 - **EdgeOnly** | All the boundary edge-chains are sharp creases; boundary |
| | vertices are not affected |
| | |
+------------------------+----------------------------------------------------------+
On a quad example:
.. image:: images/vertex_boundary.png
:align: center
:target: images/vertex_boundary.png
Facevarying Data
****************
The following rule sets can be applied to facevarying data interpolation:
+--------+----------------------------------------------------------+
| Mode | Behavior |
+========+==========================================================+
| 0 | Bilinear interpolation (no smoothing) |
+--------+----------------------------------------------------------+
| 1 | Smooth UV |
| | |
| | |
+--------+----------------------------------------------------------+
| 2 | Same as (1) but does not infer the presence of corners |
| | where two facevarying edges meet at a single faceA |
| | |
+--------+----------------------------------------------------------+
| 3 | Smooths facevarying values only near vertices that are |
| | not at a discontinuous boundary; all vertices on a |
| | discontinuous boundary are subdivided with a sharp rule |
| | (interpolated through). |
| | This mode is designed to be compatible with ZBrush and |
| | Maya's "smooth internal only" interpolation. |
+--------+----------------------------------------------------------+
Unwrapped cube example:
.. image:: images/fvar_boundaries.png
:align: center
:target: images/fvar_boundaries.png
Propagate Corners
+++++++++++++++++
Facevarying interpolation mode 2 (*EdgeAndCorner*) can further be modified by the
application of the *Propagate Corner* flag.
----
Semi-Sharp Creases
==================
It is possible to modify the subdivision rules to create piecewise smooth surfaces
containing infinitely sharp features such as creases and corners. As a special
case, surfaces can be made to interpolate their boundaries by tagging their boundary
edges as sharp.
However, we've recognized that real world surfaces never really have infinitely
sharp edges, especially when viewed sufficiently close. To this end, we've added
the notion of semi-sharp creases, i.e. rounded creases of controllable sharpness.
These allow you to create features that are more akin to fillets and blends. As
you tag edges and edge chains as creases, you also supply a sharpness value that
ranges from 0-10, with sharpness values >=10 treated as infinitely sharp.
It should be noted that infinitely sharp creases are really tangent discontinuities
in the surface, implying that the geometric normals are also discontinuous there.
Therefore, displacing along the normal will likely tear apart the surface along
the crease. If you really want to displace a surface at a crease, it may be better
to make the crease semi-sharp.
.. image:: images/gtruck.jpg
:align: center
:height: 300
:target: images/gtruck.jpg
----
Chaikin Rule
============
Chaikin's curve subdivision algorithm improves the appearance of multi-edge
semi-sharp creases with vayring weights. The Chaikin rule interpolates the
sharpness of incident edges.
.. image:: images/chaikin.png
:align: center
:target: images/chaikin.png
----
Hierarchical Edits
==================
To understand the hierarchical aspect of subdivision, we realize that subdivision
itself leads to a natural hierarchy: after the first level of subdivision, each
face in a subdivision mesh subdivides to four quads (in the Catmull-Clark scheme),
or four triangles (in the Loop scheme). This creates a parent and child relationship
between the original face and the resulting four subdivided faces, which in turn
leads to a hierarchy of subdivision as each child in turn subdivides. A hierarchical
edit is an edit made to any one of the faces, edges, or vertices that arise anywhere
during subdivision. Normally these subdivision components inherit values from their
parents based on a set of subdivision rules that depend on the subdivision scheme.
A hierarchical edit overrides these values. This allows for a compact specification
of localized detail on a subdivision surface, without having to express information
about the rest of the subdivision surface at the same level of detail.
.. image:: images/hedit_example1.png
:align: center
:height: 300
:target: images/hedit_example1.png
----
Hierarchical Edits Paths
************************
In order to perform a hierarchical edit, we need to be able to name the subdivision
component we are interested in, no matter where it may occur in the subdivision
hierarchy. This leads us to a hierarchical path specification for faces, since
once we have a face we can navigate to an incident edge or vertex by association.
We note that in a subdivision mesh, a face always has incident vertices, which are
labelled (in relation to the face) with an integer index starting at zero and in
consecutive order according to the usual winding rules for subdivision surfaces.
Faces also have incident edges, and these are labelled according to the origin
vertex of the edge.
.. image:: images/face_winding.png
:align: center
:target: images/face_winding.png
.. role:: red
.. role:: green
.. role:: blue
In this diagram, the indices of the vertices of the base face are marked in :red:`red`;
so on the left we have an extraordinary Catmull-Clark face with five vertices
(labeled :red:`0-4`) and on the right we have a regular Catmull-Clark face with four
vertices (labelled :red:`0-3`). The indices of the child faces are :blue:`blue`; note that in
both the extraordinary and regular cases, the child faces are indexed the same
way, i.e. the subface labeled :blue:`n` has one incident vertex that is the result of the
subdivision of the parent vertex also labeled :red:`n` in the parent face. Specifically,
we note that the subface :blue:`1` in both the regular and extraordinary face is nearest
to the vertex labelled :red:`1` in the parent.
The indices of the vertices of the child faces are labeled :green:`green`, and
this is where the difference lies between the extraordinary and regular case;
in the extraordinary case, vertex to vertex subdivision always results in a vertex
labeled :green:`0`, while in the regular case, vertex to vertex subdivision
assigns the same index to the child vertex. Again, specifically, we note that the
parent vertex indexed :red:`1` in the extraordinary case has a child vertex :green:`0`,
while in the regular case the parent vertex indexed :red:`1` actually has a child
vertex that is indexed :green:`1`. Note that this indexing scheme was chosen to
maintain the property that the vertex labeled 0 always has the lowest u/v
parametric value on the face.
.. image:: images/hedit_path.gif
:align: center
:target: images/hedit_path.gif
By appending a vertex index to a face index, we can create a vertex path
specification. For example, (:blue:`655` :green:`2` :red:`3` 0) specifies the 1st.
vertex of the :red:`3` rd. child face of the :green:`2` nd. child face of the of
the :blue:`655` th. face of the subdivision mesh.
----
Vertex Edits
************
Vertex hierarchical edits can modify the value or the sharpness of primitive variables for vertices
and sub-vertices anywhere in the subdivision hierarchy.
.. image:: images/hedit_example1.png
:align: center
:height: 300
:target: images/hedit_example1.png
The edits are performed using either an "add" or a "set" operator. "set" indicates the primitive
variable value or sharpness is to be set directly to the values specified. "add" adds a value to the
normal result computed via standard subdivision rules. In other words, this operation allows value
offsets to be applied to the mesh at any level of the hierarchy.
.. image:: images/hedit_example2.png
:align: center
:height: 300
:target: images/hedit_example2.png
----
Edge Edits
**********
Edge hierarchical edits can only modify the sharpness of primitive variables for edges
and sub-edges anywhere in the subdivision hierarchy.
.. image:: images/hedit_example4.png
:align: center
:height: 300
:target: images/hedit_example4.png
----
Face Edits
**********
Face hierarchical edits can modify several properties of faces and sub-faces anywhere in the
subdivision hierarchy.
Modifiable properties include:
* The "set" or "add" operators modify the value of primitive variables associated with faces.
* The "hole" operation introduces holes (missing faces) into the subdivision mesh at any
level in the subdivision hierarchy. The faces will be deleted, and none of their children
will appear (you cannot "unhole" a face if any ancestor is a "hole"). This operation takes
no float or string arguments.
.. image:: images/hedit_example5.png
:align: center
:height: 300
:target: images/hedit_example5.png
----
Limitations
***********
XXXX
----
Uniform Subdivision
===================
Applies a uniform refinement scheme to the coarse faces of a mesh. This is the most
common solution employed to apply subdivision schemes to a control cage. The mesh
converges closer to the limit surface with each iteration of the algorithm.
.. image:: images/uniform.gif
:align: center
:width: 300
:target: images/uniform.gif
----
Feature Adaptive Subdivision
============================
Generates bi-cubic patches on the limit surface and applies a progressive refinement
scheme in order to isolate non-C2 continuous extraordinary features.
.. image:: images/adaptive.gif
:align: center
:width: 300
:target: images/adaptive.gif
----
Uniform or Adaptive ?
=====================
Main features comparison:
+-------------------------------------------------------+--------------------------------------------------------+
| Uniform | Feature Adaptive |
+=======================================================+========================================================+
| | |
| * Bilinear approximation | * Bicubic limit patches |
| * No tangents / no normals | * Analytical tangents / normals |
| * No smooth shading around creases | |
| * No animated displacements | |
| | |
+-------------------------------------------------------+--------------------------------------------------------+
| * Exponential geometry Growth | * Feature isolation growth close to linear |
| | |
+-------------------------------------------------------+--------------------------------------------------------+
| * Boundary interpolation rules supported: | * Boundary interpolation rules supported: |
| * All vertex & varying rules supported dynamically| * All vertex & varying rules supported dynamically |
| * All face-varying rules supported \ | * Bilinear face-varying interpolation \ |
| statically at vertex locations (there is no \ | supported statically |
| surface limit) | * Bi-cubic face-varying interpolation \ |
| | currently not supported |
| | |
+-------------------------------------------------------+--------------------------------------------------------+
| * No GPU shading implications | * Requires GPU composable shading |
| | |
+-------------------------------------------------------+--------------------------------------------------------+
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