On Mesh Editing, Manifold Learning, and Diffusion Wavelets
citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.604.6871...
by RM Rustamov - Cited by 9 - Related articles
Second, there is an interpretation of learningregularized manifold schemes which translates into a remarkable reading of Laplacian editing. To explain, we.
On Mesh Editing, Manifold Learning, and
Diffusion Wavelets
Raif M. Rustamov
Drew University, Madison NJ 07940, USA
rrustamov@drew.edu
Abstract. We spell out a formal equivalence between the naive Laplacian
editing and semi-supervised learning by bi-Laplacian Regularized
Least Squares. This allows us to write the solution to Laplacian mesh
editing in a closed form, based on which we introduce the Generalized
Linear Editing (GLE). GLE has both naive Laplacian editing and gradient
based editing as special cases. GLE allows using diffusion wavelets for
mesh editing. We present preliminary experiments, and shortly discuss
connections to segmentation.
1 Introduction
A remarkable similarity exists between semi-supervised manifold learning and
mesh editing: both seek to extrapolate data attached at some points to the whole
manifold.
Given a set of labeled samples, extrapolating labels throughout the entire
sample space is the task of semi-supervised learning. The qualification “manifold”
is added if the samples are assumed to belong to a manifold embedded into
a high-dimensional space – attaching labels is equivalent to defining a function
on this manifold.
Editing a mesh involves determining the new locations of vertices given new
locations of some of the vertices – the handles. The displacement vectors – the
differences between new and old vertex positions – can be considered to define a
function on the mesh. Thus, given the values of this function at the handles we
are trying to extrapolate to the whole mesh – a task that would otherwise qualify
as semi-supervised learning. If rotations at handles are also given, propagating
them throughout the mesh is again an instance of semi-supervised learning.
Does this similarity of the two fields extend beyond the objectives sought?
Laplacian based approaches to mesh editing start by extracting the surface’s
differential coordinates, and then reconstruct the surface by imposing the handle
constraints and requiring that the differential coordinates are preserved as much
as possible. The differential coordinates capture the local detail, so the more
they are preserved, the more the shape is preserved. When viewed from this
angle, Laplacian mesh editing seems to bear no resemblance to the methods of
semi-supervised learning.
Semi-supervised learning seeks a function f ∗ that minimizes X l i=1 (f(pi) − di) 2 + βkfk 2 E + γkfk 2 I . (1) The first term tries to enforce the function to take values prescribed at labeled vertices. The last two terms are called regularization terms, and they aim to make the function smooth in extrinsic and intrinsic senses. To clarify, extrinsic smoothness could mean that function takes close values at points that are close in Euclidean space; intrinsic smoothness would mean close function values for points that are geodetically close. We will be mostly interested in the first and last terms, dropping the middle terms of (1) in what follows.
Semi-supervised learning seeks a function f ∗ that minimizes X l i=1 (f(pi) − di) 2 + βkfk 2 E + γkfk 2 I . (1) The first term tries to enforce the function to take values prescribed at labeled vertices. The last two terms are called regularization terms, and they aim to make the function smooth in extrinsic and intrinsic senses. To clarify, extrinsic smoothness could mean that function takes close values at points that are close in Euclidean space; intrinsic smoothness would mean close function values for points that are geodetically close. We will be mostly interested in the first and last terms, dropping the middle terms of (1) in what follows.
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citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.604.6871...
by RM Rustamov - Cited by 9 - Related articles
Second, there is an interpretation of regularized manifold learning schemes which translates into a remarkable reading of Laplacian editing. To explain, we.
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