1
Variants 1-2

In this chapter, we introduce the classical Whitney extension problem. Thereafter, we introduce the near distorted Whitney extension problem and two variants of it. The first, via a purely harmonic analysis problem and the second, translated into a problem related to non-rigid alignment and interpolation of data in . We discuss the Procrustes rigid alignment problem.

1.1 The Whitney Extension Problem

Given a real valued function on an arbitrary compact set in , the classical Whitney extension problem asks how can one decide whether extends to a function in , the space of real valued functions on whose derivatives of order are continuous and bounded? Whitney [114, 115] first studied this problem in 1934. He solved the real line case () and proved the classic Whitney extension theorem. See [64, 63] and the references cited therein for an interesting account of this problem.

1.2 Variants (1-2) [39, 40]

Problems (1-2) are examples of Variants (1-2).

Problem 1. Let us be given a positive constant small enough depending on . Does there exist a positive constant small enough depending on so that the following holds? Given two sets of distinct points in , and . Suppose for every ,

• (1) Does there exist a -distortion which obeys ?
• (2) Is it possible that at the same time agrees with Euclidean motions as well?
• (3) Can one say something about how are related?

Problem 2. Let us be given a positive constant small enough depending on . Does there exist a positive constant small enough depending on so that the following holds? Given two sets of distinct labeled points in , and . Suppose for every , (1.1) holds.

• (1) Is it possible to find a Euclidean motion which obeys is close to for every . Here close depends on and the points and is measured in the Euclidean norm.
• (2) Can one say something about how are related?

Remark 1. A central remark, at this juncture, is needed moving forward. Problem 1 and Problem 2 are fundamentally different in the sense that Problem 1 is a problem dealing with the existence of extensions. Problem 2 does not ask for an extension. It asks for a Euclidean motion only. This fact translates itself in many ways, for example in how the constants relate to each other.

In the case of isometry, Remark 1 is far less subtle: indeed, the following result is well-known, see for example [112].

Let and be two collections of distinct points in . Suppose that the pairwise distances between the points are equal, that is, the two sets of points are isometric. That is

Then, there exists a Euclidean motion, with

1.3 Variant 2

In this section, we provide some perspective on Variant 2. We briefly discuss the interpolation problem in the sense of manifold learning in Chapter 4.

1.4 Visual Object Recognition and an Equivalence Problem in

Visual object recognition is the ability to perceive properties (such as shape, color and texture) of a visual object in and to apply semantic attributes to it (such as identifying the visual object). This process includes the understanding of the visual object's use, previous experience with the visual object, and how it relates to the containing space . Regardless of the object's position or illumination, the ability to effectively identify an object, makes the object a "visual" object.

One significant aspect of visual object recognition is the ability to recognize a visual object across varying viewing conditions. These varying conditions include object orientation, lighting, object variability, for example, size, color, and other within-category differences to name just a few. Visual object recognition includes viewpoint-invariant, viewpoint-dependent and multiple view theories to name just a few examples. Visual information gained from an object is often divided into simple geometric components, then matched with the most similar visual object representation that is stored in its memory to provide the object's identification. See the following references and the many cited therein [1, 2, 4-6, 19, 39-42, 59-62, 66, 67, 69, 75, 82, 85-88, 91-93, 100, 101, 107, 111, 118, 120].

With this in mind, we define what we mean by an equivalence problem in . Imagine we are given two visual objects and in . An equivalence and symmetry function ?:??, when well defined, is an element of a group. See [98].

Some examples of vision maps are:

• Affine functions: a function is an affine function if there exists a linear transformation and so that for every , . Affine functions preserve area (volume) ratios. If is invertible (i.e., is then invertible affine), then is either proper or improper. If is not invertible, the function is neither proper nor improper.
• Euclidean motions.
• Reflections: a reflection is an isometry with a hyperplane as a set of fixed points.
• Similarity functions: a Euclidean motion plus a scaling. Similarity functions preserve length ratios.
• Projective motions, with

• Camera rotations, projective orthogonal transformation:

  .
• Motion tracking (video group). .
  • Here, in (e-g), are certain real constants. See for example [98].

1.5 Procrustes: The Rigid Alignment Problem

The recent advances in ?=? data acquisition and the increasing interest in augmented and virtual reality have led to an explosion of volumetric data, as exemplified by point clouds. Point cloud data is prevalent in numerous applications, including robotics, autonomous driving, medical imaging, neuroscience, social science and computer graphics. In many of these applications, the captured point clouds correspond to noisy observations of an object/scene undergoing different deformations. One of the core challenges in these applications is to perform point cloud registration, which refers to finding a transformation that aligns or partially aligns the source and target point sets. At a high level, any point cloud registration algorithm must solve two problems: 1) finding accurate correspondences between the points in the source and target point clouds (implicitly or explicitly), and 2) modeling the deformation to match corresponding source and target points. The existing methods then propose different correspondence estimation algorithms and/or propose novel deformation modeling approaches.

The registration/deformation map (i.e., the transformation) could be rigid, or non-rigid. Most of the existing works in the literature have focused on rigid registration of point clouds, as it is a more prevalent problem in classic computer vision tasks like simultaneous localization and mapping (SLAM). The core innovations in these approaches are often with regards to finding the right correspondences between the points. For instance, the classic iterative closest point (ICP) algorithm relies on nearest-neighbor correspondences as measured via the Euclidean distance between points. This section deals with rigid matching. See [1]. The best way to understand the rigid alignment problem is via Procrustes alignment. The classical rigid Procrustes problem is to find a rigid motion that best aligns two given point-sets in the least-squares sense. More precisely, given two sets of distinct points in , say and . The rigid Procrustes problem is the optimization problem

where is the norm. Here, we recall that is the orthogonal group of orthogonal matrices. The alignment is label-wise. Unlabeled problems are challenging, given it is often unclear which point to map to which. See for example our work in Chapter 21.

The rigid Procrustes optimization problem (See Figures 1.1-1.2), has a closed form solution obtained by applying a singular value decomposition...