Measure of Image of Set Under Continuous Function

Continuous Image

A continuous image of an Eberlein compact is an Eberlein compact, a continuous image of a uniform Eberlein compact is a uniform Eberlein compact, a continuous image of a Corson compact is a Corson compact and a continuous image of a scattered compact is a scattered compact.

From: Handbook of the Geometry of Banach Spaces , 2003

Geometric Shortest Paths and Network Optimization

Joseph S.B. Mitchell , in Handbook of Computational Geometry, 2000

1.3 Geometric preliminaries

Throughout the survey, we will have need of some basic terminology, which we outline in this section.

First, a path is a continuous image of an interval. A polygonal s-t path is a path from point s to point t consisting of a finite number of line segments (edges, or links) joining a sequence of points (vertices).

The length of an s-t path is a nonnegative number associated with the path, measuring its total cost according to some prescribed metric. Unless otherwise specified, the length will be the Euclidean length of the path.

A shortest path is then a path of minimum length among all paths that are feasible (satisfying all imposed constraints). We often refer to a shortest path also as an "optimal path" or a "geodesic path". (The word "geodesic" is sometimes used differently, to refer to paths that are "locally optimal", as defined below.)

The shortest-path problem induces a metric, the shortest path metric, in which the distance between two points s and t is given by the length of a shortest s-t path; in many geometric contexts, this metric is also referred to as geodesic distance.

A simple polygon, P, having n vertices, is a closed, simply-connected region whose boundary is a union of n (straight) line segments (edges), whose endpoints are the vertices of P. A polygonal domain, P, having n vertices and h holes, is a closed, multiply-connected region whose boundary is a union of n line segments, forming h +   1 closed (polygonal) cycles. (A simple polygon is a polygonal domain with h =   0.)

A triangulation of P is a decomposition of P into triangles such that any two triangles either intersect in a common vertex, a common edge, or not at all. A triangulation of a simple polygon P can be computed in O(n) time [92]; a polygonal domain can be triangulated in time O(n log n) [326] or O(n  + h log^{1   + ε} h) [55] time. (See the chapter of Bern and Plassman [67] in this handbook, or the survey by Bern [64] for more information on triangulations.)

We will use the term obstacle to refer to any region of space whose interior is forbidden to paths. The complement of the set of obstacles is the free space. If the free space is a polygonal domain P, the obstacles are the h +   1 connected components of the complement of P (h holes, plus the face at infinity).

A path that cannot be improved by making a small change to it that preserves its combinatorial structure (e.g., the ordered sequence of triangles visited, for some triangulation of a polygonal domain P) is called a locally shortest or locally optimal path. It is also known as a taut-string path in the case of a shortest obstacle-avoiding path.

The visibility graph, VG(P), is a graph whose nodes are the vertices of P and whose edges join pairs of nodes for which the corresponding segment lies inside P. An example is shown in Figure 2.

Fig. 1. A shortest path map with respect to source point s within a polygonal domain with h =   3. The heavy dashed path indicates the shortest s-t path, which reaches t via the root r of its cell. Bisector curves are shown in narrow solid curves; extension segments are shown thin and dashed.

Fig. 2. The visibility graph VG(P): Edges of VG(P) are of two types — (1) the heavy dark boundary edges of P, and (2) the edges that intersect the interior of P, shown with thin dashed segments. A shortest s-t path is highlighted.

Given a source point, s, a shortest path tree, SPT(s, P), is a spanning tree of s and the vertices of P such that the (unique) path in the tree between s and any vertex of P is a shortest path in P.

A single-source query is a type of shortest path problem in which a source point, s, is fixed, and for each query (goal) point, t, one requests the length of a shortest path from the source point s to t. The query may also require the retrieval of an actual instance of a shortest s-t path; in general, this can be reported in additional time O(k), where k is the complexity of the output (e.g., number of edges).

One method of handling the single-source query problem is to construct a shortest path map, SPM(s), which is a decomposition of free space into regions (cells) according to the "combinatorial structure" of shortest paths from a fixed source point s to points in the regions. Specifically, for shortest paths in a polygonal domain, SPM(s) is a decomposition of P into cells such that for all points t interior to a cell, the sequence of obstacle vertices along an s-t path is fixed. In particular, the last obstacle vertex along a shortest s-t path is the root of the cell containing t. Each cell is star-shaped with respect to its root, which lies on the boundary of the cell, meaning that the root can "see" all points within the cell. Typically, we will store with each vertex, v, of P the geodesic distance, d(s, v), from s to v, as well as a pointer to the predecessor of v, which is the vertex (possibly s) preceding v in a shortest path from s to v. (The predecessor pointers provide an encoding of the SPT(s, P).) Note that v will appear on the boundary of the star-shaped cell rooted at its predecessor. The boundaries of cells consist of portions of obstacle edges, extension segments (extensions of visibility graph edges incident on the root), and bisector curves. The bisector curves are, in general, hyperbolic arcs that are the locus of points p that are (geodesically) equidistant from two roots, u and v: they satisfy d(s, u)   + d ₂(u, p)   = d(s, v)   + d ₂(v, p), where d ₂(·,·) denotes Euclidean distance. (Extension segments can be considered to be degenerate cases of bisector curves.) In Figure 1, the root of the cell containing t is labeled r. If SPM(s) is preprocessed for point location (see the chapter by Goodrich [180] in this handbook), then single-source queries can be answered efficiently by locating the query point t within the decomposition: If t lies in the cell rooted at r, the geodesic distance to t is given by d(s, t)   = d(s, r)   + d ₂(r, t). A shortest s-t path can then be output in time O(k), where k is the number of vertices along the path, by simply following predecessor pointers back from r to s.

In a two-point query problem, we are asked to construct a data structure that allows us to answer efficiently a query that specifies two points, s and t, and requests the length of a shortest path between them. In all cases discussed here, an actual instance of a shortest path can be reported in additional time O(k), where k is the complexity of the output (e.g., number of edges).

A geodesic Voronoi diagram (VD) is a Voronoi diagram for a set of sites, in which the underlying metric is the geodesic distance. See the chapter of Aurenhammer and Klein [46] in this handbook for details about Voronoi diagrams.

The geodesic center of P is a point within P that minimizes the maximum of the shortest-path lengths to any other point in P. The geodesic diameter of P is the maximum of the lengths of the shortest paths joining pairs of vertices of P.

Finally, we remark that in most of the algorithmic results reported here, the model of computation assumed has been the real RAM, which assumes that exact operations on real numbers can be done in constant time per operation. We acknowledge that this model is not, in general, realistic. At a couple places in the survey, we will point to results involving bit complexity models.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444825377500164

Morphological Amoebas and Partial Differential Equations

Martin Welk , Michael Breuß , in Advances in Imaging and Electron Physics, 2014

3.1 Continuous Amoeba Construction

Here, we consider a space-continuous image; i.e., a sufficiently smooth function $u:\Omega \to \mathrm{ℝ}.$ The amoeba construction then will rely on the representation of this image by its (rescaled) graph:

(14) $\mathit{\Gamma }\left(u\right):=\{(x,y,\sigma u(x,y))|(x,y)\in \Omega \}\subseteq {\mathrm{ℝ}}^3.$

As mentioned previously, this embedding has been used in the image-processing context in the Beltrami framework by Kimmel et al. (1997) and Yezzi (1998).

On the 2-D manifold Γ embedded in ℝ³, a continuous amoeba metric is constructed as an infinitesimal (Riemannian or Finslerian) metric ds ². It is again combined with the standard (Euclidean) metric $\text{d}{s}_{\mathit{\Omega }}^2$ on Ω and the standard metric $\text{d}{s}_{\mathrm{ℝ}}^2$ on the gray-value range via

(15) $\text{d}s=\phi (\text{d}{s}_{\Omega },\text{d}{s}_{\mathrm{ℝ}}),$

where $\phi$ is as in subsection 2.1, such that the length of a curve $C:\left[0,1\right]\to \mathit{\Gamma }$ is measured as

(16) $\begin{array}{c}L\left(C\right)=\underset{0}{\overset{1}{\int }}\phi \left(\sqrt{{\left(\frac{\text{d}x\left(p\right)}{\text{d}p}\right)}^2+{\left(\frac{\text{d}y\left(p\right)}{\text{d}p}\right)}^2},\sigma |\frac{\text{d}u(x\left(p\right),y\left(p\right))}{\text{d}p}|\right)\text{d}p\\ =\underset{0}{\overset{1}{\int }}\phi \left(\sqrt{x^{\prime }{\left(p\right)}^2+y^{\prime }{\left(p\right)}^2},\sigma |u_x{x}^{\prime }\left(p\right)+u_y{y}^{\prime }\left(p\right)|\right)\text{d}p,\end{array}$

and the distance between two points ${\mathit{p}}_1,{\mathit{p}}_2\in \Gamma$ is the length of the shortest curve between them:

(17) $d({\mathit{p}}_1,{\mathit{p}}_2)=\underset{C\left(1\right)={\mathit{p}}_2}{\underset{C\left(0\right)={\mathit{p}}_1}{\underset{C:[0,1]\to \Gamma }{\min }}}L\left(C\right).$

As in the discrete case, two choices of particular interest for $\phi$ are ${\phi }_2\left(s,t\right)=\sqrt{s^2+t^2},$ for which the amoeba metric d ≡ d ₂ is induced by the Euclidean metric of the embedding space ${\mathrm{ℝ}}^3$ , and ${\phi }_1\left(s,t\right)=s+t$ with the resulting amoeba metric d ≡ d ₁. Typical amoeba shapes for both cases are shown in Figure 1. Note that in the case of d ₂, the contour of the amoeba is smooth. In contrast, for d ₁, the contour has a digon-like overall shape, with kinks at the very points where it is hit by the level line through the point of reference (x ₀,y ₀).

Figure 1. Amoeba structuring elements. (a) Typical amoeba with metric d ≡ d ₂. (b) Typical amoeba with metric d ≡ d ₁.

From Welk, Breuß, and Vogel (2011), ©Springer 2010. With kind permission from Springer Science and Business Media.

Then an amoeba-structuring element $\mathcal{A}$ (x ₀,y ₀) is obtained by taking a closed ϱ-neighborhood of (x ₀,y ₀,u(x ₀,y ₀)) on Γ with regard to the metric ds, and projecting this set back onto Ω. Such neighborhoods have also been used in the construction of short-time Beltrami kernels by Spira, Kimmel, and Sochen (2007). By construction, the structuring elements are compact sets in the image domain $\Omega$ .

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780128001448000033

Infinite Words

In Pure and Applied Mathematics, 2004

Theorem 6.11

Let X be a nonempty subset of A^ω. The following conditions are equivalent:

(1)

X is Suslin,

(2)

X is the continuous image of some Polish space,

(3)

X is the projection of some Π₂-subset of {a, b} ^ω × A^ω ,

(4)

X is the continuous image of some closed subset of ℕ ^ω ,

(5)

X is the continuous image of ℕ ^ω .

Proof

We prove the implications (1) ⇒ (4) ⇒ (2) ⇒ (5) ⇒ (3) ⇒ (1) in this order.

(1)

implies (4) by Proposition 6.7.

(4)

implies (2) since every closed subset of ℕ ^ω is Π₂ by Proposition 2.5 and every Π₂-subset of ℕ ^ω is Polish by Propositions 2.7 and 3.4.

(2)

implies (5) is a consequence of Proposition 6.10.

(5)

implies (3). By assumption, there is a continuous map f from N^ω onto X. Consider the continuous map g : (a ^* b) ^ω → ℕ ^ω defined by

$g\left(a^{n_0}b{a}^{n_1}b{a}^{n_k}\cdots \right)=\left(n_0,n_1,\dots ,n_k,\dots \right)$

and let

$Y\left\{\left(x,y\right)\in {\left\{a,b\right\}}^{\omega }\times A^{\omega }|y=f\left(g\left(x\right)\right)\text{and}x\in {\left(a*b\right)}^{\omega }\right\}$

Since g ∘ f is continuous, its graph is a closed subset of (a ^* b) ^ω × A^ω . Now since (a ^* b) ^ω is a countable intersection of open sets of {a, b} ^ω , it follows that Y itself is a countable intersection of open sets of {a, b} ^ω × A^ω . Finally, X is the projection of Y, since g maps (a ^* b) ^ω onto ℕ ^ω .

(3)

implies (1). Suppose that X is the projection of some Π₂-subset of {a, b} ^ω × A^ω . By Proposition 3.14, {a, b} ^ω × A^ω is homeomorphic to {a, b} ^ω or to ℕ ^ω . Since the projection from {a, b} ^ω × A^ω onto A^ω is continuous, X is the continuous image of some Borel subset of either { a, b} ^ω or ℕ ^ω . Therefore, X is Suslin.

We now come to the general abstract definition of a Suslin set. A topological space X is said to be Suslin if it is Hausdorff and if there exists a Polish space E and a continuous map from E onto X. In particular, every Polish set is Suslin. Condition (2) of Theorem 6.11 shows that our new definition is compatible with the one given for the subsets of A^ω . Suslin sets are closed under continuous images and pre-images.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S0079816904800047

FN-Topologies and Group-Valued Measures

Hans Weber , in Handbook of Measure Theory, 2002

PROOF

We may assume that the μ-topology is Hausdorff. Otherwise consider $\widehat{\mu }$ : $\widehat{R}$ → G on the quotient $\widehat{R}$ = R/N(μ) defined by $\widehat{\mu }$ ( $\widehat{x}$ ) = μ(x) for x ∈ $\widehat{x}$ ∈ $\widehat{R}$ .

(a)

Under the assumptions of (a), R is by Theorem 9.1 connected with respect to the μ-topology. Therefore the continuous image μ(R) is connected.

(b)

Let ( $\tilde{R}$ , $\tilde{u}$ ) be the completion of (R, μ-topology) and $\tilde{\mu }$ : ( $\tilde{R}$ , $\tilde{u}$ ) → G the continuous extension of μ on $\tilde{R}$ . If R is μ-chained, then $\tilde{R}$ is atomless by Theorem 6.16(a), i.e., $\overline{\mu }$ . is atomless. Therefore $\tilde{\mu }$ ( $\tilde{R}$ ) is connected by (a). Hence $\overline{\mu (R)}=\overline{\tilde{\mu }(\tilde{R})}$ is connected.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444502636500178

Spline Basics

Carl de Boor , in Handbook of Computer Aided Geometric Design, 2002

6.13 SPLINE FUNCTIONS VS SPLINE CURVES

So far, we have only dealt with spline functions, even though CAGD is mainly concerned with spline curves. The distinction is fundamental.

Every spline function f = ∑ _j α _j B _jk gives rise to (planar) curve, namely its graph, i.e., the pointset

$\left\{\left(x,f\left(x\right)\right);x\in I_{k,t}\right\}.$

Assuming that #t _j < k for all interior knots t _j, this is indeed a curve in the mathematical sense, i.e., the continuous image of an interval. Its natural parametrization is the spline curve

(6.13.1) $x\mapsto \left(x,f\left(x\right)\right)={\displaystyle \sum _j{P}_j{B}_{jk}\left(x\right)},$

with

$P_j:=\left(t_{jk,}^{*}{\alpha }_j\right)$

its j th control point, and the equality in (6.13.1) is justified by (6.6.9).

However, spline curves are not restricted to control points of this specific form. By choosing the control points P_j in (6.13.1) in any manner whatsoever as d-vectors, we obtain a spline curve in ℝ ^d that smoothly follows the shape outlined by its control polygon, which is the broken line that connects these points P_j in order.

Note the CAGD-standard use of the term 'spline curve' to denote both, a curve that can be parametrized by a spline, and the (vector-valued) spline that provides this parametrization.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444511041500071

Generalized Axiomatic Scale-Space Theory

Tony Lindeberg , in Advances in Imaging and Electron Physics, 2013

4.1 Structural Scale-Space Axioms

Let us initially restrict ourselves to static (time-independent) data and focus on the spatial aspects: If we regard the incoming image intensity f as defined on an N -dimensional continuous image plane $f:{\mathrm{ℝ}}^N\to \mathrm{ℝ}$ , with Cartesian image coordinates denoted by $x={\left(x_1,\dots ,x_N\right)}^T$ , then the problem of defining a set of early visual operations can be formulated as finding a family of operators ${\mathcal{T}}_s$ that act on f to produce a family of new intermediate image representations: ¹

(2) $L\left(·;s\right)={\mathcal{T}}_{s}f(·),$

which are also defined as functions on ${\mathrm{ℝ}}^N$ ; i.e., $L\left(·;s\right):{\mathrm{ℝ}}^N\to \mathrm{ℝ}$ . These intermediate representation may be dependent on some parameter $s\in {\mathrm{ℝ}}^M$ , which in the simplest case may be 1-D or, under more general circumstances, multi-dimensional.

On a spatial domain where the smoothing operation is required to be rotationally symmetric, a 1-D parameter s with $M=1$ may be regarded as sufficient, whereas a higher dimensionality of the parameter s is needed to account for different amounts of smoothing along different directions in space as will be needed in the presence of general affine image transformations.

Linearity

If we want these initial visual processing stages to make as few irreversible decisions as possible, it is natural to initially require ${\mathcal{T}}_s$ to be a linear operator, such that

(3) ${\mathcal{T}}_s\left(a_1{f}_1+a_2{f}_2\right)=a_1{\mathcal{T}}_s{f}_1+a_2{\mathcal{T}}_s{f}_2$

holds for all functions $f_1,f_2:{\mathrm{ℝ}}^N\to \mathrm{ℝ}$ and all scalar constants $a_1,a_2\in \mathrm{ℝ}$ .

Linearity also implies that a number of special properties of receptive fields (to be developed below) will transfer to spatial derivatives of these and therefore imply that different types of image structures will be treated in a similar manner irrespective of what types of linear filters they are captured by.

Derivative operators are essential for modeling visual operations, since they respond to relative differences between image intensities in a local neighborhood and are therefore less sensitive to illumination variations than zero-order (undifferentiated) image intensities. (see Lindeberg (2012b) for a more precise statement).

Translational invariance

Let us also require ${\mathcal{T}}_s$ to be a shift-invariant operator in the sense that it commutes with the shift operator ${\mathcal{S}}_{\text{Δ}x}$ , defined by $\left({\mathcal{S}}_{\text{Δ}x}f\right)\left(x\right)=f\left(x-\text{Δ}x\right)$ , such that

(4) ${\mathcal{T}}_s\left({\mathcal{S}}_{\text{Δ}x}f\right)={\mathcal{S}}_{\text{Δ}x}\left({\mathcal{T}}_{s}f\right)$

holds for all $\text{Δ}x\in {\mathrm{ℝ}}^N$ . The motivation behind this assumption is the basic requirement that the representation of a visual object should be similar irrespective of its position in the image plane. ² Alternatively stated, the operator ${\mathcal{T}}_s$ can be said to be homogeneous across space.

Convolution structure

From a general result in linear systems theory, it follows from the assumptions of linearity and shift-invariance that the internal representations $L\left(·;s\right)$ are given by convolution transformations (Hirschmann & Widder 1955):

(5) $L\left(x;s\right)=\left(T\left(·;s\right)\ast f\right)\left(x\right)=\underset{\xi \in {\mathrm{ℝ}}^N}{\int }T\left(\xi ;s\right)f\left(x-\xi \right)d\xi ,$

where $T\left(·;s\right)$ denotes some family of convolution kernels. These convolution kernels and their spatial derivatives can also be referred to as (spatial) receptive fields.

Regularity

To be able to use tools from functional analysis, we will initially assume that both the original signal f and the family of convolution kernels $T\left(·;s\right)$ are in the Banach space $L^2\left({\mathrm{ℝ}}^N\right)$ ; i.e., that $f\in L^2\left({\mathrm{ℝ}}^N\right)$ and $T\left(·;s\right)\in L^2\left({\mathrm{ℝ}}^N\right)$ with the norm

(6) ${⋮f⋮}_2^2=\underset{x\in {\mathrm{ℝ}}^N}{\int }{\left|f\left(x\right)\right|}^{2}dx.$

Then, also the intermediate representations $L\left(·;s\right)$ will be in the same Banach space and the operators ${\mathcal{T}}_s$ can be regarded as well defined.

Positivity (non-negativity)

Concerning the convolution kernels, one may require them to be non-negative in order to constitute smoothing transformations:

(7) $T\left(x;s\right)\ge 0.$

Normalization

Furthermore, it may be natural to require the convolution kernels to be normalized to unit mass

(8) $\underset{x\in {\mathrm{ℝ}}^N}{\int }T\left(x;s\right)dx=1$

to leave a constant signal unaffected by the smoothing transformation.

Quantitative measurement of the spatial extent and the spatial offset of non-negative scale-space kernels

For a non-negative convolution kernel, we can measure its spatial offset by the mean operator

(9) $m=\overline{x}=M\left(T\left(·;s\right)\right)=\frac{{\int }_{x\in {\mathrm{ℝ}}^N}xT\left(x;s\right)dx}{{\int }_{x\in {\mathrm{ℝ}}^N}T\left(x;s\right)dx}$

and its spatial extent by the spatial covariance matrix

(10) $\text{Σ}=C\left(T\left(·;s\right)\right)=\frac{{\int }_{x\in {\mathrm{ℝ}}^N}(\left(x-\overline{x}\right){\left(x-\overline{x}\right)}^{T}T\left(x;s\right)dx}{{\int }_{x\in {\mathrm{ℝ}}^N}T\left(x;s\right)dx}.$

Using the additive properties of mean values and covariance matrices under convolution, that hold for non-negative distributions, it follows that

(11) $m=M\left(T\left(·;s_1\right)\ast T\left(·;s_2\right)\right)=M\left(T\left(·;s_1\right)\right)+M\left(T\left(·;s_2\right)\right)=m_1+m_2,$

(12) $\text{Σ}=C\left(T\left(·;s_1\right)\ast T\left(·;s_2\right)\right)=C\left(T\left(·;s_1\right)\right)+C\left(T\left(·;s_2\right)\right)={\text{Σ}}_1+{\text{Σ}}_2$

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780124077010000017

Biomimetic Vision Sensors

Cameron H.G. Wright , Steven F. Barrett , in Engineered Biomimicry, 2013

1.2.2.4 Detector Arrays

With a scene imaged at the focal plane, the next step is to use some photosensitive devices to convert the light energy to electrical energy. One of the most common techniques is to use a rectangular, planar array of photodetectors (typically CCD or CMOS) at the focal plane; this is often called a focal plane array (FPA). Use of the FPA introduces two more effects on the image: the spatial integration of light over the finite photosensitive area of each photodetector, and the spatial sampling of the continuous image. For compactness, a one-dimensional approach is used where appropriate for the explanation that follows.

Each photodetector of the FPA must have a large enough photosensitive area to capture a sufficient number of photons to obtain a useable signal above the noise floor. This finite area results in spatial integration that produces a blurring or smearing effect. In the most common case, the sensitivity of the photodetector is relatively constant over the entire photosensitive area, so the effective PSF of the FPA is just a rectangular (or top-hat) function with a width determined by the size of the photosensitive area in the given dimension for the individual photodetectors. The OTF of the FPA is the Fourier transform of a rectangular function, which is the well-known sinc function. Thus, the MTF of the FPA is the magnitude of the associated sinc:

(1.8) ${\mathrm{MTF}}_{\text{FPA}}=\left|\frac{\sin (\pi x_{d}u)}{\pi x_{d}u}\right|=|\mathrm{sinc}(x_{d}u)|\text{,}$

where $x_d$ is the size of the photosensitive area in the x direction (Figure 1.10). While the magnitude of a sinc function extends to infinity (so technically there is no associated cutoff frequency with this MTF), it is common to consider the MTF of an FPA only up to the first zero of the sinc, which occurs at $1/x_d$ . Thus, a larger photosensitive area may gather more light but results in the first zero occurring at a lower spatial frequency, which induces more blurring in the image. Note that the units of spatial frequency in Eq. (1.8) for the MTF of an FPA are linear units (e.g., cycles/meter); when optical MTFs were previously discussed, we followed the common convention of using angular units. Angular spatial frequency divided by the focal length of the optical system equals linear spatial frequency at the focal plane, assuming the small angle approximation holds.

Figure 1.10. Normalized MTF of a typical focal plane array with a square photosensitive area, where $x_d=y_d$ . The spatial frequency units of the horizontal axes ( $u\text{,}v$ directions) are the reciprocal of the spatial dimension of a single photodetector in that direction, as in $1/x_d$ or $1/y_d$ , respectively.

A commonly used mathematical description of image sampling is to convolve the point-spread function of all the nonideal aspects of the imaging system (optics, sensor array, and associated electronics) with an ideal continuous image source, followed by sampling via an ideal basis function such as a train of delta functions [15, 26]. Thus, for an image source sampled in the x direction,

(1.9) $g[n]=[h(x)\ast f(x)]{\displaystyle \sum _{n\in z}}\delta (x-{\mathit{nX}}_s)\text{,}$

where $g[n]$ is the discrete space–sampled image result, $h(x)$ is the combined point-spread function of all the nonideal effects, $f(x)$ is the ideal continuous space image, and $X_s$ is the center-to-center spacing of the photodetectors. ⁸ Sampling in the y direction has a similar form. The top-hat PSF due the FPA would contribute to the overall $h(x)$ in Eq. (1.9). But what of the effect of sampling? Since $X_s$ is the spatial sampling interval, then $F_s=1/X_s$ is the spatial sampling frequency. From the sampling theorem, we can conclude that any spatial frequencies sampled by the FPA that are higher than one-half the sampling frequency, or $F_s/2=1/2X_s$ , will be aliased as described by Eq. (1.3). With no dead space between detectors (i.e., a fill factor of 1), the first zero of the FPA's MTF occurs at $F_s$ , so the FPA will respond to spatial frequencies well above $F_s/2$ , which means aliasing is likely. Lower fill factors exacerbate the potential for aliasing, since a smaller detector size moves the first zero of the MTF to a higher spatial frequency.

This may or may not present a problem, depending on the application. Monochrome aliasing tends to be less objectionable to human observers than color aliasing, for example. ⁹ Real-world images are often bandlimited only by the optical cutoff frequency of the optics used to form the image. This optical cutoff frequency is often considerably higher than $F_s/2$ , and in that case aliasing will be present. However, some FPAs come with an optical low-pass filter in the form of a birefringent crystal window on the front surface of the array.

Note that the spatial sampling can be implemented in various ways to meet the requirements of the application, as depicted in Figure 1.11, but the treatment of spatial integration (which leads to the MTF) and the spatial sampling (with considerations of aliasing) remain the same. Ideal sampling (as shown in Figure 1.11a) is a mathematical construct, useful for calculations [as in Eq. (1.9)] but not achievable in practice. The most common type of sampling is top-hat sampling on a planar base, as shown in Figure 1.11b, where the fill factor implied by the figure is 50%. The MTF associated with top-hat sampling was given in Eq. (1.8). Some FPAs do not exhibit constant sensitivity over the photosensitive area of each photodetector, with the most common variation being an approximation to the Gaussian shape, as shown in Figure 1.11c. The MTF of this type of array is Gaussian, since the Fourier transform of a Gaussian is a scaled Gaussian. Figures 1.11d and e show Gaussian sampling with an intentional overlap between adjacent samples, on a planar and on a spherical base; these specific variations will be discussed further in the case study describing a biomimetic vision sensor based on Musca domestica, the common housefly.

Figure 1.11. Various types and geometries of spatial sampling for vision sensors: (a) ideal, (b) top-hat, (c) Gaussian, (d) overlapping Gaussian, and (e) overlapping Gaussian with a nonplanar base.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780124159952000015

Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2

Ke Chen , ... Jan Modersitzki , in Handbook of Numerical Analysis, 2019

2 Mathematical background

The goal of image registration is to find a suitable map y that maps a template T to a reference image R, such that T(y) and R are aligned, that is corresponding points are placed at the same position. We also aim to provide a mathematical framework that is capable to cover as many image registration approaches as possible. This section will introduce this framework, modelling ideas and notation but leaving details to Sections 3–7.

2.1 Continuous and discrete images

From reading this volume of the handbook, the reader should be clear that mathematical imaging can offer powerful modelling tools (namely the variational framework) mainly because we can assume that the input images T, R are continuous. Then the full power of functional analysis can be applied to imaging and it is natural to define and discuss geometry (such as gradients, curvature, H ₁ norm) of T, R. Also, a continuous interpretation of image data is much more convenient if geometrical transformations are to be applied.

In the continuous interpretation that is used in this chapter, a generic image is defined as a mapping I : Ω → G. Here, $\mathrm{\Omega }\subset {\mathbb{R}}^d$ denotes the image domain and d is the image dimensionality, where typically Ω is a one-dimensional interval with d = 1, and a square with d = 2 or a cube with d = 3. The image range is denoted by G and may indicate binary images $G=\left\{0,1\right\}$ , grey scale images $G=\left\{0,\dots ,255\right\}$ , real images $G=\mathbb{R}$ , or multispectral images such as colour images, mass spectroscopy, or tensors. In this chapter, we focus on real images mainly with d = 2. However, the discussed models extend to other images.

In applications, the image data I is typically given as a d-array of size m ₁ × ⋯ × m _d , which may be naturally viewed as sampled from a continuous image function I. Interpolation or approximation techniques must be used to embed the data into a function space; see, e.g., Modersitzki (2004) and Chen (2013). In this sense, the notation I(y) implies that an interpolation must be implemented (unless y takes integer values only). This aspect is unique in IR and SR, different from other imaging problems. However, this chapter emphasizes the "existence" of a continuous image I (via interpretation) that is assumed available for discrete images.

If the continuous image I needs to be visualized, the domain Ω is partitioned into a number n of cells with cell-centres $x_j=(x_j^1,\dots ,x_j^d)$ , and the discrete image [I(x _j ), j = 1, …, n] is displayed. In imaging, a pixelized interpretation of image data is often used for processing; see Fig. 1 for an illustration.

Fig. 1. An image (A) overlaid with a pixel grid (B), a rotated version of the image (D) overlaid with a pixel grid (E); a nonlinearly deformed version of the image overlaid with a pixel grid (F). Part (C) displays a partitioned image domain Ω with 6-by-4 cells with cell-centres $x_j=(x_j^1,x_j^2)$ , j = 1, …, n = 24.

Finally we remark that another popularly used notation for the deformed template T(y) is T ∘ y, where the transform y may also be written as y = x + u(x) later when we apply regularization to the deformation field u.

2.2 A mathematical framework for image registration

As common for ill-posed problems, we use a variational framework that aims to minimize a joint energy functional constituting a data fidelity term or an image distance measure and a regularizer. Details on distance measures $\mathcal{D}$ and the regularizer $\mathcal{R}$ are provided in Sections 3 and 4.

The objective is to determine the wanted transformation $y:{\mathbb{R}}^d\to {\mathbb{R}}^d$ as a minimizer of a joint energy $\mathcal{J}$ over a feasible set of transformation $\mathcal{A}$ , i.e.

(1) $\begin{array}{l}\hfill \mathcal{J}:\mathcal{A}\to \mathbb{R},\mathcal{J}(y):=\mathcal{D}(T\circ y,R)+\mathcal{R}(y),\hfill \end{array}$

where the data fidelity term $\mathcal{D}$ , the regularization term $\mathcal{R}$ , and the feasible set $\mathcal{A}$ are discussed below.

We remark that, though the approach applies to both, mono- and multimodal images, we focus on mono-modality for ease of presentation. We also remark that is often convenient to consider the deformation u rather than the transformation y, where the deformation or displacement denotes the change, y(x) = x + u(x). Note, that in contrast to the differences of transformation and deformation the notations transformed image and deformed image are used synonymously in the literature.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S1570865919300146

Geometric Function Theory

Ch. Pommerenke , in Handbook of Complex Analysis, 2002

2.1 Jordan curves and locally connected sets

The most important result in this area is Carathéodory's theorem [10]. By Jordan arc we mean the homeomorphic image of [0, 1]. By Jordan curve we mean the homeomorphic image of $\mathbb{T}$ ; it bounds two Jordan domains.

Theorem 2.1

A conformal map φ of a Jordan domain F onto a Jordan domain G can be extended to a homeomorphism of ${\displaystyle \overline{F}}$ onto ${\displaystyle \overline{G}}$ .

By the factorization φ = g ∘ f ⁻¹ in (1.1.1) it is sufficient to show that a conformal map f from $\mathbb{D}$ onto a Jordan domain F has a continuous and injective extension to ${\displaystyle \overline{\mathbb{D}}}$ . The continuity will follow from Theorem 2.2 (i) and the injectivity from Theorem 2.10 because Jordan curves have no cut points.

A consequence is the purely topological Schoenflies theorem: A bijective continuous map of $\mathbb{T}$ onto a Jordan curve in $\mathbb{C}$ can be extended to a homeomorphism of $\mathbb{C}$ onto $\mathbb{C}$ .

First we consider continuous extensions which need not be injective. We understand continuity in the spherical metric if F is unbounded.

A compact set A in ${\displaystyle \widehat{\mathbb{C}}}$ is called locally connected if, for every ε > 0, there exists δ > 0 such that, for every a, b ∈ A with dist(a, b) < δ, we can find a connected compact set B with

$\begin{array}{ll}a,b\in B\subset A,\hfill & \text{diam}B<\epsilon \hfill \end{array}.$

See, e.g., [93, p. 88], [125, p. 20 ]. The continuous image of a locally connected compact set is again locally connected and compact. A curve is, by definition, the continuous image of a segment and is therefore locally connected.

Theorem 2.2

Let f map $\mathbb{D}$ conformally onto $F\subset \mathbb{C}$ . Then the following four conditions are equivalent:

(i)

f has a continuous extension to ${\displaystyle \overline{\mathbb{D}}}$ ;

(ii)

∂F is a curve;

(iii)

∂F is locally connected;

(iv)

$\mathbb{C}\backslash F$ is locally connected.

The only difficult part is the implication (iv) ⇒ (i). We now sketch the proof restricting ourselves to bounded domains.

Let $\zeta \in \mathbb{T}$ and consider the circular arc

$\begin{array}{ll}C(r)=\left\{z\in \mathbb{D}:|z-\zeta |=r\right\}\hfill & (0<r<1)\hfill \end{array}.$

By the Schwarz inequality, the length $\mathcal{l}(r)$ of its image f(C(r)) satisfies

$\mathcal{l}{(r)}^2={\left({\int }_{C(r)}\left|f^{\prime }(z)\right||dz|\right)}^2<\pi r{\int }_{C(r)}{\left|f^{\prime }(z)\right|}^2|dz|$

because C(r) has length < πr. It follows that

${\int }_0^1\frac{\mathcal{l}{(r)}^2}{\pi r}dr<\int {{\int }_{\mathbb{D}}\left|f^{\prime }(z)\right|}^{2}dxdy=\text{area}F<\infty .$

Hence there exist r _n → 0 such that $\mathcal{l}(r_n)\to 0$ as n → ∞.

Since $\mathcal{l}(r_n)<\infty$ the curve f(C(r _n )) has definite endpoints w _n and $w_n^{\prime }$ on ∂F. By (iv), there are connected compact sets B _n with

(2.1.1) $\begin{array}{ll}{w}_n,w_n^{\prime }\in B_n\subset \mathbb{C}\backslash F,\hfill & \begin{array}{ll}\text{diam}{B}_n\to 0\hfill & (n\to \infty ).\hfill \end{array}\hfill \end{array}$

Since $V_n:=\{f(z):z\in \mathbb{D},|z-\zeta |<r_n\}$ lies in the domain between f(C(r _n )) and B _n , it follows that

(2.1.2) $\begin{array}{ll}\text{diam}{V}_n\le \max \left(\mathcal{l}(r_n),\text{diam}{B}_n\right)\to 0\hfill & (n\to \infty )\hfill \end{array}$

which implies that f(z) tends to a limit as z → ζ for every $\zeta \in \mathbb{T}$ , and it is easy to deduce that (i) holds.

The reasoning that led from (2.1.1) to (2.1.2) was somewhat vague. Statements like this can often be made precise by a useful topological theorem [93, p. 110], [25, p. 362].

Janiszewski's Theorem or Alexander's Lemma

Let A and B be closed sets in ${\displaystyle \widehat{\mathbb{C}}}$ whose intersection A ⋂ B is connected. If two points are separated neither by A nor by B, then they are not separated by the union A ⋃ B.

We mention some further theorems from plane topology that are often useful in geometric function theory. See [128,85,73], [125, p. 108] for the first result. A compact set T is called totally disconnected if each of its components is a point. This is, e.g., true if T is countable.

Plane Separation Theorem

Let $A\subset \mathbb{C}$ and $B\subset {\displaystyle \widehat{\mathbb{C}}}$ be compact and let T = A ⋂ B be totally disconnected. Given a ∈ A \ T, b ∈ B \ T and ε > 0 there exists a Jordan curve J with

$J\cap (A\cup B)\subset T$

that separates a and b and lies in an ε-neighbourhood of A.

See [105, p. 36] for a "colour" version of the next result [86]. A triod consists of three Jordan arcs that intersect only at their common junction point.

Moore Triod Theorem

Every collection of disjoint plane triods is countable.

Torhorst Theorem

([117], [125, p. 106]). Let $E\subset {\displaystyle \widehat{\mathbb{C}}}$ be compact and locally connected and let G be a component of ${\displaystyle \widehat{\mathbb{C}}}\backslash E$ . Then ∂G is connected and locally connected.

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/S187457090280004X

Luitzen Egbertus Jan Brouwer

Dirk van Dalen , in History of Topology, 1999

In the same address he mentioned some open problems, e.g.:

"In how far are spaces of distinct dimension-number different for our group [of homeomorphisms]". He added "It is very likely that this is always the case, but it seems most difficult to provide a proof, and it will probably remain an unsolved problem for a long time."

… one has no certainty that the 3-dimensional Cartesian space is split into two domains by a closed Jordan surface, i.e. the one–one continuous image of a sphere.

Brouwer ended his lecture with a plea for basing mathematical theories on analysis situs, the prime example being the topological treatment of geometry (as found in Hilbert's "Über die Grundlagen der Geometrie" [65]. See also Grundlagen der Geometrie, Anhang IV). In the case of geometry, coordinates can be introduced afterwards by using Van Staudt's techniques.

"And so", Brouwer concluded, "coordinates will not have to be banned from other theories, if one succeeds in founding them on analysis situs, but the formula-free 'geometric' treatment will be the point of departure, the analytic one will become a dispensable tool.

It is this possibility and desirability of this priority of the geometric treatment, also in parts of mathematics where it does not yet exist, that I have mainly wished to point out in the above lines".

Read full chapter

URL:

https://www.sciencedirect.com/science/article/pii/B9780444823755500365

mathewsigntearame.blogspot.com

Source: https://www.sciencedirect.com/topics/mathematics/continuous-image