Taking advantage of the new developments in mathematical statistics, a multiscale approach is designed to detect filament or filament-like features in noisy images. The major contribution is to introduce a general framework in cases when the data is digital. Our detection method can detect the presence of an underlying curvilinear feature with the lowest possible strength that are still detectible in theory. Simulation results on synthetic data will be reported to illustrate its effectiveness in finite digital situations.

Let {(Zi,Wi) : i = 1,...,n} be uniformly distributed in [0,1] d × G(k,d), where G(k,d) denotes the space of k-dimensional linear subspaces of R d. For a differentiable function f: [0,1] k → [0,1] d, we say that f interpolates (z,w) ∈ [0,1] d × G(k,d) if there exists x ∈ [0,1] k such that f(x) = z and ⃗ f(x) = w, where ⃗ f(x) denotes the tangent space at x defined by f. For a smoothness class F of Hölder type, we obtain probability bounds on the maximum number of points a function f ∈ F interpolates. 1

We are given a set of n points that might be uniformly distributed in the unit square [0,1] 2. We wish to test whether the set, although mostly consisting of uniformly scattered points, also contains a small fraction of points sampled from some (a priori unknown) curve with C α-norm bounded by β. An asymptotic detection threshold exists in this problem; for a constant T−(α,β)> 0, if the number of points sampled from the curve is smaller than T−(α,β)n 1/(1+α) , reliable detection is not possible for large n. We describe a multiscale significantruns algorithm that can reliably detect concentration of data near a smooth curve, without knowing the smoothness information α or β in advance, provided that the number of points on the curve exceeds T∗(α,β)n 1/(1+α). This algorithm therefore has an optimal detection threshold, up to a factor T∗/T−. At the heart of our approach is an analysis of the data by counting membership in multiscale multianisotropic strips. The strips will

Connect-The-Dots (CTD) is our name for a class of geometric optimization problems in which, given a point cloud, one finds the curve f from a given class F which passes through the maximum possible number of points in the cloud. Examples in 2D include monotone increasing graphs, curves of bounded curvature or length, and so on. In higher dimen-sions we may generalize graphs/curves to surfaces of fixed co-dimension, or bounded surface area. Other generalizations include replacing the “passing through points ” condition by “passing through points in specified directions. ” Potential applications arise in, for exam-ple, image processing (tracking amid clutter) and understanding human vision (perceptual psychophysics), and many more. This report describes a family of dynamic programming algorithms for various CTD problems. Some are their first appearance: e.g., the algorithms for the Hölder-2 functions and the nondecreasing function in high dimension. A software library—CTDLab—that implements many of these algorithms is available on the Internet. Simulations that are based on these tools give insights and new conjectures regarding the asymptotic behavior of solutions when the point (or vector) clouds are i.i.d. uniform.

We are given a set of n points that appears uniformly distributed in the unit square [0, 1] 2. We wish to test whether the set actually is generated from a non-uniform distribution having a small fraction of points concentrated on some (a priori unknown) curve with C α-norm bounded by β. An asymptotic detection threshold exists in this problem; for a constant T−(α, β)> 0, if the number of points on the curve is smaller than T−(α, β)n 1/(1+α) , reliable detection is not possible for large n. We describe a Multiscale Significant-Runs Algorithm; it can reliably detect concentration of data near a smooth curve, without knowing the smoothness information α or β in advance, provided that the number of points on the curve exceeds T∗(α, β)n 1/(1+α). This algorithm therefore has an optimal detection threshold, up to a factor T∗/T−. At the heart of our approach is an analysis of the data by counting membership in multiscale multi-anisotropic strips. The strips have an area of C/n and exhibit a variety of lengths, orientations and anisotropies. The strips are partitioned into anisotropy classes; each class is organized as a directed graph whose vertices are strips all of the same anisotropy and whose edges link such strips to their ‘good continuations’. The point cloud data are reduced to counts measuring membership in strips. Each anisotropy graph is reduced to a subgraph consisting of strips with ‘significant ’ counts. The algorithm rejects H0 whenever some such subgraph contains a path connecting many consecutive ‘significant ’ counts.