Related papers: Why are images smooth?
We propose and study quantitative measures of smoothness which are adapted to anisotropic features such as edges in images or shocks in PDE's. These quantities govern the rate of approximation by adaptive finite elements, when no constraint…
As borders between different regions, lines are an important element of natural images. Already at the level of the mammalian primary visual cortex (V1), neurons respond best to lines of a given orientation. We reduce a set of images to…
The protein folding problem must ultimately be solved on all length scales from the atomic up through a hierarchy of complicated structures. By analyzing the stability of the folding process using physics and mathematics, this paper shows…
With the improvement of social life quality and the real needs of daily work, images are more and more all around us. Image blurring due to camera shake, human movement, etc. has become the key to affecting image quality. How to remove…
There is an intricate relation between the properties of an image and how humans behave while describing the image. This behavior shows ample variation, as manifested in human signals such as eye movements and when humans start to describe…
Continuous-domain visual signals are usually captured as discrete (digital) images. This operation is not invertible in general, in the sense that the continuous-domain signal cannot be exactly reconstructed based on the discrete image,…
The description of complex physical phenomena often involves sophisticated models that rely on a large number of parameters, with many dimensions and scales. One practical way to simplify that kind of models is to discard some of the…
Splashing occurs when a liquid drop hits a dry solid surface at high velocity. This paper reports experimental studies of how the splash depends on the roughness and the texture of the surfaces as well as the viscosity of the liquid. For…
Spectrograms are 2D representations of sound that look very different from the images found in our visual world. And natural images, when played as spectrograms, make unnatural sounds. In this paper, we show that it is possible to…
Algorithmic randomness theory starts with a notion of an individual random object. To be reasonable, this notion should have some natural properties; in particular, an object should be random with respect to image distribution if and only…
Natural image matting is a fundamental and challenging computer vision task. Conventionally, the problem is formulated as an underconstrained problem. Since the problem is ill-posed, further assumptions on the data distribution are required…
We present a concise derivation for several influential score-based diffusion models that relies on only a few textbook results. Diffusion models have recently emerged as powerful tools for generating realistic, synthetic signals --…
Many classical geometric inequalities on functionals of convex bodies depend on the dimension of the ambient space. We show that this dimension dependence may often be replaced (totally or partially) by different symmetry measures of the…
Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is…
To understand the true nature of black holes, fundamental theoretical developments should be linked all the way to observational features of black holes in their natural astrophysical environments. Here, we take several steps to establish…
Imaging is an important means by which information is gathered regarding the physical world. Spatial resolution and signal-to-noise ratio are underpinning concepts. There is a paucity of rigorous definitions for these quantities, which are…
A description of physical reality in which wholeness is the foundation is discussed along with the motivation for such an attempt. As a possible mathematical framework within which a physical theory based on wholeness may be expressed,…
We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…
With the development and widespread application of digital image processing technology, image splicing has become a common method of image manipulation, raising numerous security and legal issues. This paper introduces a new splicing image…
What does it mean to be flat? We propose to define it by measuring the maximal variation around a point, or from a dual perspective, the distance to neighboring level sets. After developing some calculus rules, we show how flat minima,…