Measures That Are Translation Invariant In One Coordinate

This chapter discusses a case for single-parameter singular integral operators, where ρ‎ is the usual distance on ℝn. There, we obtain the most classical theory of singular integrals, which is useful for studying elliptic partial differential operators. The chapter defines singular integral operators in three equivalent ways. This trichotomy can be seen three times, in increasing generality: Theorems 1.1.23, 1.1.26, and 1.2.10. This trichotomy is developed even when the operators are not translation invariant (many authors discuss such ideas only for translation invariant, or nearly translation invariant operators). It also presents these ideas in a slightly different way than is usual, which helps to motivate later results and definitions.

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High-Performance Image Filters via Sparse Approximations

Proceedings of the ACM on Computer Graphics and Interactive Techniques ◽

10.1145/3406182 ◽

2020 ◽

Vol 3 (2) ◽

pp. 1-19

Author(s):

Kersten Schuster ◽

Philip Trettner ◽

Leif Kobbelt

Keyword(s):

High Performance ◽

Hardware Acceleration ◽

Optimization Method ◽

Translation Invariant ◽

Approximation Quality ◽

Trade Offs ◽

Sparse Approximations ◽

Image Filters ◽

Good Trade ◽

And Performance

We present a numerical optimization method to find highly efficient (sparse) approximations for convolutional image filters. Using a modified parallel tempering approach, we solve a constrained optimization that maximizes approximation quality while strictly staying within a user-prescribed performance budget. The results are multi-pass filters where each pass computes a weighted sum of bilinearly interpolated sparse image samples, exploiting hardware acceleration on the GPU. We systematically decompose the target filter into a series of sparse convolutions, trying to find good trade-offs between approximation quality and performance. Since our sparse filters are linear and translation-invariant, they do not exhibit the aliasing and temporal coherence issues that often appear in filters working on image pyramids. We show several applications, ranging from simple Gaussian or box blurs to the emulation of sophisticated Bokeh effects with user-provided masks. Our filters achieve high performance as well as high quality, often providing significant speed-up at acceptable quality even for separable filters. The optimized filters can be baked into shaders and used as a drop-in replacement for filtering tasks in image processing or rendering pipelines.

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Isomorphisms between determinantal point processes with translation-invariant kernels and Poisson point processes

Ergodic Theory and Dynamical Systems ◽

10.1017/etds.2020.123 ◽

2020 ◽

pp. 1-14

Author(s):

SHOTA OSADA

Keyword(s):

Point Processes ◽

Duke Math ◽

Random Point ◽

Determinantal Point Processes ◽

Fredholm Determinants ◽

Poisson Point Processes ◽

Translation Invariant ◽

Ergodic Properties ◽

Tree Representations ◽

Invariant Kernels

Abstract We prove the Bernoulli property for determinantal point processes on $ \mathbb{R}^d $ with translation-invariant kernels. For the determinantal point processes on $ \mathbb{Z}^d $ with translation-invariant kernels, the Bernoulli property was proved by Lyons and Steif [Stationary determinantal processes: phase multiplicity, bernoullicity, and domination. Duke Math. J.120 (2003), 515–575] and Shirai and Takahashi [Random point fields associated with certain Fredholm determinants II: fermion shifts and their ergodic properties. Ann. Probab.31 (2003), 1533–1564]. We prove its continuum version. For this purpose, we also prove the Bernoulli property for the tree representations of the determinantal point processes.

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