Pointwise Operations Research Articles

PolyMage

This paper presents the design and implementation of PolyMage, a domain-specific language and compiler for image processing pipelines. An image processing pipeline can be viewed as a graph of interconnected stages which process images successively. Each stage typically performs one of point-wise, stencil, reduction or data-dependent operations on image pixels. Individual stages in a pipeline typically exhibit abundant data parallelism that can be exploited with relative ease. However, the stages also require high memory bandwidth preventing effective utilization of parallelism available on modern architectures. For applications that demand high performance, the traditional options are to use optimized libraries like OpenCV or to optimize manually. While using libraries precludes optimization across library routines, manual optimization accounting for both parallelism and locality is very tedious. The focus of our system, PolyMage, is on automatically generating high-performance implementations of image processing pipelines expressed in a high-level declarative language. Our optimization approach primarily relies on the transformation and code generation capabilities of the polyhedral compiler framework. To the best of our knowledge, this is the first model-driven compiler for image processing pipelines that performs complex fusion, tiling, and storage optimization automatically. Experimental results on a modern multicore system show that the performance achieved by our automatic approach is up to 1.81x better than that achieved through manual tuning in Halide, a state-of-the-art language and compiler for image processing pipelines. For a camera raw image processing pipeline, our performance is comparable to that of a hand-tuned implementation.

PolyMage

This paper presents the design and implementation of PolyMage, a domain-specific language and compiler for image processing pipelines. An image processing pipeline can be viewed as a graph of interconnected stages which process images successively. Each stage typically performs one of point-wise, stencil, reduction or data-dependent operations on image pixels. Individual stages in a pipeline typically exhibit abundant data parallelism that can be exploited with relative ease. However, the stages also require high memory bandwidth preventing effective utilization of parallelism available on modern architectures. For applications that demand high performance, the traditional options are to use optimized libraries like OpenCV or to optimize manually. While using libraries precludes optimization across library routines, manual optimization accounting for both parallelism and locality is very tedious. The focus of our system, PolyMage, is on automatically generating high-performance implementations of image processing pipelines expressed in a high-level declarative language. Our optimization approach primarily relies on the transformation and code generation capabilities of the polyhedral compiler framework. To the best of our knowledge, this is the first model-driven compiler for image processing pipelines that performs complex fusion, tiling, and storage optimization automatically. Experimental results on a modern multicore system show that the performance achieved by our automatic approach is up to 1.81x better than that achieved through manual tuning in Halide, a state-of-the-art language and compiler for image processing pipelines. For a camera raw image processing pipeline, our performance is comparable to that of a hand-tuned implementation.

Space-Time Regularization for Video Decompression

We consider the problem of reconstructing frames from a video which has been compressed using the video compressive sensing (VCS) method. In VCS data, each frame comes from first subsampling the original video data in space and then averaging the subsampled sequence in time. This results in a large linear system of equations whose inversion is ill-posed. We introduce a convex regularizer to invert the system, where the spatial component is regularized by the total variation seminorm, and the temporal component is regularized by enforcing sparsity on the difference between the spatial gradients of each frame. Since the regularizers are $L^1$-like norms, the model can be written in the form of an easy-to-solve saddle point problem. The saddle point problem is solved by the primal-dual algorithm, whose implementation calls for nearly pointwise operations (i.e., no direct linear inversion) and has a simple parallel version. Results show that our model decompresses videos more accurately than other popular mod...

A conservative, skew-symmetric finite difference scheme for the compressible Navier–Stokes equations

We present a fully conservative, skew-symmetric finite difference scheme on transformed grids. The skew-symmetry preserves the kinetic energy by first principles, simultaneously avoiding a central instability mechanism and numerical damping. In contrast to other skew-symmetric schemes no special averaging procedures are needed. Instead, the scheme builds purely on point-wise operations and derivatives. Any explicit and central derivative can be used, permitting high order and great freedom to optimize the scheme otherwise. This also allows the simple adaption of existing finite difference schemes to improve their stability and damping properties. Numerical examples covering acoustic phenomena and shocks are presented.

Naturalness of the class of Lebesgue-Borel-Hausdorff measurable functions

It is well known that the family of all continuous functions on a topological space contains all constant functions and is closed with respect to the usual pointwise operations (addition, multiplication, finite supremum and infimum, division) and uniform convergence. The complete description of such function families (normal families) was given by Borel, Lebesgue, and Hausdorff. The normal families turned out to be exactly the families of all functions measurable with respect to multiplicative σ-additive families of sets. In 1914, Hausdorff also described the normal envelope of an arbitrary family of functions. If uniform convergence is replaced by pointwise convergence, then the notions of completely normal family and completely normal envelope arise. In 1977, Regoli described all completely normal families. They turned out to be precisely the families of all functions measurable with respect to σ-algebras of sets. Moreover, Regoli described the completely normal envelope of a specific family of functions. The present paper gives descriptive and some constructive characterizations of the completely normal envelope of an arbitrary family of functions.

Rings with kernel inclusion quasiorder

Building on previous work for semigroups of functions and binary relations, we axiomatize structures consisting of endomorphisms of abelian groups equipped with composition, the usual pointwise operations, and the quasi-order of kernel inclusion. The resulting structures are associative rings enriched by a quasiorder satisfying a finite set of laws. More generally, we axiomatize the kernel inclusion quasi-order on the ring R induced by a right R-module, and we call the resulting abstract structures rings with ker-order. A characterisation of the possible ker-orders on a fixed ring is given in terms of certain families of its right ideals. The lattice of all ker-orders on the ring of rational integers is described.

Algebraic and topological structure of some spaces of set-valued maps

In this paper we consider possible topological and algebraic structures on some spaces of set-valued maps. In particular, we introduce algebraic operations on the set M(X,Y) of all minimal upper semi-continuous compact-valued maps from a topological space X into a topological group Y. It is shown that, under suitable assumptions on the spaces X and Y, we may equip the set M(X,Y) with a group structure. This structure extends the usual pointwise operations on the set of point-valued continuous functions. We also introduce convergence structures on certain sets of set-valued maps. In particular, we consider the continuous convergence structure on sets of upper semi-continuous maps, as well as a convergence structure on M(X,Y) derived through it, which is compatible with the mentioned algebraic structure. It is also shown that the generalized compact-open topology is compatible with the algebraic structure introduced on M(X,Y).

Operational semantics of proto

The Proto spatial computing language in Beal and Bachrach (2006) [1] simplifies the creation of scalable, robust, distributed programs by abstracting a network of locally communicating devices as a continuous geometric manifold. However, Proto’s successful application in a number of domains is challenging its coherence across different platforms and distributions. We thus present a complete operational semantics for the Proto language, as executed asynchronously on a network of fast message-passing devices. This semantics covers all of the operations of the three space–time operator families unique to Proto—restriction, feedback, and neighborhood—as well as the current pointwise operations that it shares with most other languages. This formalization will provide a reference to aid implementers in preserving language coherence across platforms, domains, and distributions. The formalization process has also advanced the Proto language in several ways, which we explain in detail.

Relative version of the Titchmarsh convolution theorem

We consider the algebra Cu = Cu(ℝ) of uniformly continuous bounded complex functions on the real line ℝ with pointwise operations and sup-norm. Let I be a closed ideal in Cu invariant with respect to translations, and let ahI (f) denote the minimal real number (if it exists) satisfying the following condition. If λ > ahI (f), then \(\left. {\left( {\hat f - \hat g} \right)} \right|_V = 0\) for some g ∈ I, where V is a neighborhood of the point λ. The classical Titchmarsh convolution theorem is equivalent to the equality ahI(f1 · f2) = ahI (f1) + ahI (f2), where I = {0}. We show that, for ideals I of general form, this equality does not generally hold, but ahI (fn) = n · ahI (f) holds for any I. We present many nontrivial ideals for which the general form of the Titchmarsh theorem is true.

Topological stable rank of H ∞(Ω) for circular domains Ω

Let Ω be a circular domain, that is, an open disk with finitely many closed disjoint disks removed. Denote by H ∞(Ω) the Banach algebra of all bounded holomorphic functions on Ω, with pointwise operations and the supremum norm. We show that the topological stable rank of H ∞(Ω) is equal to 2. The proof is based on Suárez’s theorem that the topological stable rank of H ∞(\( \mathbb{D} \)) is equal to 2, where \( \mathbb{D} \) is the unit disk. We also show that for circular domains symmetric to the real axis, the Bass and topological stable ranks of the real-symmetric algebra H ∞ℝ (Ω) are 2.

Some algebraic properties of theWiener–Laplace algebra

We denote by W +(ℂ +) the set of all complex-valued functions defined in the closed right half plane ℂ + D {s ∈ ℂ | Re(s) ≥ 0} that differ from the Laplace transform of functions from L 1(0, ∞) by a constant. Equipped with pointwise operations, W +(C +) forms a ring. It is known that W +(ℂ +) is a pre-Bezout ring. The following properties are shown for W +(ℂ +): W +(ℂ +) is not a GCD domain, that is, there exist functions F 1, F 2 in W +(ℂ +) that do not possess a greatest common divisor in W +(ℂ +). W +(ℂ +) is not coherent, and in fact, we give an example of two principal ideals whose intersection is not finitely generated. We will also observe that W +(ℂ +) is a Hermite ring, by showing that the maximal ideal space of W +(ℂ +), equipped with the Gelfand topology, is contractible.

Noncoherence of a Causal Wiener Algebra Used in Control Theory

Let , and let denote the ring of all functions such that , where , and equipped with pointwise operations. (Here denotes the Laplace transform.) It is shown that the ring is not coherent, answering a question of Alban Quadrat. In fact, we present two principal ideals in the domain whose intersection is not finitely generated.

Optimal ‘Tomography’ of 2-Layered Targets: 3D Parameters Reconstruction from Shadow Images

A method of ICF targets parameters reconstruction from the set of backlit shadowgraph images was developed. Proposed approach can be used for nondestructive inner (DT ice in the case of cryotarget) surface quality characterization of single- and double-layered targets and shells.Previously designed computer 3D ray-tracing model allowed us to carry out detailed investigation of the target shadowgraph image formation, to localize rays forming bright ring and to infer analytical description of this rays’ group. Having been guided by this experience we designed an algorithm of inner surface shape determination using bright ring location on target’s image and developed corresponding software package.This package provides a wide set of image processing tools: both general processing (pointwise operations, spatial filtering, maximums and edges localization, etc.) and specific methods (3D reconstruction, inner and outer surfaces RMS and power spectra estimation, results’ visualization in different forms, etc).Proposed method and its software implementation were tested using two kinds of image sets – set of backlit photographs of real one-layered shells and set of digitally synthesized shadowgraph images.

Enhanced sensitivity digital holographic interferometry

Corona discharges are generally characterized by a low optical density whose detection is often near or under the limits of interferometric techniques. In this paper, we propose a method of digital holographic interferometry that enables detection with enhanced sensitivity. This sensitivity increase is obtained by post-processing the digital holographic recordings with a set of point-wise image operations. The procedure is described mathematically and illustrated experimentally. Examples are given for an opaque object and for DC corona discharges generated in the symmetrical point-plane geometry.

Revisiting the free 2-generator abelian [formula omitted]-group

The free n -generator abelian ℓ -group ( F n , P ) was characterized by G. Birkhoff as the smallest ℓ -group of real-valued functions over R n containing the n identity functions, and closed under the pointwise operations of addition, subtraction, max and min. It follows that F n is a subdirect product of integers. Since F n is countable, by a theorem of Baer and Specker, F n is a free abelian group, F n = ⨁ i ∈ N 〈 χ i 〉 for some generators χ i . There is no direct description in the literature of those elements n 1 χ 1 + ⋯ + n k χ k which belong to the monoid P of positive elements of F n . A simple description is given in this elementary note for the case n = 2 .

Grey Colour Sharpening

We propose a colour to greyscale algorithmproviding colour separation as well as edge and texture enhancement. An image dependent grey-axis is computed based on the colour distribution in the image. An initial greyscale image is created by a point-wise operation where the grey value is the magnitude of the RGB coordinates re-mapped to the grey axis. The resulting greyscale image is enhanced by applying a novel correction mask. This mask, resembling an unsharp mask, is the sum of the difference between each of the colour components and a blurred version of the greyscale image. The resulting greyscale images are rich in detail without undesirable artifacts.

A Banach algebra of functions of several variables of finite total variation and Lipschitzian superposition operators. I

It is shown that the space of functions of n real variables with finite total variation in the sense of Vitali, Hardy and Krause, defined on a rectangle I a b ⊂ R n , is a Banach algebra under the pointwise operations and Hildebrandt–Leonov's norm. This result generalizes the classical case of functions of bounded Jordan variation on an interval I a b = [ a , b ] for n = 1 and a previous result of the author in [Monatsh. Math. 137(2) (2002) 99–114] for n = 2 .

A Characterization of the free n-generated MV-algebra

An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1−x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free n is the algebra of [0,1]-valued functions over the n-cube [0,1] n generated by the coordinate functions ξ i ,i=1, . . . ,n, with pointwise operations. Any such function f is a McNaughton function, i.e., f is continuous, piecewise linear, and each piece has integer coefficients. Conversely, McNaughton proved that all McNaughton functions f: [0,1] n →[0,1] are in Free n . The elements of Free n are logical equivalence classes of n-variable formulas in the infinite-valued calculus of Łukasiewicz. The aim of this paper is to provide an alternative, representation-free, characterization of Free n .

Superposition Operators in the Algebra of Functions of Two Variables with Finite Total Variation

We show that the Hardy space of functions of two variables with finite total variation is a Banach algebra under the pointwise operations and a suitably chosen norm. Then we characterize Nemytskii superposition operators, which map the Hardy space into itself and satisfy the global Lipschitz condition.

Blind inverse gamma correction

The luminance nonlinearity introduced by many imaging devices can often be described by a simple point-wise operation (gamma correction). This paper presents a technique for blindly estimating the amount of gamma correction in the absence of any calibration information or knowledge of the imaging device. The basic approach exploits the fact that gamma correction introduces specific higher-order correlations in the frequency domain. These correlations can be detected using tools from polyspectral analysis. The amount of gamma correction is then estimated by minimizing these correlations.