Modulo Normalization Research Articles

MTF Measurement by Slanted-Edge Method Based on Improved Zernike Moments.

The modulation transfer function (MTF) is an important parameter for performance evaluation of optical imaging systems in photogrammetry and remote sensing; the slanted-edge method is one of the main methods for measuring MTF. To solve the problem of inaccurate edge detection by traditional methods under the conditions of noise and blur, this paper proposes a new method of MTF measurement with a slanted-edge method based on improved Zernike moments, which firstly introduces the Otsu algorithm to automatically determine the Zernike moment threshold for sub-pixel edge detection to precisely locate the edge points, then obtains LSF through edge point projection, ESF sampling point acquisition, smoothing, fitting, taking ESF curve differential and Gaussian fitting, and finally, accurately obtaining MTF by LSF Fourier transform and modulo normalization. Based on simulation experiments and outdoor target experiments, the reliability of the proposed algorithm is verified by the deviations of slanted-edge angle and MTF measurement, and the tolerance degree of edge detection to noise and ambiguity are analyzed. The results show that compared with ISO 12233, OMNI-sine method, Hough transform method and LSD method, this algorithm has the highest edge detection accuracy, the maximum tolerance of noise and ambiguity, and also improves the accuracy of MTF measurement.

Estimates of $p$-harmonic functions in planar sectors

Suppose that $p \in (1,\infty]$, $v \in [1/2,\infty)$ and $\mathcal{S}_v = \left\{ (x, y) \in \mathbb{R}^2 \setminus \{(0, 0)\}; |\phi| < \frac{\pi}{2v}\right\}$, where $\phi$ is the polar angle of $(x,y)$. Let $R>0$ and $\omega_p(x)$ be the $p$-harmonic measure of $\partial B(0,R) \cap \mathcal{S}_v$ at $x$ with respect to $B(0, R)\cap \mathcal{S}_v$. We prove that there exists a constant $C$ such that \begin{align*} C^{-1}\left(\frac{|x|}{R}\right)^{k(v,p)} \, \leq \omega_p(x) \, \leq C \left(\frac{|x|}{R}\right)^{k(v,p)} \end{align*} where the exponent $k(v,p)$ is given explicitly as a function of $v$ and $p$. Using this estimate we derive local growth estimates for $p$-sub- and $p$-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of $p$-harmonic measure we also derive a sharp Phragmen-Lindel\of theorem for $p$-subharmonic functions in the unbounded sector $\mathcal{S}_v$. Moreover, if $p = \infty$ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in $\mathbb{R}^n$. Finally, when $v \in (1/2, \infty)$ and $p\in (1,\infty)$ we prove a uniqueness result (modulo normalization) for positive $p$-harmonic functions in $\mathcal{S}_v$ vanishing on $\partial\mathcal{S}_v$.

A Simple Shape Descriptor Merging Arithmetical Wrap Around Technique with Absolute Localized Pixel Differences

The quest for computationally simple, highly accurate and precise shape descriptors supporting retrieval continues to be an active research area in computer vision. In this paper, a simple feature descriptor is realized by blending Modulo Arithmetic (MA) with Local Absolute Pixel Differences (LAPD) labelled as MA-LAPD. MA initially refines edges of images through modulo normalization and later operated by LAPD to capture local texture patterns. Subsequently, LAPD encodes the local intensity transitions in eight directions with regard to center pixel. The two prominent directional indices are converted into unique decimal codes that represent each pixel position, thus, transforming each image into a collection of LAPD codes. The ensuing LAPD image is then fabricated into histograms for characterizing the distribution of local features, used for matching and retrieval. Quantitative and qualitative investigations on Kimia’s 99, MPEG-7 Part-B and Tari-1000 datasets reveal consistent Bull’s Eye Retrieval (BER) scores above 91%. Also, relative analysis exposes the superiority of MA-LAPD with its predecessors in majority of the datasets.

Classification of Nonnegative g −Harmonic Functions in Half-Spaces

In this paper we present a short proof of the following classification Theorem for g −harmonic functions in half-spaces. Assume that u is a nonnegative solution to Δgu = 0 in {xn > 0} that continuously vanishes on the flat boundary {xn = 0}. Then, modulo normalization, u(x) = xn in {xn ≥ 0}. Our proof depends on a recent quantitative version of the Hopf-Oleinik Lemma proven by the authors in Braga and Moreira (Adv. Math.334, 184–242, 2018). Moreover, in this paper, we show how to adapt the proofs in the literature to extend Carleson Estimate, Boundary Harnack Inequality and Schwartz Reflection Principle to the context of nonnegative g −harmonic functions. These results are also ingredients for the proof of the main result.

Entanglement and the double copy

We construct entangled states of gluons that scatter exactly as if they were gravitons. Operationally, these objects implement the double copy at the level of the wave function. Our analysis begins with a general ansatz for a wave function characterizing gluons in two copies of SU(N ) gauge theory. Given relatively minimal assumptions following from permutation invariance and dimensional analysis, the three- and four-particle wave functions generate scattering amplitudes that automatically coincide exactly with gravity, modulo normalization. For five-particle scattering the match is not automatic but imposing certain known selection rules on the amplitude is sufficient to uniquely reproduce gravity. The resulting amplitudes exhibit a color-dressed and permutation-invariant form of the usual double copy relations. We compute the entanglement entropy between the two gauge theory copies and learn that these states are maximally-entangled at large N . Moreover, this approach extends immediately to effective field theories, where Born-Infeld photons and Galileons can be similarly recast as entangled gluons and pions.

A trace formula for rigid varieties, and motivic Weil generating series for formal schemes

We establish a trace formula for rigid varieties X over a complete discretely valued field, which relates the set of unramified points on X to the Galois action on its étale cohomology. Next, we show that the analytic Milnor fiber of a morphism f at a point x completely determines the formal germ of f at x. We develop a theory of motivic integration for formal schemes of pseudo-finite type over a complete discrete valuation ring R, and we introduce the Weil generating series of a regular formal R-scheme $${\mathfrak{X}}$$ of pseudo-finite type, via the construction of a Gelfand-Leray form on its generic fiber. When $${\mathfrak{X}}$$ is the formal completion of a morphism f from a smooth irreducible variety to the affine line, then its Weil generating series coincides (modulo normalization) with the motivic zeta function of f. When $${\mathfrak{X}}$$ is the formal completion of f at a closed point x of the special fiber $${f^{-1}(0)}$$ , we obtain the local motivic zeta function of f at x.

Gödelization in the lambda calculus

Gödelization is a meta-linguistic encoding of terms in a language. While it is impossible to define an operator in the lambda calculus which encodes all closed lambda expressions, it is possible to construct restricted versions of such an encoding operator modulo normalization. In this paper, we propose such an encoding operator for proper combinators.

An Extension of Plancherel's Formula to Separable Unimodular Groups

We show that for any f square-integrable on a separable unimodular locally compact group G (relative to Haar measure), the integral over G of the square of the absolute value of f equals the integral over a certain measure space of the square of the (at each point of the space) of the Fourier transform of f. By relative norm we mean that defined by von Neumann [5] for factors, and for the present we define the Fourier transform thru the use of the von Neumann reduction theory [6]. The formula which is obtained here has as instances, though not as direct corollaries, the formula of Plancherel, its generalization to separable locally compact abelian groups, the Peter-Weyl theorem, and a formula recently obtained by Gelfand and Neumark [2] for the case of the Lorentz group. The measure space in question has its measure ring isomorphic to the Boolean ring B of closed linear manifolds in L2(G) invariant under both left and right translations, and the corresponding measure is uniquely determined, modulo normalization of the at each point of the space. This measure ring acts as a kind of measure-theoretic to the group. For example, making the so-called standard normalization of the norm, if G is abelian, then B as a measure ring is abstractly identical with the measure ring of the character group of G, under Haar measure; if G is compact, then B is abstractly the measure ring of a discrete set of points, the set being in one-to-one correspondence with the collection of equivalence classes of continuous irreducible representations of G, the measure of a point being proportional to the degree of the corresponding representation. Basic in our work is a certain countably-additive non-negative function, which we call the dual of G, defined on the lattice of all projections in the weak closure V of the algebra generated by left translations in L2(G),-or alternatively, on the lattice of closed linear manifolds in L2(G) invariant under right translations. The gage, which is defined in an intrinsic fashion, is invariant under transformation by unitary operators in V, and is in fact a weight function in the terminology of von Neumann [6]. His reduction theory consequently yields a representation of the gage as an integral over the measure space described above of constituents (relative dimension functions) arising from factors. This representation together with the facts that the convolution operator on L2(G) defined by a self-adjoint element of L2(G) is hypermaximal symmetric (Ambrose [1]) and that every bounded linear operator on L2(G) which commutes with all right translations is in V (Segal [4]), are the principal known results used in our derivation of the generalized Plancherel formula.