Accurate Quaternion Polar Harmonic Transform for Color Image Analysis

Polar harmonic transforms (PHTs) have been applied in pattern recognition and image analysis. But the current computational framework of PHTs has two main demerits. First, some significant color information may be lost during color image processing in conventional methods because they are based on RGB decomposition or graying. Second, PHTs are influenced by geometric errors and numerical integration errors, which can be seen from image reconstruction errors. This paper presents a novel computational framework of quaternion polar harmonic transforms (QPHTs), namely, accurate QPHTs (AQPHTs). First, to holistically handle color images, quaternion-based PHTs are introduced by using the algebra of quaternions. Second, the Gaussian numerical integration is adopted for geometric and numerical error reduction. When compared with CNNs (convolutional neural networks)-based methods (i.e., VGG16) on the Oxford5K dataset, our AQPHT achieves better performance of scaling invariant representation. Moreover, when evaluated on standard image retrieval benchmarks, our AQPHT using smaller dimension of feature vector achieves comparable results with CNNs-based methods and outperforms the hand craft-based methods by 9.6% w.r.t mAP on the Holidays dataset.

The primitive equations in the scaling-invariant space $$L^{\infty }(L^1)$$

Consider the primitive equations on $$\mathbb {R}^2\times (z_0,z_1)$$ R 2 × ( z 0 , z 1 ) with initial data a of the form $$a=a_1+a_2$$ a = a 1 + a 2 , where $$a_1 \in \mathrm{BUC}_\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$ a 1 ∈ BUC σ ( R 2 ; L 1 ( z 0 , z 1 ) ) and $$a_2 \in L^\infty _\sigma (\mathbb {R}^2;L^1(z_0,z_1))$$ a 2 ∈ L σ ∞ ( R 2 ; L 1 ( z 0 , z 1 ) ) . These spaces are scaling-invariant and represent the anisotropic character of these equations. It is shown that for $$a_1$$ a 1 arbitrary large and $$a_2$$ a 2 sufficiently small, this set of equations admits a unique strong solution which extends to a global one and is thus strongly globally well posed for these data provided a is periodic in the horizontal variables. The approach presented depends crucially on mapping properties of the hydrostatic Stokes semigroup in the $$L^\infty (L^1)$$ L ∞ ( L 1 ) -setting. It can be seen as the counterpart of the classical iteration schemes for the Navier–Stokes equations, now for the primitive equations in the $$L^\infty (L^1)$$ L ∞ ( L 1 ) -setting.

Hubble parameter and related formulas for a Weyl scaling invariant dark energy action

We extend our previous analysis of a model for “dark energy” based on a Weyl scaling invariant dark energy action. We reexpress all prior results in terms of proper time, using the fluctuation amplitude [Formula: see text] without approximation, and derive a compact formula for the squared effective Hubble parameter. This formula involves effective dark energy and matter densities that differ from their expressions in the standard [Formula: see text] cosmology. We also give new analytic results for the function [Formula: see text] and discuss their implications.

Regularity criterion for 3D nematic liquid crystal flows in terms of finite frequency parts in $\dot{B}_{\infty,\infty }^{-1}$

In this paper, we establish the regularity criterion for the weak solution of nematic liquid crystal flows in three dimensions when the $L^{\infty }(0,T;\dot{B}_{\infty,\infty }^{-1})$ L ∞ ( 0 , T ; B ˙ ∞ , ∞ − 1 ) -norm of a suitable low frequency part of $(u,\nabla d)$ ( u , ∇ d ) is bounded by a scaling invariant constant and the initial data $(u_{0},\nabla d_{0})$ ( u 0 , ∇ d 0 ) . Our result refines the corresponding one in (Liu and Zhao in J. Math. Anal. Appl. 407:557-566, 2013) and that in (Ri in Nonlinear Anal. TMA 190:111619, 2020).