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For refinable function-based affine bi-frames, nonhomogeneous ones admit fast algorithms and have extension principles as homogeneous ones. But all extension principles are based on some restrictions on refinable functions. So it is natural to ask what are expected from general refinable functions. In this paper, we introduce the notion of weak nonhomogeneous affine bi-frame (WNABF). Under the setting of reducing subspaces of L 2(R d ), we characterize WNABFs and obtain a mixed oblique extension principle for WNABFs based on general refinable functions.

In this paper, we introduce the notion of partial affine system that is a subset of an affine system. It has potential applications in signal analysis. A general affine system has been extensively studied; however, the partial one has not. The main focus of this paper is on partial affine system–based frames and dual frames. We obtain a necessary condition and a sufficient condition for a partial affine system to be a frame and present a characterization of partial affine system–based dual frames. Some examples are also provided.

We prove that generically, for a self-affine set in $\mathbb{R}^d$, removing one of the affine maps which defines the set results in a strict reduction of the Hausdorff dimension. This gives a partial positive answer to a folklore open question., Comment: arXiv admin note: text overlap with arXiv:1609.07360

Estimating homography using only one affine correspondence.The proposed theory makes multi-homography estimation less ambiguous.Stochastic sampling can be omitted from robust homography estimation.Affine-covariant detectors are compared w.r.t. quality of estimated homographies.Equivalence of affine and perspective-invariances for known epipolar geometry. We propose a method, called HAF, to estimate planar homography from an affine correspondence satisfying the epipolar constraint in an image pair. An affine correspondence consists of a point pair and the related local affine transformation mapping the pixels infinitely close to the point locations from the first to the second images. As a minimal solver, it estimates the homography from only one correspondence, however, it is generalized for the over-determined case as well. The required local affinities are obtained by affine-covariant feature detectors accurately. As a side-effect of the tests, the state-of-the-art affine-covariant detectors are compared to each other w.r.t. the accuracy of the estimated point-wise homographies. The proposed method is validated both on the publicly available AdelaideRMF dataset and in a synthetic testing environment.

In the present paper we obtain a new proof of the fundamental equations of F-planar mappings of manifolds with affine connections. We discuss alternative ways in the definition of F-planar mappings.

Nonlinear single-input affine systems represented in a canonical (normal) form are considered. The control resource is assumed to be constrained. For a closed-loop system obtained by applying a linearizing feedback, the problem of finding an estimate of the attraction domain is set. A method for constructing an ellipsoidal estimate that is based on results of absolute stability theory is suggested. Construction of the estimate reduces to solving a system of linear matrix inequalities. The discussion is illustrated by numerical examples.

In this paper we introduce a notion of a para-complex affine hypersphere. We give a complete local classification of such hypersurfaces and give several examples. It turns out that every para-complex affine hypersphere can be constructed from (real) affine hyperspheres. As an application, we classify all 2-dimensional para-complex affine hyperspheres.

Reduced affine arithmetic (RAA) eliminates the main deficiency of the standard affine arithmetic (AA), i.e. a gradual increase of the number of noise symbols, which makes AA inefficient in a long computation chain. To further reduce overestimation in RAA computation, a new algorithm for the Chebyshev minimum-error multiplication of reduced affine forms is proposed. The algorithm yields the minimum Chebyshev-type bounds and works in linear time, which is asymptotically optimal. We also propose a simplified \(\mathcal {O}(n\log n)\) version of the algorithm, which performs better for low dimensional problems. Illustrative examples show that the presented approach significantly improves solutions of many numerical problems, such as the problem of solving parametric interval linear systems or parametric linear programming, and also improves the efficiency of interval global optimisation.

This paper theoretically analyzes the Mean Square Error (MSE) on the steady-state frequency estimation realized by the all-pass-based adaptive notch filtering algorithms with the normalized lattice structure. The adaptive algorithms to be considered are the Simplified Lattice Algorithm (SLA) proposed by Regalia and the Affine Combination Lattice Algorithm (ACLA) proposed by the authors. For these two algorithms, we derive the frequency estimation MSE in closed-form. The derivation is based on construction of a linear time-invariant model for generation of frequency estimation error, and division of this model into two submodels of which output signals are statistically uncorrelated to each other. This strategy leads to more accurate theoretical MSE expressions than the direct use of the existing analysis methods. Simulation results demonstrate that our theoretical MSE expressions are in very good agreement with the simulated MSE values. HighlightsThe frequency estimate MSEs are derived in closed-form for SLA and ACLA.The proposed MSE expressions show higher accuracy than the conventional methods.The key point in MSE derivation is division of error generation models.

In this paper, a modified Hestenes–Stiefel conjugate gradient method for unconstrained problems is developed, which can achieves the twin goals of generating sufficient descent direction at each iteration as well as being close to the Newton direction. In our methods, the hybridization parameter can also be obtained based on other kinds of conjugacy conditions. Under mild condition, we establish their global convergence for general objective functions. Numerical experimentation with the new method indicates that it efficiently solves the test problems and therefore is promising.

Dimensional reduction is a primary way to analyze and work with complex and large amount of multidimensional data by avoiding the effect of curse of dimensionality. This problem of constructing low dimensional embedding gains importance in number of fields like artificial intelligence, image processing, geographical research and lot more. In this paper, we introduce a modified locally linear embedding, an unsupervised learning algorithm that computes low dimensional data from complex high dimensional data using affine transformation and neighborhood preserving embedding. Unlike novel locally linear embedding, our method is affine invariant where each point is being represented by an affine combination of its neighboring points. At the end, we conduct the experiment to evaluate our proposed method and compare its performance with existing methods. Results show that our proposed method is unaffected by affine transformation, specifically shear while existing methods fail to produce correct results in case of shear.

In this paper we study fundamental equations of geodesic mappings of manifolds with affine connection onto symmetric manifolds. We obtain fundamental equations of this problem. At the end of our paper we demonstrate example of studied mappings.

We study affine Bessel sequences in connection with the spectral theory and the multishift structure in Hilbert space. We construct a non-Besselian affine system fun(x)g1 n=0 generated by continuous periodic function u(x). The result is based on Nikishin?s example concerning convergence in measure. We also show that affine systems fun(x)g1 n=0 generated by any Lipchitz function u(x) are Besselian.

This paper proposes a robust set-membership affine projection algorithm with coefficient vector reuse (RSM-APA-CVR) for high background noise environment. In the proposed algorithm, the sum of the squared $$L_{2}$$ norms of the differences between the updated weight vector and past weight vectors is minimized and a new robust error bound is designed. Simulation results in acoustic echo cancellation context show that the proposed algorithm has faster convergence rate and smaller steady-state misalignment as compared to the conventional RSM-APA.

We present a novel method, called Simplex Assembly , to compute inversion-free mappings with low or bounded distortion on simplicial meshes. Our method involves two steps: simplex disassembly and simplex assembly. Given a simplicial mesh and its initial piecewise affine mapping, we project the affine transformation associated with each simplex into the inversion-free and distortion-bounded space. The projection disassembles the input mesh into disjoint simplices. The disjoint simplices are then assembled to recover the original connectivity by minimizing the mapping distortion and the difference of the disjoint vertices with respect to the piecewise affine transformations, while the piecewise affine mapping is restricted inside the feasible space. Due to the use of affine transformations as variables, our method explicitly guarantees that no inverted simplex occurs, and that the mapping distortion is below the bound during the optimization. Compared with existing methods, our method is robust to an initialization with many inverted elements and positional constraints. We demonstrate the efficiency and robustness of our method through a variety of geometric processing tasks.

Correlation filter achieves promising performance with high speed in visual tracking. However, conventional correlation filter based trackers cannot tackle affine transformation issues such as scale variation, rotation and skew. To address this problem, in this paper, we propose a part-based representation tracker via kernelized correlation filter (KCF) for visual tracking. A Spatial-Temporal Angle Matrix (STAM), severed as confidence metric, is proposed to select reliable patches from parts via multiple correlation filters. These stable patches are used to estimate a 2D affine transformation matrix of the target in a geometric method. Specially, the whole combination scheme for these stable patches is proposed to exploit sampling space in order to obtain numerous affine matrices and their corresponding candidates. The diversiform candidates would help to seek for the optimal candidate to represent the object's accurate affine transformation in a higher probability. Both qualitative and quantitative evaluations on VOT2014 challenge and Object Tracking Benchmark (OTB) show that the proposed tracking method achieves favorable performance compared with other state-of-the-art methods.

Affine invariant measures are powerful tools to develop robust shape descriptors that can be applied, for example, to shape matching, shape retrieval, or symmetry detection problems. In this work we introduce estimators for the affine structure of surfaces represented by triangle meshes, i.e. affine co-normal and normal vectors, affine curvature tensors, affine mean and Gaussian curvatures, and affine principal directions and curvatures. The proposed method estimates the affine normal using a finite differences scheme together with a least-squares approximation, followed by a weighted average strategy to approach discrete affine curvature tensors. When compared to the exact geometric measures of analytic models, experiments on regular meshes obtain small error, which decreases for finer meshes, and outperforms the state-of-the-art method in some cases. Experiments to evaluate affine invariance show that the difference between measures before and after equi-affine transformations remains small even after large deformations.

We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. Numerical results are shown for optimal control problems subject to a nonlinear heat equation.

A generalized polygon is an ordered set of vertices. This notion generalizes the concept of the boundary of a polygonal shape because self-intersections are allowed. In this paper we study the problem of matching generalized polygons under affine transformations. Our approach is based on invariants. Firstly we associate an ordered set of complex numbers with each polygon and construct a collection of complex scalar functions on the space of plane polygons. These invariant functions are defined as quotients of the so-called Fourier descriptors, also known as discrete Fourier transforms.Each one of these functions is invariant under similarity transformations; that is, the function associates the same complex number to similar polygons. Moreover, if two polygons are affine related (one of them is the image of the other under an affine transformation), the pseudo-hyperbolic distance between their associated values is a constant that depends only on the affine transformation involved, but independent of the polygons.More formally, given a collection { Z 1 , Z 2 , ź , Z m } of n-sided polygons in the plane and a query polygon W, we give algorithms to find all Z ź such that f ( Z ź ) = W + Δ W , where f is an unknown affine transformation and Δ W = ( Δ w 1 , ź , Δ w n ) with | Δ w k | ź ź , where ź is certain tolerance.

In this paper, we introduce a Bayesian framework to perform model selection for nonlinear acoustic echo cancellation. This is especially important for scenarios where the functional form of the underlying nonlinear distortion is time-varying and/or is unknown, e.g., nonlinear distortions that vary with the volume level of the loudspeakers. To this end, the proposed method evaluates the model probabilities, or what is known as the evidence density, in a Bayesian manner. Thus, unlike convex and affine combination schemes of adaptive filters, the proposed method optimizes both the model complexity as well as the model performance by a single criterion. Moreover, by using the significance-aware principle, the proposed framework is realized in a computationally efficient way. The method is validated by three experiments using synthesized time-invariant nonlinearities, synthesized time-varying nonlinearities, and using real recorded nonlinearities.

This work presents a data driven control strategy able to track a set point without steady state error. The control sequence is computed as an affine combination of past control signals, which belong to a set of past closed loop trajectories stored in a process historian database. This affine combination is computed so that the variance of the tracking error is minimized. It is shown that offset free control (zero mean tracking error) is achieved under the assumption that the underlying dynamics are linear and the closed loop trajectories of the database are in turn offset free. That is, the proposed strategy inherits the offset free tracking capability of the stored past closed loop trajectories. No prior or subsequent knowledge about the process dynamics is required. The procedure to build the database is to store only the best trajectories that meet a design criteria, chosen from a series of iteratively tuned controllers. In this way the proposed controller will learn how to obtain a well tuned control in spite of the different operating conditions.

Part I Affine geometry: vector spaces matrices systems of linear equations some linear algebra rank determinants affine space - (I) - (II) geometry of affine planes geometry of affine space linear maps linear maps and matrices, affine changes of coordinates linear operators transformation groups. Part II Euclidean geometry: bilinear and quadratic forms diagonalizing quadratic forms scalar product vector product Euclidean space unitary operators and isometries isometries of the plane and of three-dimensional space the complex case.

This work presents a data driven control strategy able to track a set point without steady-state error. The control sequence is computed as an affine combination of past control signals, which belong to a set of trajectories stored in a process historian database. This affine combination is computed so that the variance of the tracking error is minimised. It is shown that offset free control, that is zero mean tracking error, is achieved under the assumption that the state is measurable, the underlying dynamics are linear and the trajectories of the database share the same error dynamics and are in turn offset free. The proposed strategy learns the underlying controller stored in the database while maintaining its offset free tracking capability in spite of differences in the reference, disturbances and operating conditions. No training phase is required and newly obtained process data can be easily taken into account. The proposed strategy, related to direct weight optimisation learning techniques, is tested on a process control trainer. MINECO-Spain and FEDER Funds project DPI2016-76493-C3-1-R University of Seville(Spain) grant 2014/425

Outlier detection is an important task in data mining, with many practical applications ranging from fraud detection to public health. However, with the emergence of more and more multi-source data in many real-world scenarios, the task of outlier detection becomes even more challenging as traditional mono-source outlier detection techniques can no longer be suitable for multi-source heterogeneous data. In this paper, a general framework based the consistent representations is proposed to identify multi-source heterogeneous outlier. According to the information compatibility among different sources, Manifold learning are combined in the proposed method to obtain a shared representation space, in which the information-correlated representations are close along manifold while the semantic-complementary instances are close in Euclidean distance. Furthermore, the multi-source outliers can be effectively identified in the affine subspace which is learned through affine combination of shared representations from different sources in the feature-homogeneous space. Comprehensive empirical investigations are presented that confirm the promise of our proposed framework.

The Straight-Through Estimator (STE) is widely used for back-propagating gradients through the quantization function, but the STE technique lacks a complete theoretical understanding. We propose an alternative methodology called alpha-blending (AB), which quantizes neural networks to low-precision using stochastic gradient descent (SGD). Our method (AB) avoids STE approximation by replacing the quantized weight in the loss function by an affine combination of the quantized weight w_q and the corresponding full-precision weight w with non-trainable scalar coefficient $\alpha$ and $1-\alpha$. During training, $\alpha$ is gradually increased from 0 to 1; the gradient updates to the weights are through the full-precision term, $(1-\alpha)w$, of the affine combination; the model is converted from full-precision to low-precision progressively. To evaluate the method, a 1-bit BinaryNet on CIFAR10 dataset and 8-bits, 4-bits MobileNet v1, ResNet_50 v1/2 on ImageNet dataset are trained using the alpha-blending approach, and the evaluation indicates that AB improves top-1 accuracy by 0.9%, 0.82% and 2.93% respectively compared to the results of STE based quantization., Comment: conference version accepted by IJCAI-2019

This paper investigates affine control systems with jumps for which the ideal If(g1, …, gm) generated by the drift vector field f in the Lie algebra L(f, g1, …, gm) can be imbedded as a kernel of a linear first-order partial differential equation. It will lead us to uncontrollable affine control systems with jumps for which the corresponding reachable sets are included in explicitly described differentiable manifolds.

In this work we give a systemic study of affine translation surfaces in affine 3-dimensional space. Specifically, we obtain the complete classification of minimal affine translation surfaces. Moreover, we consider affine translation surfaces with some natural geometric conditions, such as constant affine mean curvature and constant Gauss–Kronecker curvature. Some characterization results with these geometric conditions are also obtained.

Recent subspace clustering algorithms, which use sparse or low-rank representations, conduct clustering by considering the errors and noises into their objective functions. Then, the similarity matrix is solved via alternating direction method of multipliers. However, these approaches are subject to the restriction that the characteristic of errors and outliers in sample points should be known as the prior information. Furthermore, these algorithms are time-consuming during the iterative process. Motivated by this observation, this paper proposes a new subspace clustering algorithm: an affine subspace clustering algorithm based on ridge regression. The method introduces ridge regression as objective function which applies affine criteria into subspace clustering. An analytic solution to the problem has been determined for the coefficient matrix. Experimental results obtained on face datasets demonstrate that the proposed method not only improves the accuracy of the clustering results, but also enhances the robustness. Furthermore, the proposed method reduces the computational complexity.

The linear semidefinite programming problem is considered. The dual affine scaling method in which all current iterations belong to the feasible set is proposed for its solution. Moreover, the boundaries of the feasible set may be reached. This method is a generalization of a version of the affine scaling method that was earlier developed for linear programs to the case of semidefinite programming.

The goal of this paper is to clarify when a stochastic partial differential equation with an affine realization admits affine state processes. This includes a characterization of the set of initial points of the realization. Several examples, as the HJMM equation from mathematical finance, illustrate our results., 27 pages

Sufficient conditions for the stability of a nonlinear affine system subject to interval initial conditions are established. These conditions are based on new estimates of the norms of the solutions of the systems of perturbed equations of motion. This stabilization method is used to analyze an electromechanical system with permanent magnet

Since unity step size could guarantee the fastest convergence and more detailed analysis for the affine projection (AP) algorithm, a statistical tracking behavior of AP algorithm is discussed in this paper. Deterministic recursive equations are derived for the mean weight error and mean-square error. All the possible correlations between the adaptive filtering coefficients and the past measurement noise are considered as well.

In this paper we present an affine combination strategy for two adaptive filters. One filter is designed to handle sparse impulse responses and the other one performs better if impulse response is dispersive. Filter outputs are combined using an adaptive mixing parameter and the resulting output shows a better performance than each of the combining filters separately. We also demonstrate that affine combination results in faster convergence than a convex combination of two adaptive filters.

We investigate an algorithm to find a point of an affine subspace in the positive orthant such that it is the closest one to the original point with respect to the Kullback---Leibler distance. This problem is solved by means of the classical Darroch---Ratcliff algorithm (see [1]), while we use ideas of the information geometry founded by Chentsov (see [2]) and Csiszar (see [3]). The main theorem of the present work proves the convergence of that algorithm (the method of the proof is different from previous ones). The proposed algorithm can be applied, e.g., to find the maximum likelihood estimates in an exponential family (see the last section of the paper).

Optimal Sobolev norms under volume preserving affine transformations are considered. It turns out that this minimal transform is equivalent to the ( p , 2 ) Fisher information matrix defined by Lutwak, Lv, Yang, and Zhang. Furthermore, some analytic inequalities regarding to the L p affine and L p sine energies for the optimal function are investigated.

Using the classical Elie Cartan method, we construct a canonical frame for a smooth curve in multidimensional affine space with canonical and non-canonical parameters.

This paper aims to solve the keypoint correspondence problem in lunar surface images, a typical correspondence task under point ambiguity. Point ambiguity may be caused by repetitive patterns, cluttered scenes, and outliers in the images, which makes the local descriptors less discriminative. In this paper we introduce locally affine-invariance constraints on graphs to tackle the keypoint correspondence problem under point ambiguity. The key idea is that each point can be represented with the affine combination of its neighbors. It is suitable for our problem because it is not only invariant to scale and rotational change, but also more resistant to outliers. Specifically, we introduce the locally affine-invariance constraints into the subgraph matching problem and the common subgraph matching problem. The locally affine-invariance constraint is not directly applicable on common subgraph matching due to its dependency on awareness of selected keypoints. This problem is approximately addressed by solving a series of reliable matching identification and rematching problems. In the experiments, we first apply the proposed method on standard graph matching datasets to evaluate its effectiveness on general correspondence problem under point ambiguity, and second validate the applicability on the lunar surface image dataset.

Quantum circuits of the generalized affine transform are devised based on the novel enhanced quantum representation of digital images. A novel quantum image encryption algorithm combining the generalized affine transform with logistic map is suggested. The gray-level information of the quantum image is encrypted by the XOR operation with a key generator controlled by the logistic map, while the position information of the quantum image is encoded by the generalized affine transform. The encryption keys include the independent control parameters used in the generalized affine transform and the logistic map. Thus, the key space is large enough to frustrate the possible brute-force attack. Numerical simulations and analyses indicate that the proposed algorithm is realizable, robust and has a better performance than its classical counterpart in terms of computational complexity.

We prove some general stability results for a family of equations, which generalizes the equation of p-Wright affine functions. In this way we obtain some hyperstability properties for those equations, as well. We also provide some applications of those outcomes in proving inequalities characterizing the inner product spaces and stability of *-homomorphisms of C*-algebras. The main tool in the proofs is a fixed point result in Brzd?k, Chudziak, Pales (2011).

Shape is considered to be one of the most promising tools to represent and recognise an object. In this study, an effective and rigorous shape matching algorithm is developed based on a new descriptor and relaxation labelling technique. For each contour point, the descriptor captures the distribution of all points within the shape region along the vector perpendicular to that from the centroid to the point. In addition to stable affine invariance, the descriptor is robust to noise since it makes use of all points in the shape region. The descriptor distance is used to initialise the contour point matching probability, and relaxation labelling technique is utilised to update the matching probability using a new compatibility coefficient function, which is defined based on the shape projection preserving characteristic. The experiments on synthetic and real remote sensing data are provided to test the performance of the authors’ proposed algorithm. Compared to other four state-of-the-art contour-based shape matching algorithms, their algorithm is more robust and capable of shape matching under affine transformations and noise.

In this paper, the relative p p -affine capacities are introduced, developed, and subsequently applied to the trace theory of affine Sobolev spaces. In particular, we geometrically characterize such a nonnegative Radon measure μ \mu given on an open set O ⊆ R n \mathcal {O}\subseteq \mathbb R^n that naturally induces an embedding of the p p -affine Sobolev class W 0 , d 1 , p ( O ) {W}^{1,p}_{0,d}(\mathcal {O}) into the Lebesgue space L q ( O , μ ) L^q(\mathcal {O},\mu ) (under 1 ≤ p ≤ q > ∞ 1\le p\le q>\infty ) and the exponentially-integrable Lebesgue space exp ⁡ ( ( n ω n 1 n | f | ) n / ( n − 1 ) ) ∈ L 1 ( O , μ ) \exp \big ((n\omega _n^\frac 1n|f|)^{n/(n-1)}\big )\in L^1(\mathcal {O},\mu ) (under p = n p=n ) as well as the Lebesgue space L ∞ ( O , μ ) L^\infty (\mathcal {O},\mu ) (under n > p > ∞ n>p>\infty ) with μ ( O ) > ∞ \mu (\mathcal {O})>\infty . The results discovered here are new and nontrivial.

In this note, we determined the distance signatures of the incidence matrices of affine resolvable designs. This proves a conjecture by Kohei Yamada., Comment: 5 pages

The statistical affine shape theory is set in this work in the context of real normed division algebras. The general densities apply for every field: real, complex, quaternion, octonion, and for any noncentral and non-isotropic elliptical distribution; then the separated published works about real and complex shape distributions can be obtained as corollaries by a suitable selection of the field parameter and univariate integrals involving the generator elliptical function. As a particular case, the complex normal affine density is derived and applied in brain magnetic resonance scans of normal and schizophrenic patients. Statistical affine shape theory in the context of real normed division algebras is studied.Affine shape density is obtained.The complex normal affine density is derived and applied in brain magnetic resonance scans of normal and schizophrenic patients.

The aim in this paper is to construct an affine transformation using the classical physics analogy between the fields of optics and mechanics. Since optics and mechanics both have symplectic structures, the concept of optics can be replaced by that of mechanics and vice versa. We list the four types of eikonal (generating functions). We also introduce a unitary operator for the affine transformation. Using the unitary operator, the kernel (propagator) is calculated and the wavization (quantization) of the Gabor function is discussed. The dynamic properties of the affine transformed Wigner function are also discussed.