Order Directional Derivatives Research Articles

A dual number formulation to efficiently compute higher order directional derivatives

This contribution proposes a new formulation to efficiently compute directional derivatives of order one to fourth. The formulation is based on automatic differentiation implemented with dual numbers. Directional derivatives are particular cases of symmetric multilinear forms; therefore, using their symmetric properties and their coordinate representation, we implement functions to calculate mixed partial derivatives. Moreover, with directional derivatives, we deduce concise formulas for the velocity, acceleration, jerk, and jounce/snap vectors. The utility of our formulation is proved with three examples. The first example presents a comparison against the forward mode of finite differences to compute the fourth-order directional derivative of a scalar function. To this end, we have coded the finite differences method to calculate partial derivatives until the fourth order, to any order of approximation. The second example presents efficient computations of the velocity, acceleration, jerk, and jounce/snap. Finally, the third example is related to the computation of some partial derivatives. The implemented code of the proposed formulation and the finite differences method is proportioned as additional material to this article.

Convexity of nonlinear mappings between bounded linear operator spaces

<abstract><p>Motivated by the work <sup>[<xref ref-type="bibr" rid="b7">7</xref>]</sup>, in which the author studied the convexity of nonlinear mappings defined between bounded linear operator spaces, our research extends this inquiry. In this work, we continue the study of the convexity of nonlinear mappings defined between bounded linear operator spaces and we establish a characterization in terms of the second order directional derivative. We apply the main result to prove the convexity and the nonconvexity of well-known nonlinear mappings. The case of nondifferentiable mappings is also treated in the last section.</p></abstract>

QRFODD: Quaternion Riesz fractional order directional derivative for color image edge detection

Edge detection is the prominent method to determine the discontinuities present in an image. There exists an issue of loss of information and correlation during the extraction of edges in color images. To resolve it, a novel quaternion domain based Riesz fractional order directional derivative approach is presented in this paper to extract more edge details in color images. The Riesz fractional masks obtained by utilizing Aitken interpolation are applied on color images in the quaternion domain for yielding the edge map as there is no need for recalculation of basis polynomials in case of the addition of any data point. The performance of the proposed quaternion based technique is analyzed on the benchmark BSDS300 and BSDS500 datasets in the terms of Figure of Merit and F-Score. The results obtained exhibit the efficiency of proposed technique over the classical and state-of-the-art edge detection approaches. The robustness of this approach is further established by taking into account uncontrolled conditions such as noise, JPEG compression, and varying illumination as it provided more features than the classical integer-order detectors. Moreover, the superiority of the proposed quaternion based technique is validated for medical images as well as its application in image enhancement.

Analysis of directional higher order jump discontinuities with trigonometric shearlets

<p style='text-indent:20px;'>In a recent article, we showed that trigonometric shearlets are able to detect directional step discontinuities along edges of periodic characteristic functions. In this paper, we extend these results to bivariate periodic functions which have jump discontinuities in higher order directional derivatives along edges. In order to prove suitable upper and lower bounds for the shearlet coefficients, we need to generalize the results about localization- and orientation-dependent decay properties of the corresponding inner products of trigonometric shearlets and the underlying periodic functions.</p>

Integration of Probability Based Ridge Variation Information with Local Ridge Orientation for Fingerprint Liveness Detection

Fingerprints are commonly used in biometric systems. However, the authentication of these systems became an open challenge because fingerprints can easily be fabricated. In this paper, a hybrid feature extraction approach named Integration of Probability Weighted Spatial Gradient with Ridge Orientation (IPWSGRo) has been proposed for fingerprint liveness detection. IPWSGRo integrates intensity variation and local ridge orientation information. Intensity variation is computed by using probability-weighted moments (PWM) and second order directional derivative filter. Moreover, the ridge orientation is estimated using rotation invariant Local Phase Quantization (LPQri) by retaining only the significant frequency components. These two feature vectors are quantized into predefined intervals to plot a 2-D histogram. The support vector machine classifier (SVM) is then used to determine the validity of fingerprints as either live or spoof. Results are obtained by applying the proposed technique on three standard databases of LivDet competition 2011, 2013, and 2015. Experimental results indicate that the proposed method is able to reduce the average classification error rates (ACER) to 5.7, 2.1, and 5.17% on LivDet2011, 2013, and 2015, respectively.

Texture Feature Extraction of Lumbar Spine Trabecular Bone Radiograph Image using Laplacian of Gaussian Filter with KNN Classification to Diagnose Osteoporosis

The human bones are categorized based on elemental micro architecture and porosity. The porosity of the inner trabecular bone is high that is 40-95% and the nature of the bone is soft and spongy whereas the cortical bone is harder and is less porous that is 5 to 15%. Osteoporosis is a disease that normally affects women usually after their menopause. It largely causes mild bone fractures and further stages lead to the demise of an individual. The detection of Osteoporosis in Lumbar Spine has been widely recognized as a promising way to frequent fractures. Therefore, premature analysis of osteoporosis will estimate the risk of the bone fracture which prevents life threats. The paper is systematized in two different sections to classify normal (non-osteoporosis) and abnormal(osteoporosis)Lumbar spine trabecular bone. In this method, the first section is based on discriminating the lumbar spine trabecular bone micro-architecture predisposing by means of first and second order directional derivative of Laplacian of Gaussian filter with different standard deviation to acquire the minimum and maximum responses. The dimension reduction of texture features, quantization and adjacent scale coding with weighted multipliers are used to lessen the intensity variations of texture features. The second section is based on the reduction of histogram features as a training data set for classification of normal and osteoporotic images of lumbar spine (L1-L4) using K-Nearest Neighborhood (KNN) classifier. The tested dataset result gives effective classification accuracy of 97.22% with lesser texture feature dimension. The usage of weight multiplier as well as quantization technique plays a major role for the improvement of accuracy to diagnose osteoporosis for an input noisy and noiseless image.

Second order directional shape derivatives of integrals on submanifolds

<p style='text-indent:20px;'>We compute first and second order shape sensitivities of integrals on smooth submanifolds using a variant of shape differentiation. The result is a quadratic form in terms of one perturbation vector field that yields a second order quadratic model of the perturbed functional. We discuss the structure of this derivative, derive domain expressions and Hadamard forms in a general geometric framework, and give a detailed geometric interpretation of the arising terms.</p>

Optimization in curbing risk contagion among financial institutes

Financial institutions are interconnected by holding debt claims against each other. A default bank may cause its creditors to default, and the risk may be further propagated to up-stream institutes (risk contagion). Such interconnection is a key contributing factor to the past worldwide financial crisis. We show that a good mechanism of default liquidation may improve the total wealth of the financial system and therefore may curb the risk contagion. We formulate this problem as a nonlinear optimization problem with constraints and propose an optimal liquidation policy to minimize the system’s loss. We show that the problem resembles a Markov decision problem (MDP) and therefore we can apply the direct-comparison based optimization approach to solve this problem. Higher order directional derivatives and some optimality properties are obtained. Furthermore, we derive an iterative algorithm which combines both the policy iteration and the gradient based approach to find a local optimal policy, and under some conditions, a global optimal policy. Our work provides a new direction in curbing the risk contagion in financial networks; and it illustrates the advantages of the direct-comparison based approach, originated in the field of discrete event dynamic system, in nonlinear optimization problems.

Directional derivatives and higher order chain rules for abelian functor calculus

In this paper, we consider abelian functor calculus, the calculus of functors of abelian categories established by the second author and McCarthy. We carefully construct a category of abelian categories and suitably homotopically defined functors, and show that this category, equipped with the directional derivative, is a cartesian differential category in the sense of Blute, Cockett, and Seely. This provides an abstract framework that makes certain analogies between classical and functor calculus explicit. Inspired by Huang, Marcantognini, and Young's chain rule for higher order directional derivatives of functions, we define a higher order directional derivative for functors of abelian categories. We show that our higher order directional derivative is related to the iterated partial directional derivatives of the second author and McCarthy by a Faà di Bruno style formula. We obtain a higher order chain rule for our directional derivatives using a feature of the cartesian differential category structure, and with this provide a formulation for the nth layers of the Taylor tower of a composition of functors F∘G in terms of the derivatives and directional derivatives of F and G, reminiscent of similar formulations for functors of spaces or spectra by Arone and Ching. Throughout, we provide explicit chain homotopy equivalences that tighten previously established quasi-isomorphisms for properties of abelian functor calculus.

Retinal Microaneurysms Detection Using Gradient Vector Analysis and Class Imbalance Classification.

Retinal microaneurysms (MAs) are the earliest clinically observable lesions of diabetic retinopathy. Reliable automated MAs detection is thus critical for early diagnosis of diabetic retinopathy. This paper proposes a novel method for the automated MAs detection in color fundus images based on gradient vector analysis and class imbalance classification, which is composed of two stages, i.e. candidate MAs extraction and classification. In the first stage, a candidate MAs extraction algorithm is devised by analyzing the gradient field of the image, in which a multi-scale log condition number map is computed based on the gradient vectors for vessel removal, and then the candidate MAs are localized according to the second order directional derivatives computed in different directions. Due to the complexity of fundus image, besides a small number of true MAs, there are also a large amount of non-MAs in the extracted candidates. Classifying the true MAs and the non-MAs is an extremely class imbalanced classification problem. Therefore, in the second stage, several types of features including geometry, contrast, intensity, edge, texture, region descriptors and other features are extracted from the candidate MAs and a class imbalance classifier, i.e., RUSBoost, is trained for the MAs classification. With the Retinopathy Online Challenge (ROC) criterion, the proposed method achieves an average sensitivity of 0.433 at 1/8, 1/4, 1/2, 1, 2, 4 and 8 false positives per image on the ROC database, which is comparable with the state-of-the-art approaches, and 0.321 on the DiaRetDB1 V2.1 database, which outperforms the state-of-the-art approaches.

High-order local ternary patterns with locality preserving projection for smoke detection and image classification

It is a challenging task to recognize smoke from visual scenes due to large variations in the color, texture, shapes of smoke. To improve detection accuracy, we propose a novel feature extraction method by encoding high order directional derivatives at each pixel. We first quantize the directional derivatives into ternary values to generate Local Ternary Patterns (LTP). For the sake of simplification, each LTP code is usually decomposed into an upper LBP code and a lower LBP code, but this leads to loss of information. Hence, we use joint histograms to preserve the co-occurrence of upper and lower LBP codes for each order LTP. Then we concatenate all joint histograms from different orders to propose High-order Local Ternary Patterns (HLTP). To improve computational efficiency, we apply Locality Preserving Projection (LPP) to reduce the dimension of HLTP. To further improve performance, we present a noise resistant mechanism to remove noisy derivatives, and then propose HLTP based on Magnitudes of noise removed derivatives and values of Center pixels (HLTPMC). Finally, the Support Vector Machine (SVM) is used for training and classification. Experiments on large scale smoke data sets show that our method can achieve detection rates above 94% with false alarm rates below 1.33%. Experiments on a multi-class Brodatz texture data set also achieved good performance with low dimensional features. So our method has powerful discriminative capabilities and compact feature representation for multi-class image classification.

Voice pathology detection using interlaced derivative pattern on glottal source excitation

Abstract In this paper, we propose a voice pathology detection and classification method using an interlaced derivative pattern (IDP), which involves an n -th order directional derivative, on a spectro-temporal description of a glottal source excitation signal. It is shown previously that directional information is useful to detect pathologies due to its encoding ability along time, frequency, and time-frequency axes. The IDP, being an n -th order derivative, is capable of describing more information than a first order derivative pattern by combining all the directional information into one. In the IDP, first-order derivatives are calculated in four directions, and these derivatives are thresholded with the center value of each directional channel to produce the final IDP. A support vector machine is used as a classification technique. Experiments are conducted using three different databases, which are the Massachusetts Eye and Ear Infirmary database, Saarbrucken Voice Database, and Arabic Voice Pathology Database. Experimental results show that the IDP based features give higher accuracy than that using other related features in all the three databases. The accuracies using cross-databases are also high using the IDP features.

Variational Analysis of the Ky Fan k-norm

In this paper, we will study some variational properties of the Ky Fan k-norm 𝜃 = ∥⋅∥(k) of matrices, which are closed related to a class of basic nonlinear optimization problems involving the Ky Fan k-norm. In particular, for the basic nonlinear optimization problems, we will introduce the concept of nondegeneracy, strict complementarity and the critical cones associated with the generalized equations. Finally, we present the explicit formulas of the conjugate function of the parabolic second order directional derivative of 𝜃, which will be referred to as the sigma term of the second order optimality conditions. The results obtained in this paper provide the necessary theoretical foundations for future work on sensitivity and stability analysis of the nonlinear optimization problems involving the Ky Fan k-norm.

Higher order optimality conditions with an arbitrary non-differentiable function

In this paper, we introduce a higher order directional derivative and higher order subdifferential of Hadamard type of a given proper extended real function. We obtain necessary and sufficient optimality conditions of order n (n is a positive integer) for unconstrained problems in terms of them. We do not require any restrictions on the function in our results. In contrast to the most known directional derivatives, our derivative is harmonized with the classical higher order Fréchet directional derivative of the same order in the sense that both of them coincide, provided that the last one exists. A notion of a higher order critical direction is introduced. It is applied in the characterizations of the isolated local minimum of order n. Higher order invex functions are defined. They are the largest class such that the necessary conditions for a local minimum are sufficient for global one. We compare our results with some previous ones. As an application, we improve a result due to V. F. Demyanov, showing that the condition introduced by this author is a complete characterization of isolated local minimizers of order n.

Integrating Feature Direction Information with a Level Set Formulation for Image Segmentation

AbstractA feature-dependent variational level set formulation is proposed for image segmentation. Two second order directional derivatives act as the external constraint in the level set evolution, with the directional derivative across the image features direction playing a key role in contour extraction and another only slightly contributes. To overcome the local gradient limit, we integrate the information from the maximal (in magnitude) second-order directional derivative into a common variational framework. It naturally encourages the level set function to deform (up or down) in opposite directions on either side of the image edges, and thus automatically generates object contours. An additional benefit of this proposed model is that it does not require manual initial contours, and our method can capture weak objects in noisy or intensity-inhomogeneous images. Experiments on infrared and medical images demonstrate its advantages.

Mathematical Foundations and Generalisations of the Census Transform for Robust Optic Flow Computation

In recent years, the popularity of the census transform has grown rapidly. It provides features that are invariant under monotonically increasing intensity transformations. Therefore, it is exploited as a key ingredient of various computer vision problems, in particular for illumination-robust optic flow models. However, despite being extensively applied, its underlying mathematical foundations are not well understood so far. The goal of our paper is to provide these missing insights, and in this way to generalise the concept of the census transform. To this end, we transfer the inherently discrete transform to the continuous setting and embed it into a variational framework for optic flow estimation. This uncovers two important properties: the strong reliance on local extrema and the induced anisotropy of the data term by acting along isolines of the image. These new findings open the door to generalisations of the census transform that are not obvious in the discrete formulation. To illustrate this, we introduce and analyse second order census models that are based on thresholding the second order directional derivatives. Last but not least, we constitute links of census-based approaches to established data terms such as gradient constancy, Hessian constancy, and Laplacian constancy. We confirm our findings by means of experiments.

Robust method for infrared small-target detection based on Boolean map visual theory.

In this paper, we present an infrared small target detection method based on Boolean map visual theory. The scheme is inspired by the phenomenon that small targets can often attract human attention due to two characteristics: brightness and Gaussian-like shape in the local context area. Motivated by this observation, we perform the task under a visual attention framework with Boolean map theory, which reveals that an observer's visual awareness corresponds to one Boolean map via a selected feature at any given instant. Formally, the infrared image is separated into two feature channels, including a color channel with the original gray intensity map and an orientation channel with the orientation texture maps produced by a designed second order directional derivative filter. For each feature map, Boolean maps delineating targets are computed from hierarchical segmentations. Small targets are then extracted from the target enhanced map, which is obtained by fusing the weighted Boolean maps of the two channels. In experiments, a set of real infrared images covering typical backgrounds with sky, sea, and ground clutters are tested to verify the effectiveness of our method. The results demonstrate that it outperforms the state-of-the-art methods with good performance.

English

Super-resolution image reconstruction provides an effective way to increase image resolution from a single or multiple low resolution images. There exists various single image super-resolution based on different assumptions, amongst which edge adaptive algorithms are particularly used to improve the accuracy of the interpolation characterizing the edge features in a larger region. A recent algorithm for image iterative curvature based interpolation (ICBI) performs iterative procedure of the interpolated pixels obtained by the 2 nd order directional derivative of the image intensity. ICBI in comparison with bicubic interpolation and the other interpolation algorithm such as improved new edge directed interpolation (INEDI) provides notably higher values in terms of qualitative and quantitative analysis. Comparative analysis of these algorithms performed on number of test images on the basis of PSNR and RMSE metrics show effectiveness of edge based techniques.

Multidimensional fractional advection-dispersion equations and related stochastic processes

In this paper we study multidimensional fractional advection-dispersion equations involving fractional directional derivatives both from a deterministic and a stochastic point of view. For such equations we show the connection with a class of multidimensional Levy processes. We introduce a novel Levy-Khinchine formula involving fractional gradients and study the corresponding infinitesimal generator of multi-dimensional random processes. We also consider more general fractional transport equations involving Frobenius-Perron operators and their stochastic solutions. Finally, some results about fractional power of second order directional derivatives and their applications are also provided.

Rotation and scale invariant local binary pattern based on high order directional derivatives for texture classification

Local Binary Pattern (LBP) only encodes the first order directional derivatives of a center pixel but it does not consider higher order derivatives. This paper proposes a rotation and scale invariant local binary pattern by jointly taking into account high order directional derivatives, circular shift sub-uniform, and scale space. Each order directional derivatives are independently encoded in a similar way of the first order derivatives to generate a code for the center pixel. Different order derivatives produce different codes that result in several histograms over an image, and then all the histograms multiplied by weights are concatenated together to fully utilize information of different order derivatives. To further improve performance, circular shift sub-uniform and scale space techniques are used to obtain rotation and scale invariant local binary patterns. Extensive experiments show that the high order derivatives based LBP can achieve good performance and obviously outperforms existing methods.