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In this paper, we present a method to increase invariance against affine deformations in feature based object detection systems. We use the gradient distribution of an image region to calculate two non-orthogonal basis vectors defining an affine invariant coordinate system, which is used to normalize the image region. The proposed method is an intermediate processing step subsequent to the feature detection and can be combined with any feature detector and descriptor combination. Its performance is evaluated on locally affine transformed as well as on real world images and compared to state of the art methods for affine invariant feature description. The observed results outperform the results obtained by SIFT, ASIFT or the Harris-Affine based feature normalization method, without introducing significant additional demands on the memory requirement or the computational complexity.

The affine projection algorithm (APA) attracts a wide attention for its advantages such as simple frame, fast convergence rate, and so on. This paper is based on the AP algorithm to introduce nonlinear update equation for the weight error vector. So the affine projection mixed-norm (APMN) algorithm is proposed. In this paper, we use a new approach to analyse the APMN algorithm. In this thesis, the stability and convergence rate of APMN algorithm are analyzed in theory and, simulation and verification of theoretical analysis is also performed.

This paper investigates the identification of continuous piecewise affine systems in state space form with jointly unknown partition and subsystem matrices. The partition of the system is generated by the so-called centers. By representing continuous piecewise affine systems in the max-form and using a recursive Gauss-Newton algorithm for a suitable cost function, we derive adaptive laws to online estimate parameters including both subsystem matrices and centers. The effectiveness of the proposed approach is demonstrated with a numerical example.

International audience; This paper presents a new method for exemplar-based image inpainting using transformed patches. We build upon a recent affine invariant self-similarity measure which automatically transforms patches to compare them in an appropriate manner. As a consequence, it intrinsically extends the set of available source patches to copy information from. When comparing two patches, instead of searching for the appropriate patch transformation in a highly dimensional parameter space, our approach allows us to determine a single transformation from the texture content in both patches. We incorporate the affine invariant similarity measure in a variational formulation for inpainting and present an algorithm together with experimental results illustrating this approach.

Image correspondence is one of the critical tasks across various applications of image processing and computer vision. The simple correspondence is being studied from many years for the purpose of image stitching and stereo correspondence. The images assumed for the simple correspondence have same pixel value even after applying the geometric transformations. In this work, we try to correspond images sharing similar content but vary due to change in acquisition like view angle, scale, and illumination. The use of features for flow computation was proposed in SIFT flow, which was used for correspondence across fields. In this method, dense SIFT descriptors are extracted and flow is estimated for matching the SIFT descriptors between two images. In this work, we applied SIFT flow algorithm on the affine transformations of images to be aligned.

In this paper we consider convex improper affine maps of the 3-dimensional affine space and classify their singularities. The main tool developed is a generating family with properties that closely resembles the area function for non-convex improper affine maps., 11 pages, 2 figures

Feature point based image watermarking against geometric distortions has attracted great attention in recent years. However, for the state-of-the-art intensity based feature points detectors, the feature points often gather at textured portions of the image or on the edges where the change of intensity is significant, so that many feature points capture the same portion of the image, which makes the watermark be vulnerable to local geometric distortions. In this paper, we propose an affine invariant image watermarking scheme with good visual quality and reasonable resistance toward local geometric distortions, which utilizes the intensity probability density-based Harris-Laplace detector. Firstly, the uniform and robust feature points are extracted by utilizing modified Harris-Laplace detector, in which the intensity probability density gradient is used instead of intensity gradient. Then, the affine invariant local ellipse regions (LERs) are constructed adaptively according to the variation of local intensity probability density. Finally, the digital watermark is embedded into the affine invariant LERs in nonsubsampled contourlet transform (NSCT) domain by modulating the lowpass NSCT coefficients. By binding the watermark with the affine invariant LERs, the watermark detection can be done without synchronization error. Experimental results show that the proposed image watermarking is not only invisible and robust against common image processing operations such as sharpening, noise adding, and JPEG compression, but also robust against the global affine transforms and local geometric distortions.

An approach of the edge feature matching of remote sensing images based on parameter decomposition of the affine transformation model is presented. First, the adaptability of affine transformation model used in image matching is analyzed, and parameters of the affine transformation model are decomposed. Second, the sum of squared gradient (SSG) similarity is constructed based on the edge feature in the images, and the parameters of the affine transformation model between the images are solved with the Genetic Algorithms (GA) which is used to obtain the global optimum solution of the similarity. Finally, the performance of the approach is analyzed in theory and validated with remote sensing image matching experiments.

A new technical framework for remote sensing image matching by integrating affine invariant feature extraction and RANSAC is presented. The novelty of this framework is an automatic optimization strategy for affine invariant feature matching based on RANSAC. An automatic way to determine the distance threshold of RANSAC is proposed, which is a key problem to implement this RANSAC-based automatic optimization. Since affine invariant feature matching technology has been successfully applied to remote sensing image matching, we design an experiment to compare the proposed framework (with optimization) with the standard affine invariant feature matching (without optimization). By using three pairs with different types of imagery, the experimental results indicate that the proposed framework can always get higher correctness of image matching in automatic way, compared to the standard affine invariant feature matching technology.

Canonical correlation analysis (CCA) is invariant with regard to affine transformation, but it cannot be directly applied to affine invariant pattern recognition. The reason mainly lies in that many existing CCA-based schemes represent the pattern by matrix-to-vector method, as a result, the structure and spatial information of the original pattern is discarded. In this paper, an affine invariant discriminate analysis (AIDA) method is developed for pattern recognition. Dislike the matrix-to-vector representation, an object is first converted to a projection matrix by central projection transform (CPT). After a point matching process, CCA is performed to projection matrices of the object and the model, and two vectors will be derived. Therefore, the object is classified to a model by the smallest distance between the obtained vectors. Comparisons of experimental results are given with respect to some existing methods, which demonstrate the effectiveness of the proposed AIDA method.

Fast and robust feature extraction is crucial for many computer vision applications such as image matching. The representative and the state-of-the-art image features include Scale Invariant Feature Transform (SIFT), Speeded Up Robust Features (SURF), and Affine SIFT (ASIFT). However, neither of them is fully affine invariant and computation efficient at the same time. To overcome this problem, we propose in this paper a fully affine invariant SURF algorithm. The proposed algorithm makes full use of the affine invariant advantage of ASIFT and the efficient merit of SURF while avoids their drawbacks. Experimental results on applications of image matching demonstrate the robustness and efficiency of the proposed algorithm.

The processing of the images simultaneously degraded by blur and affine transformation has become a key task in many applications and many novel methods are designed specifically for it in which the moment-based methods play an important role. However, the existing moment-based methods all resort to non-orthogonal moments invariants which have problem of information redundancy and are sensitive to noise. In this paper, we construct a new set of combined invariants of orthogonal Legendre moments which hold for blur and affine transformation together. The experimental results show that the proposed invariants have better discriminative power and robustness to noise with the comparison to other invariants.

Within Hybrid systems, piecewise affine systems are a common class to be identified from input/output data. In this paper an improved algorithm for identifying piecewise affine systems is developed. The algorithm stems from clustering-based system identification. An affine output error algorithm is used to identify final models. The performance of the new Piecewise Affine Output Error (PWA-OE) algorithm is demonstrated using experimental data from a Radio Frequency MicroElectroMechanical Systems switch. Compared to the existing state-of-the-art, the PWA-OE algorithm generates a potential 62% improvement in model coefficient accuracy. Furthermore the PWA-OE algorithm is less sensitive to two additional input parameter selections.

Usually, state-of-the-art interest point detectors tend to over-represent the local image structures associating several interest points for each local image structure. This fact avoids interest points to clearly stand out against its neighborhood, losing the ability to clearly describe the global uniqueness of each local image structure. In order to solve this problem we propose a sparse affine invariant blob detector, which tries to describe each blob structure with a single interest point. The proposed detector is carried out in two stages: an initial stage, where a set of scale invariant interest points are located by means of the idea of blob movement and blob evolution (creation, annihilation and merging) along different scales by using a precise description of the image provided by the Gaussian curvature, providing a global bottom-up estimation of the image structure. During the second stage, the shape and location of each scale invariant interest point is refined by fitting an anisotropic Gaussian function, which minimizes the error with the underlying image and simultaneously estimates both the shape and location, by means of a non-linear least squares approach. A comparative evaluation of affine invariant detectors is presented, showing that our approach outperforms state-of-the-art affine invariant detectors in terms of precision and recall, and obtains a similar performance to that of the best ones in terms of repeatability and matching. In addition we demonstrate that our detector does not over-represent blob structures and provides a sparse detection that improves distinctiveness and reduces drastically the computational cost of matching tasks. In order to verify the accuracy and the reduction in the computational cost we have evaluated our detector in image registration tasks.

Hough transform is an effective way in object recognition and applied to many industrial processes. Based on the principle of Hough transform, a new algorithm which can detect objects through an affine transform was proposed in this paper. First, application of Hough transform to extract straight lines in a model image and a scene image, got these coordinates of the lines, sorted according to the direction angle. Because of affine transform and the periodic direction angle, the direction order of the lines on scene image were different from those on the model image, these lines on scene image were expanse a cycle. Finally affine transform parameters were applied to objects detection. The results showed the effectiveness of the algorithm.

Affine Differential Geometry provides a set of measures invariant under a larger set of transformations compared to rigid motions. This leads to several applications using robust shape descriptors. Although affine-invariant operations are already used for surfaces, they do not intend to approximate the definitions of Affine Differential Geometry, which are the basis for further differential invariants. In this work we propose estimators for the local affine structure of an implicit surface, i.e. the affine metric, the co-normal and normal vectors, and the affine Gaussian and mean curvatures. The direct derivation of the formulae from the implicit function theorem lead to very intensive computations and numerical instabilities. This work further proposes a geometrical reduction allowing a much simpler and more stable formulae, and compares the results by incorporating the proposed estimators in Marching Cubes based algorithms.

In this paper we investigate analytic affine control systems = X + uY , u ∈ [ a,b ] , where X,Y is an orthonormal frame for a generalized Martinet sub-Lorentzian structure of order k of Hamiltonian type. We construct normal forms for such systems and, among other things, we study the connection between the presence of the singular trajectory starting at q 0 on the boundary of the reachable set from q 0 with the minimal number of analytic functions needed for describing the reachable set from q 0 .

The size of 3D data stored around the web has become bigger. Therefore the development of recognition applications and retrieval systems of 3D models is important. This paper deals with invariants for 3D models recognition. Thus under general affine transform we propose in a first time determinants of three points to realize invariance under affinity. To solve starting point problem we needs Fourier Series (FS) to extract affine invariant descriptors, called Fourier Series Descriptor (FSD). The difference between first and second approaches: in first approach determinants are computed on cartesian coordinates directly while in the second one determinants are computed from FS coefficients to eliminate dependency on starting point. The FS are also applied on 2D slices to generate affine invariants for 3D volume. FS can be computed based on hole points of volume, but this technique. The principal advantages of proposed approaches are the possibility to handle affine transform and 3D volume. Two types of 3D objects are used in the experimentations: mesh and volume, the Princeton Shape Benchmarek (PSB) is also used to test our descriptor based on FSD.

Affine invariant functions are constructed in spatial domain. Unlike the previous affine representation functions in transform domain, these functions are constructed directly on the object contour without any transformation. To eliminate the effect of the choice of points on the contour, an affine invariant function using seven points on the contour is constructed. For objects with several separable components, a closed curve is derived to construct the affine invariant functions. Several experiments have been conducted to evaluate the performance of the proposed method. Experimental results show that the constructed affine invariant functions can be used for object classification.

In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.

The extraction of affine invariant features plays an important role in many fields of image processing. In this paper, the original image is transformed into new images to extract more affine invariant features. To construct new images, the original image is cut in two areas by a closed curve, which is called general contour (GC). GC is obtained by performing projections along lines with different polar angles. New image is obtained by changing gray value of pixels in inside area. The traditional affine moment invariants (AMIs) method is applied to the new image. Consequently, cutting affine moment invariants (CAMIs) are derived. Several experiments have been conducted to evaluate the proposed method. Experimental results show that CAMIs can be used in object classification tasks.

This paper presents a unified way to estimate the parameters of affine transformation in the absence of original image. With 2-D cyclostationary characterization, we analytically show that the covariance of affine transformed image is periodic with the affine transformation matrix. Based on the relationship between the affine transformation matrix and the position of resampling-caused striking peaks in the 2-D spectrum of the image’s edge map, we further study how to estimate the parameters of several typical affine transformations, e.g., the scaling factor, the rotation angle, the joint scaling and rotation transformation parameters. Examples of the output of our algorithm are shown and comparative results are presented to evaluate the performance of the proposed algorithm.

We propose a method for estimating multiple-hypothesis affine regions from a keypoint by using an anisotropic Laplacian-of-Gaussian (LoG) filter. Although conventional affine region detectors, such as Hessian/Harris-Affine, iterate to find an affine region that fits a given image patch, such iterative searching is adversely affected by an initial point. To avoid this problem, we allow multiple detections from a single keypoint. We demonstrate that the responses of all possible anisotropic LoG filters can be efficiently computed by factorizing them in a similar manner to spectral SIFT. A large number of LoG filters that are densely sampled in a parameter space are reconstructed by a weighted combination of a limited number of representative filters, called "eigenfilters", by using singular value decomposition. Also, the reconstructed filter responses of the sampled parameters can be interpolated to a continuous representation by using a series of proper functions. This results in efficient multiple extrema searching in a continuous space. Experiments revealed that our method has higher repeatability than the conventional methods.

Scale Invariant Feature Transform (SIFT) is a method to detect and match invariant feature points on images, and is robust against contrast, rotation, and scale changes. However, SIFT cannot find many correct matching points between affine transformed images because this method employs Gaussian function for scale parameter which specifies a circle area on image planes. In this paper, we propose a method to use Bi-dimensional Empirical Mode Decomposition (BEMD) for keypoint detection, where a given image is decomposed into Intrinsic Mode Functions (IMFs). Our method also employs Affine Moment Invariants (AMIs) instead of SIFT's feature values. As a result, the proposed method detects more matching points than SIFT in a steep affine transformed image.

This paper proposes a novel method via dynamic tree to solve the non-rigid registration of point sets with large shape difference which is a difficult problem for existing methods. Affine ICP algorithm with bidirectional distance is employed to evaluate the similarity between two point sets, and then non-rigid registration is conducted on similar models and subjects. Subjects with accurate registration results are added in the dynamic tree. These steps are repeated until all subjects are linked in the tree. Therefore, large shape difference is divided into several small deformations by the intermediate point sets, and the registration results of model and every subject in the tree are satisfactory. Experimental results show that our method incredibly improves the accuracy of the registration with two point sets bearing large shape difference compared with existing approaches.

This paper presents a modification to the AffineProjection-Like I Algorithm for adaptive filtering withthe purpose of reducing the number of updates and sothe computational load. The modification is achievedby codifying the error and setting a threshold in orderthat the adaptive filter does not need to calculate thealgorithm every iteration. Simulation results showsame convergence speed than the Affine Projection-Like I algorithm, however, number of updates areconsiderably less comparing them to Affine Projection-Like I algorithm.

An object within low illumination image is difficult to distinguish and its contour needs fast computing speed in real-time visual detection systems. Aiming at above problem, a short step affine transformation (SSAT) Sobel algorithm for image edge detection is represented. First, static image needs to do affine transformation. The corresponding gradient image will be obtained after transformed image with partial derivative calculation. Then using parameter model with short step constraint, it will get the gradient different image. The last operation is to select an appropriate threshold to segment the gradient different image. Compared with one order and two order edge algorithms, SSAT Sobel algorithm can guarantee to get the right edge of image as well as meeting the real time requirements of the visual system.

Visual tracking is a challenging problem in computer vision. Most state-of-the-art visual trackers are lack of an ability to adapt affine transforms of objects. The Lucas-Kanade(L-K) algorithm is a classic gradient decent search method which has been widely applied in template matching, image alignment and visual tracking. Despite the L-K algorithm has a good property on adapting targets affine transforms, it always falls into a local optimal since the background information is accumulatively introduced to the template. In this paper, we propose a novel approach, called the Re-located L-K algorithm(RL-K), to handle affine transforms of objects. In particular, we utilize the Multiple Channel Correlation Filters(MMCFs) to deal with the translations of the object before the L-K's gradient decent search step. In addition, we integrate the gradient orientation features into the L-K algorithm to improve its robustness to visual tracking. The experimental results show that, comparing with the state-of-the-art affine tracking approaches, our approach is running at real-time without accuracy losing.

A new method dealing with recognition of partially occluded and affine distortion objects is presented. The method is designed for objects with smooth curved boundary. It divides an object into affine-invariant parts based on the feature point. And a new approach for matching each part is presented in this paper. Robust Hausdorff distance (RHD) is introduced to measure the similarity between feature points set of model and that of target. In terms of the new RHD, the optimal affine transform can be estimated. And then the sub-curve match pairs are calculated based on the optimal affine transformation. The experimental results show proposed algorithm are capable of coping with partial occlusion and affine transformation.

Image registration is a process of determining the geometrical transformation that aligns two or more images taken from different viewpoints and sensors at different times. Scale Invariant Feature Transform (SIFT) method has gained more popularity since it extracts the highest number of features and matching points compared to Speeded-Up Robust Feature (SURF) and Harris Corner Detector at little computational cost. In this paper, a combination of SIFT and Random Sample Consensus (RANSAC) is used to produce panoramic image. In order to reject outliers and estimate the transformation model, affine and projective transformations are used to study the best geometrical transformations methods to be used. The results shows that the projective transformation has a better performance in terms of accuracy.

Affine transform is widely used in the high speed image processing systems. This transform plays an important role in various high speed applications like Optical quadrature microscopy (OQM), image stabilisation in digital camera and image registration etc. In these applications, transformations of image consume most of the execution time. Hence, for high speed imaging systems, acceleration of Affine transform is very much sought for. In this paper, the pipelined architecture implementation of a proposed inherent parallel algorithm for Affine transform has been presented. The acceleration of the image transformation will help in reducing the processing time of high speed imaging systems. The architecture is mapped in Field programmable gate array (FPGA) and the result shows that the proposed algorithm is almost 4 times faster than the conventional algorithm while retaining the image quality. Using the proposed algorithm, an image of size 1,920 × 1,080 can be transformed with a frame rate of 540 frames per second and the multiplane image synthesis for image stabilisation on the same digital image can be performed with a frame rate of 65 fps.

To solve the problems that exist in present affine-invariant region detection and description methods, a new affine-invariant region detector and descriptor are proposed in this paper. First, affine-invariant regions in an image are detected using a connected-region based method. And then a vector composed of a group of affine invariant moments is adopted to descript the regions. Experiments show the effectiveness and robustness of the method. It is also very fast.

We propose to measure the quality of an affine motion by its steadiness, which we formulate as the inverse of its Average Relative Acceleration (ARA). Steady affine motions, for which ARA=0, include translations, rotations, screws, and the golden spiral. To facilitate the design of pleasing in-betweening motions that interpolate between an initial and a final pose (affine transformation), B and C , we propose the Steady Affine Morph (SAM), defined as A t ∘ B with A = C ∘ B -1 . A SAM is affine-invariant and reversible. It preserves isometries (i.e., rigidity), similarities, and volume. Its velocity field is stationary both in the global and the local (moving) frames. Given a copy count, n , the series of uniformly sampled poses, A i/n ∘ B , of a SAM form a regular pattern which may be easily controlled by changing B , C , or n , and where consecutive poses are related by the same affinity A 1/n . Although a real matrix A t does not always exist, we show that it does for a convex and large subset of orientation-preserving affinities A . Our fast and accurate Extraction of Affinity Roots (EAR) algorithm computes A t , when it exists, using closed-form expressions in two or in three dimensions. We discuss SAM applications to pattern design and animation and to key-frame interpolation.

The paper presents a general method of an automatic deriving affine moment invariants of any weights and orders. The method is based on representation of the invariants by graphs. We propose an algorithm for eliminating reducible and dependent invariants. This method represents a systematic approach to the generation of all relevant moment features for recognition of affinely distorted objects. We also show the difference between pseudoinvariants and true invariants.

An efficient and rigorous algorithm is proposed for contour matching invariant to the full set of affine transformations. The algorithm is based on an invariant theory of orthogonal projection matrices derived from configuration matrices of point sets. Diagonals of the orthogonal projection matrices (DOPM) are used as contour descriptors and affinity measures are deduced to act as criteria for contour matching. Perturbation analysis is performed using the theory of polar decomposition, resulting in quantitative perturbation bounds for the affine-invariant descriptors and the affinity measures. A useful schema of outlier removal based upon the monotonic property of contour correspondence is also embedded in the algorithm. Experiments for synthetic and real-world data are provided to test the algorithm and compare it with the state-of-the-art methods, validating that the algorithm is fast, robust and able to match partial contours with occlusions and outliers under affine or more complex transformations.

In this work we study a class of affine maximal surfaces with singularities that we call affine maximal maps. Our main goal is the classification of the complete affine maximal maps with one or two embedded regular ends. In particular, we obtain a new characterization of the elliptic paraboloid and a large family of global examples.

We give a complete characterization of finite-dimensional compact sets with the following property: all of their images under affine operators are symmetric (that is, have symmetry planes of certain dimensions). We also study the noncompact case; namely, we reveal a class of unbounded closed sets with this property and conjecture that this class is complete.

We describe some differential-geometric structures in combinatorial terms: namely affine connections and their torsion and curvature, and we show that torsion free affine connections may equivalently be presented in terms of some simpler combinatorial structure: midpoint formation, and point reflection (geodesic symmetry). The method employed is that of synthetic differential geometry, which is briefly explained. We describe some differential-geometric structures in combinatorial terms: namely affine connections and their torsion and curvature, and we show that torsion free affine connections may equivalently be presented in terms of some simpler combinatorial structure: midpoint formation, and point reflection (geodesic symmetry). The method employed is that of synthetic differential geometry, which is briefly explained.

This paper focuses on the proximal point regularization technique for a class of optimal control processes governed by affine switched systems. We consider switched control systems described by nonlinear ordinary differential equations which are affine in the input. The affine structure of the dynamical models under consideration makes it possible to establish some continuity/approximability properties and to specify these models as convex control systems. We show that, for some classes of cost functionals, the associated optimal control problem (OCP) corresponds to a conventional convex optimization problem in a suitable Hilbert space. The latter can be reliably solved using standard first-order optimization algorithms and consistent regularization schemes. In particular, we propose a conceptual numerical approach based on the gradient-type method and classic proximal point techniques.

In many cases, feature-matching problems can boil down to the computation of affine invariant local image features. However, many methods which are used to obtain these image features are based on affine not perspective transformation. They typically fail to get enough matching points at extreme viewpoints. In this paper, a novel method based on perspective projection to simulate all image views by sampling a camera pose in 3D space is presented. Only four variables—three angles (pan, tilt and roll) and a scale factor which are used to describe the camera pose—are involved to represent an affine transformation matrix. All these samples generated by our method are more similar to the real images captured by camera than those generated by traditional methods. We demonstrate the performance and robustness of our method by comparing it with popular SIFT, ASIFT descriptors and randomized trees method. Experimental results show that a large range of viewpoints by learning the behavior of key points patterns can be handled even when the camera is placed at some extreme positions.

Affine reconstruction is to restore the affine shape of the object. Generally, there are two ways of achieving, one is to determine the plane at infinity, another is to determine the plane homography. Using the homography which had determined the plane at infinity achieve affine reconstruction. In this paper, firstly give out the homography of infinity plane and the algorithm of affine reconstruction, then proved: if the scene contains a set of parallel planes and a infinity point, the homography of infinity plane can be obtained and affine reconstruction can be linearly got in the scene. Computer simulation and real experiments show that the linear affine reconstruction algorithm is correct, and the approach has a good precision.

The so-called Affine Projection (AP) algorithm is of large interest in many adaptive filters applications due to its considerable speed-up in convergence compared to its simpler version, the LMS algorithm. While the original AP algorithm is well understood, gradient type variants of less complexity with relaxed step-size conditions called pseudo affine projection offer still unresolved problems. This contribution shows i) local robustness properties of such algorithms, ii) global properties of these, concluding l2-stability conditions that are independent of the input signal statistics, as well as iii) steady-state values of moderate to high accuracy by relatively simple terms when applied to long filters. Of particular interest is the existence of lower step-size bounds for one of the variants, a bound that has not been observed before.

We present in this article inductive schemes that allow to classify all those affine hypersurfaces with affine normal parallel second fundamental (cubic) form, for every possible dimensional case greater or equal than two. The solution proposed is a follow up to previous works on the same topic by this author and it uses, firstly, the reduction of the problem, eminently geometric, to the classification of a certain class of solutions to an equation of Monge-Ampere type. Then it is applied the so-called “method of algorithmic sequence of coordinate changes”, in order to achieve the latter.

We present a 4f optical security scheme by using a key code mask generated by an affine transformation operation. The key encrypting code produced through a pseudo-random pattern generated from a source image depends on the affine transformation parameters and the iteration number that control its random properties. This procedure offers the advantage over the conventional double random phase mask encryption technique that we do not need to send the encrypting mask itself to the authorizer user, but the set of elements that leads to the construction of the key code mask.

This is a continuation of the investigation into the theory of wavelet frames for general affine subspaces. The main focus of this paper is on the structural properties of affine subspaces. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces, while every reducing subspace (with respect to the dilation and translation operators) is the orthogonal direct sum of two purely non-reducing ones. This result is obtained through considering the basic question as to when the orthogonal complement of an affine subspace in another one is still affine. Motivated by the fundamental question as to whether every affine subspace is singly-generated, and by a recent result that every singly generated purely non-reducing subspace admits a singly generated wavelet frame, we prove that every affine subspace can be decomposed into the direct sum of a singly generated affine subspace and some space of “small size”. As a consequence we establish a connection between the above mentioned two questions.