Hexagonal Image Processing Research Articles

Morphology for hexagonal image processing: a comprehensive simulation analysis

<p>Morphological operators for binary and grayscale images are commonly used to eliminate noise, recognize contours or specific structures, and arrange shapes in image processing for physiological modeling and biomechanics applications. Even though morphology has been substantially developed in square-pixelbased-image-processing (SIP), no effort has been made to construct morphological operators in hexagonal-pixel-based-image-processing (HIP) yet. In this paper, we transform basic SIP-domain-morphological operators such as dilation, erosion, closing, and opening into HIP-domain and compare their performance with their SIP counterparts. It is the first time to give the fundamental morphological operators in the HIP domain. The operators developed in this paper initiate the research about morphology in the HIP domain by successfully filling a significant gap by eliminating HIP’s lack of basic operators, thus capable of producing enhanced images for better analysis in anatomical models related to biology and medicine research fields.</p>

Reversible Logic-Based Hexel Value Differencing—A Spatial Domain Steganography Method for Hexagonal Image Processing

Reversible Logic-Based Hexel Value Differencing—A Spatial Domain Steganography Method for Hexagonal Image Processing

Improved exploiting modification direction steganography for hexagonal image processing

Steganography has made significant advances in the Square-pixel-based Image Processing (SIP) domain, but to our knowledge, no work has yet been done in Hexel (Hexagonal Pixel)-based Image Processing (HIP). This paper presents a HIP-domain data hiding method that exploits and improves the SIP-domain Exploiting Modification Direction (EMD) embedding scheme. The proposed method, Hexagonal EMD (HexEMD), utilizes a HIP-domain cover image's hexagonal nature and infrastructure to embed the secret message. In standard digital imaging systems, the sensor portion that converts photonic energy into an analog electrical signal and all the subunits that digitize, process, and display this signal are based on square pixel logic, so there is currently no commercial equipment available to produce HIP-domain images. Thus, the image is first transformed into the HIP domain in software using the infrastructure developed in the project. Then the HIP-domain image is partitioned into non-overlapping heptads of the standard size, each containing seven hexels. Rather than embedding segments to the independent pixel pairs as done in SIP-domain EMD, we do the embedding iteratively in each heptad. Experimental results show that the HexEMD outperforms its SIP equivalent, EMD, by improving embedding capacity and achieving low visual quality distortion.

Deep Learning-Based Hexrep Neural Network for Convergence Free with Operator’s Efficacy in Hexagonal Image Processing

The field of hexagonal image processing is concerned with the creation of image processing systems that combine the advantages of biological model-based evolutionary motivated frameworks. The structure and functionality of artificial neural networks were inspired by biological processes. The fundamental framework of recording and output devices limits their present state of the art. Prior neural networks have used square or hexagonal style input to completely connected layers, which resulted in a high coherence problem between two adjacent hexagonal kernel layers due to pooling. Previous research does not design the self-data structure to support convolution to increase computational efficiency, so it violates the convolution and pooling operator, which greatly degrades the image process performance. This paper introduces a novel paradigm Proficient Deep Learning-based Hexrep Neural Network that overcomes major significant problems in image operations structure constraint, coherence problem, and violation of convolution and pooling operator and achieves hexagonal image processing.

Integrating Hexagonal Image Processing with Evidential Probabilistic Supervised Classification Technique to Improve Image Retrieval Systems

This paper presents a suggested approach to treat a major issue in images classification namely uncertainty. Uncertainty in image classification means some pixels within each cluster are more or less likely to actually belong to this cluster. So, techniques have been used in this paper to deal with the pixels that do not belong to specific regions, helping to raise image retrieval performance. This was done by merging one of the artificial intelligence techniques, which is image processing, with one of the statistical techniques for probability, which is evidential probabilistic. In such contexts, it may be advantageous to resort to two branches: hexagonal image processing based on partial down-sampling of the image resolution in both directions by half using weighted average performance then shifting the remaining pixels in alternate rows. The other is an evidential theory which is rich and flexible formalisms for representing and manipulating uncertain information. Both hexagonal image processing and evidential theory are used to obtain high accuracy in images classification. The hierarchical nature of the hexagonal image processing addressing scheme is exploited to extract features from the image efficiently.

Simplified square to hexagonal lattice conversion based on 1-D multirate processing

Hexagonal image sampling and processing are theoretically superior to the most commonly used square lattice based sampling and processing, but due to the lack of commercial image sensors, current research mainly relies on virtually hexagonally sampled data through square to hexagonal lattice conversion, which is a typical 2-D interpolation problem. This paper presents a simplified and efficient square to hexagonal lattice conversion method. The method firstly utilizes the separable nature of the interpolation kernel to simplify the original 2-D interpolation into 1-D interpolation along the horizontal direction only, and then it applies the 1-D multirate technique to further simplify the shift-variant 1-D interpolation into shift-invariant 1-D convolutions. Compared with the original 2-D interpolation version, the proposed method becomes both simple and computationally efficient, and it is also suitable for implementation with parallel processing and hardware. Finally, experiments are performed and the results are consistent with the analysis.

A Framework for Hexagonal Image Processing Using Hexagonal Pixel-Perfect Approximations in Subpixel Resolution.

Image processing in hexagonal lattice has many advantages rather than square lattice. Researchers have addressed benefits of hexagonal structure in applications such as binarization, rotation, scaling and edge detection. Approximately all existing hardwares for capturing and displaying images are based on square lattice. Therefore, the best way for using advantages of hexagonal lattice is to find a proper software approach to convert square pixels to hexagonal ones. This paper presents a hexagonal platform based on interpolation which addresses three existing hexagonal challenges including imperfect hexagonal shape, inaccurate intensity level of hexagonal pixels and lower resolution in hexagonal space. The proposed interpolation is computed by overlaps between square and hexagonal pixels. Overlap types are formulated mathematically in 8 separate cases. Each overlap case is detected automatically and used to compute final gray-level intensity of hexagonal pixels. It is mathematically and experimentally shown that the proposed method satisfies necessary conditions for square-to-hexagonal conversion. The proposed scheme is evaluated on synthetic and real images with 10 different levels of noise in interpolation and edge detection applications. In synthetic images, the proposed method achieves the best figure of merit (FOM) 99.92% and 98.67% in high and low SNRs 100 and 20, respectively. Also, the proposed method outperforms existing state of the art hexagonal lattices with interclass correlation coefficient (ICC) 84.18% and mean rating 7.7 (out of 9) in real images.

FTSH: a framework for transition from square image processing to hexagonal image processing

This paper proposes a novel framework for transition from the ordinary square-pixel-based image processing (SIP) domain to the hexagonal-pixel-based (HIP) domain (FTSH). The conventional image acquisition and processing are based on square pixels. However, HIP can provide promising advantages in many respects, such as degrading the curse of data size and accordingly reducing the processing time. HIP did not achieve satisfactory attraction because all software, including libraries, methods and structures, as well as mathematical operations and methodologies developed to date, are aimed at SIP. In this study, we propose a framework containing the corresponding HIP equivalents of some basic SIP methods and operations. In addition, the results of these basic operations in both SIP and HIP areas are presented comparatively. Since there is no common and standardized framework or library for HIP, this study can be used by other researchers who wish to enter the HIP. Simulation results support the competitive performance of HIP, and this promising performance can be carried far beyond when properly handled and focused.

Implementation of a simulated display for hexagonal image processing

Hexagonal image processing is theoretically superior to that based on the common square lattice, and due to the lack of practical imaging devices, this paper intends to implement a simulated display. The paper investigates the sampling lattices used in the hexagonal image processing, and finds that variable lattices occur in hexagonal discrete Fourier transform (HDFT), while the most commonly used “hyperpel” approach, due to its fixed cell pattern, cannot handle such displaying tasks. Then, the paper proposes to represent each pixel with the exact Voronoi cell (VC) according to the sampling lattice. In the paper, a simple algorithm is presented to compute the VC vertices, and each simulated pixel is constructed with the VC filled with its intensity value, and then the simulated display is implemented by the tessellation of each simulated pixel on the correct lattice position. Finally, experimental results show that the proposed simulated display can display data without geometric distortion.

Three Dimensional Image Processing in Hexagonal Prism Lattice of Z3 Grid

A 2D hexagonal image is an array of what are called pixels which are the coordinates of the hexagonal lattice points decided by the linear horizontal rows and the nonlinear vertical zig-zags. A 2D hexagonal image could also be informally called as image slice or a matrix of pixel values arranged in a hexagonal array. A 3D hexagonal image is viewed as an ordered sequence of 2D hexagonal image slices arranged in the z-direction and the 3D arrangement of voxel values is called as a prism of voxel values. Most of the 3D hexagonal image processing operations are similar to those of 2D hexagonal image processing. 3D hexagonal images are processed with the help of 3D hexagonal scanning windows, whereas 2D hexagonal images are processed with the help of 2D hexagonal scanning windows. For instance, a 3D hexagonal image processing operation like 3D surface detection is carried out using analogous 2D edge detection algorithm on every image slice and the processed slices assembled to visualize 3D surface detected image. In fact, 2D contours of an image slice are called superficial features and closed surfaces of a 3D image are called volumetric features. One can always obtain surface detected version of a 3D hexagonal image by processing the 2D hexagonal image slices using 2D edge detection operation, and consequently the 3D surface detection operation is termed as 2.5D hexagonal image processing. One could also process the 3D hexagonal image data using a 3D surface detection algorithm, in which case it is termed as 3D hexagonal image processing. This is not the case with the operation of skeltonization. One cannot make use of 2.5D skeletonization operation of 2D image slices in order to get skeltonized version of the corresponding 3D image. In fact, one would come across discrepancies and differences when 2.5D skeletonization of 2D hexagonal image slices of a 3D hexagonal image is carried out instead of the direct 3D skeletonization of the 3D hexagonal image. This paper highlights certain 3D algorithms for processing 3D hexagonal images.

BACK TO BASICS: TOWARDS NOVEL COMPUTATION AND ARRANGEMENT OF SPATIAL SENSORY IN IMAGES

The current camera has made a huge progress in the sensor resolution and the lowluminance performance. However, we are still far from having an optimal camera as powerful as our eye is. The study of the evolution process of our visual system indicates attention to two major issues: the form and the density of the sensor. High contrast and optimal sampling properties of our visual spatial arrangement are related directly to the densely hexagonal form. In this paper, we propose a novel software-based method to create images on a compact dense hexagonal grid, derived from a simulated square sensor array by a virtual increase of the fill factor and a half a pixel shifting. After that, the orbit functions are proposed for a hexagonal image processing. The results show it is possible to achieve an image processing in the orbit domain and the generated hexagonal images are superior, in detection of curvature edges, to the square images. We believe that the orbit domain image processing has a great potential to be the standard processing for hexagonal images.

Tri-directional gradient operators for hexagonal image processing

Image processing has traditionally involved the use of square operators on regular rectangular image lattices. For many years the concept of using hexagonal pixels for image capture has been investigated, and several advantages of such an approach have been highlighted. We present a design procedure for hexagonal gradient operators, developed within the finite element framework, for use on hexagonal pixel based images. In order to evaluate the approach, we generate pseudo hexagonal images via resizing and resampling of rectangular images. This approach also allows us to present results visually without the use of hexagonal lattice capture or display hardware. We provide comparative results with existing gradient operators, both rectangular and hexagonal.

Multiscale Edge Detection Using a Finite Element Framework for Hexagonal Pixel-Based Images.

In recent years, the processing of hexagonal pixel-based images has been investigated, and as a result, a number of edge detection algorithms for direct application to such image structures have been developed. We build on this paper by presenting a novel and efficient approach to the design of hexagonal image processing operators using linear basis and test functions within the finite element framework. Development of these scalable first order and Laplacian operators using this approach presents a framework both for obtaining large-scale neighborhood operators in an efficient manner and for obtaining edge maps at different scales by efficient reuse of the seven-point linear operator. We evaluate the accuracy of these proposed operators and compare the algorithmic performance using the efficient linear approach with conventional operator convolution for generating edge maps at different scale levels.

Storage and addressing scheme for practical hexagonal image processing

We present a practical hexagonal storage and addressing scheme, which eliminates the difference between theory and imple- mentation in other addressing methods. This scheme employs a mid- dleware-based address-mapping module that separates the algorithm and specific data addressing; thus any hexagonal algorithm can keep the native and consistent forms as in the coordinate system through theory and implementation. The scheme simplifies the implementa- tion work and preserves all excellent features of the hexagonal lattice. Finally, we discuss the implementation issues and show that it's fea- sible and can be implemented efficiently.© 2013 SPIE and IS&T (DOI:

Vision-based hexagonal image processing using Hex-Gabor

About 97% receptive field of the neurons is very closely described as 2D Gabor wavelet and it is mostly suitable for vision system modeling. Immense work is available on texture information, especially for rectangular structures. However, there is a little work in recognizing minute details in an image on hexagonal structure by either interpolation or enhancement. In this work, the two important operations of biological visual system such as enhancement and interpolation are performed using the Hex-Gabor process. It is possible to obtain an error-free image at sigma = 2/pi using the proposed Hex-Gabor process and the significance of this sigma value is proved. For the performance analysis standard reflected images and X-ray images are considered.

BIORTHOGONAL WAVELETS WITH SIX-FOLD AXIAL SYMMETRY FOR HEXAGONAL DATA AND TRIANGLE SURFACE MULTIRESOLUTION PROCESSING

This paper discusses the construction of highly symmetric compactly supported wavelets for hexagonal data/image and triangle surface multiresolution processing. Recently, hexagonal image processing has attracted attention. Compared with the conventional square lattice, the hexagonal lattice has several advantages, including that it has higher symmetry. It is desirable that the filter banks for hexagonal data also have high symmetry which is pertinent to the symmetric structure of the hexagonal lattice. The high symmetry of filter banks and wavelets not only leads to simpler algorithms and efficient computations, it also has the potential application for the texture segmentation of hexagonal data. While in the field of computer-aided geometric design (CAGD), when the filter banks are used for surface multiresolution processing, it is required that the corresponding decomposition and reconstruction algorithms for regular vertices have high symmetry, which make it possible to design the corresponding multiresolution algorithms for extraordinary vertices. In this paper we study the construction of six-fold axial symmetric biorthogonal filter banks and the associated wavelets, with both the dyadic and [Formula: see text]-refinements. The constructed filter banks have the desirable symmetry for hexagonal data processing. By associating the outputs (after one-level multiresolution decomposition) appropriately with the nodes of the regular triangular mesh with which the input data is associated (sampled), we represent multiresolution analysis and synthesis algorithms as templates. The six-fold axial symmetric filter banks constructed in this paper result in algorithm templates with desirable symmetry for triangle surface processing.

Orthogonal and Biorthogonal FIR Hexagonal Filter Banks With Sixfold Symmetry

Recently, hexagonal image processing has attracted attention. The hexagonal lattice has several advantages in comparison with the rectangular lattice, the conventionally used lattice for image sampling and processing. For example, a hexagonal lattice needs less sampling points; it has better consistent connectivity; it has higher symmetry; and its structure is plausible to human vision systems. The multiresolution analysis method has been used for hexagonal image processing. Since the hexagonal lattice has high degree of symmetry, it is desirable that the hexagonal filter banks designed for multiresolution hexagonal image processing also have high order of symmetry, which is pertinent to the symmetry structure of the hexagonal lattice. The orthogonal or prefect reconstruction (PR) hexagonal filter banks that are available in the literature have only threefold symmetry. In this paper, we investigate the construction of orthogonal and PR finite impulse response (FIR) hexagonal filter banks with sixfold symmetry. We obtain block structures of 7-size refinement (seven-channel two-dimensional) orthogonal and PR FIR hexagonal filter banks with sixfold rotational symmetry. radic7-refinement orthogonal and biorthogonal wavelets based on these block structures are constructed. In this paper, we also consider FIR hexagonal filter banks with axial (line) symmetry, and we present a block structure of FIR hexagonal filter banks with pseudo-sixfold axial symmetry.

FIR Filter Banks for Hexagonal Data Processing

Images are conventionally sampled on a rectangular lattice. Thus, traditional image processing is carried out on the rectangular lattice. The hexagonal lattice was proposed more than four decades ago as an alternative method for sampling. Compared with the rectangular lattice, the hexagonal lattice has certain advantages which include that it needs less sampling points; it has better consistent connectivity and higher symmetry; the hexagonal structure is also pertinent to the vision process. In this paper, we investigate the construction of symmetric FIR hexagonal filter banks for multiresolution hexagonal image processing. We obtain block structures of FIR hexagonal filter banks with 3-fold rotational symmetry and 3-fold axial symmetry. These block structures yield families of orthogonal and biorthogonal FIR hexagonal filter banks with 3-fold rotational symmetry and 3-fold axial symmetry. In this paper, we also discuss the construction of orthogonal and biorthogonal FIR filter banks with scaling functions and wavelets having optimal smoothness. In addition, we present a few of such orthogonal and biorthogonal FIR filters banks.

Hexagonal tight frame filter banks with idealized high-pass filters

This paper studies the construction of hexagonal tight wavelet frame filter banks which contain three “idealized” high-pass filters. These three high-pass filters are suitable spatial shifts and frequency modulations of the associated low-pass filter, and they are used by Simoncelli and Adelson in (Proc IEEE 78:652–664, 1990) for the design of hexagonal filter banks and by Riemenschneider and Shen in (Approximation Theory and Functional Analysis, pp. 133–149, Academic Press, Boston 1991; J. Approx Theory 71:18–38 1992) for the construction of 2-dimensional orthogonal filter banks. For an idealized low-pass filter, these three associated high-pass filters separate high frequency components of a hexagonal image in 3 different directions in the frequency domain. In this paper we show that an idealized tight frame, a frame generated by a tight frame filter bank containing the “idealized” high-pass filters, has at least 7 frame generators. We provide an approach to construct such tight frames based on the method by Lai and Stockler in (Appl Comput Harmon Anal 21:324–348, 2006) to decompose non-negative trigonometric polynomials as the summations of the absolute squares of other trigonometric polynomials. In particular, we show that if the non-negative trigonometric polynomial associated with the low-pass filter p can be written as the summation of the absolute squares of other 3 or less than 3 trigonometric polynomials, then the idealized tight frame associated with p requires exact 7 frame generators. We also discuss the symmetry of frame filters. In addition, we present in this paper several examples, including that with the scaling functions to be the Courant element B 111 and the box-spline B 222. The tight frames constructed in this paper will have potential applications to hexagonal image processing.