Search results for: Bicubic interpolation

Authors: Lewis E. Hibell, Honghai Liu, David J. Brown

Abstract:

This paper presents the results of enhancing images from a left and right stereo pair in order to increase the resolution of a 3D representation of a scene generated from that same pair. A new neural network structure known as a Self Delaying Dynamic Network (SDN) has been used to perform the enhancement. The advantage of SDNs over existing techniques such as bicubic interpolation is their ability to cope with motion and noise effects. SDNs are used to generate two high resolution images, one based on frames taken from the left view of the subject, and one based on the frames from the right. This new high resolution stereo pair is then processed by a disparity map generator. The disparity map generated is compared to two other disparity maps generated from the same scene. The first is a map generated from an original high resolution stereo pair and the second is a map generated using a stereo pair which has been enhanced using bicubic interpolation. The maps generated using the SDN enhanced pairs match more closely the target maps. The addition of extra noise into the input images is less problematic for the SDN system which is still able to out perform bicubic interpolation.

Keywords: Genetic Evolution, Image Enhancement, Neuron Networks, Stereo Vision

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Authors: Fatemeh Behjati Ardakani, Fatemeh Khademian, Abbas Nowzari Dalini, Reza Ebrahimpour

Abstract:

Human activity is a major concern in a wide variety of applications, such as video surveillance, human computer interface and face image database management. Detecting and recognizing faces is a crucial step in these applications. Furthermore, major advancements and initiatives in security applications in the past years have propelled face recognition technology into the spotlight. The performance of existing face recognition systems declines significantly if the resolution of the face image falls below a certain level. This is especially critical in surveillance imagery where often, due to many reasons, only low-resolution video of faces is available. If these low-resolution images are passed to a face recognition system, the performance is usually unacceptable. Hence, resolution plays a key role in face recognition systems. In this paper we introduce a new low resolution face recognition system based on mixture of expert neural networks. In order to produce the low resolution input images we down-sampled the 48 × 48 ORL images to 12 × 12 ones using the nearest neighbor interpolation method and after that applying the bicubic interpolation method yields enhanced images which is given to the Principal Component Analysis feature extractor system. Comparison with some of the most related methods indicates that the proposed novel model yields excellent recognition rate in low resolution face recognition that is the recognition rate of 100% for the training set and 96.5% for the test set.

Keywords: Low resolution face recognition, Multilayered neuralnetwork, Mixture of experts neural network, Principal componentanalysis, Bicubic interpolation, Nearest neighbor interpolation.

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Authors: Muhammad Sajjad, Naveed Khattak, Noman Jafri

Abstract:

World has entered in 21st century. The technology of computer graphics and digital cameras is prevalent. High resolution display and printer are available. Therefore high resolution images are needed in order to produce high quality display images and high quality prints. However, since high resolution images are not usually provided, there is a need to magnify the original images. One common difficulty in the previous magnification techniques is that of preserving details, i.e. edges and at the same time smoothing the data for not introducing the spurious artefacts. A definitive solution to this is still an open issue. In this paper an image magnification using adaptive interpolation by pixel level data-dependent geometrical shapes is proposed that tries to take into account information about the edges (sharp luminance variations) and smoothness of the image. It calculate threshold, classify interpolation region in the form of geometrical shapes and then assign suitable values inside interpolation region to the undefined pixels while preserving the sharp luminance variations and smoothness at the same time. The results of proposed technique has been compared qualitatively and quantitatively with five other techniques. In which the qualitative results show that the proposed method beats completely the Nearest Neighbouring (NN), bilinear(BL) and bicubic(BC) interpolation. The quantitative results are competitive and consistent with NN, BL, BC and others.

Keywords: Adaptive, digital image processing, imagemagnification, interpolation, geometrical shapes, qualitative &quantitative analysis.

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Authors: Rongli Gai, Zhiyuan Chang, Xiaohong Wang, Jingyu Liu

Abstract:

In view of various situations in the interpolation process, most researchers use self-adaptation to adjust the interpolation process, which is also one of the current and future research hotspots in the field of CNC (Computerized Numerical Control) machining. In the interpolation process, according to the overview of the spline curve interpolation algorithm, the adaptive analysis is carried out from the factors affecting the interpolation process. The adaptive operation is reflected in various aspects, such as speed, parameters, errors, nodes, feed rates, random period, sensitive point, step size, curvature, adaptive segmentation, adaptive optimization, etc. This paper will analyze and summarize the research of adaptive imputation in the direction of the above factors affecting imputation.

Keywords: Adaptive algorithm, CNC machining, interpolation constraints, spline curve interpolation.

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Authors: A. Pashayev, C. Ardil, R. Sadiqov

Abstract:

Various methods of geofield parameters restoration (by algebraic polynoms; filters; rational fractions; interpolation splines; geostatistical methods – kriging; search methods of nearest points – inverse distance, minimum curvature, local – polynomial interpolation; neural networks) have been analyzed and some possible mistakes arising during geofield surface modeling have been presented.

Keywords: interpolation methods, geofield parameters, neural networks.

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Authors: Srijanani Anurag Prasad

Abstract:

The Coalescence Hidden-variable Fractal Interpolation Surface (CHFIS) was built by combining interpolation data from the Iterated Function System (IFS). The interpolation data in a CHFIS comprise a row and/or column of uncertain values when a single point is entered. Alternatively, a row and/or column of additional points are placed in the given interpolation data to demonstrate the node added CHFIS. There are three techniques for inserting new points that correspond to the row and/or column of nodes inserted, and each method is further classified into four types based on the values of the inserted nodes. As a result, numerous forms of node insertion can be found in a CHFIS.

Keywords: Fractal, interpolation, iterated function system, coalescence, node insertion, knot insertion.

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Authors: F. Castillo, J. Arellano, S. Sánchez

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In this paper an alternative analysis in the time domain is described and the results of the interpolation process are presented by means of functions that are based on the rule of conditional mathematical expectation and the covariance function. A comparison between the interpolation error caused by low order filters and the classic sinc(t) truncated function is also presented. When fewer samples are used, low-order filters have less error. If the number of samples increases, the sinc(t) type functions are a better alternative. Generally speaking there is an optimal filter for each input signal which depends on the filter length and covariance function of the signal. A novel scheme of work for adaptive interpolation filters is also presented.

Keywords: Interpolation, basis function, over-sampling.

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Authors: Tarun Gulati, Kapil Gupta, Dushyant Gupta

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High resolution images are always desired as they contain the more information and they can better represent the original data. So, to convert the low resolution image into high resolution interpolation is done. The quality of such high resolution image depends on the interpolation function and is assessed in terms of sharpness of image. This paper focuses on Wavelet based Interpolation Techniques in which an input image is divided into subbands. Each subband is processed separately and finally combined the processed subbandsto get the super resolution image. 

Keywords: SWT, DWTSR, DWTSWT, DWCWT.

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Authors: Jahan Zeb, Muhammad Younus Javed, Usman Qayyum

Abstract:

This research paper deals with the implementation of face recognition using neural network (recognition classifier) on low-resolution images. The proposed system contains two parts, preprocessing and face classification. The preprocessing part converts original images into blurry image using average filter and equalizes the histogram of those image (lighting normalization). The bi-cubic interpolation function is applied onto equalized image to get resized image. The resized image is actually low-resolution image providing faster processing for training and testing. The preprocessed image becomes the input to neural network classifier, which uses back-propagation algorithm to recognize the familiar faces. The crux of proposed algorithm is its beauty to use single neural network as classifier, which produces straightforward approach towards face recognition. The single neural network consists of three layers with Log sigmoid, Hyperbolic tangent sigmoid and Linear transfer function respectively. The training function, which is incorporated in our work, is Gradient descent with momentum (adaptive learning rate) back propagation. The proposed algorithm was trained on ORL (Olivetti Research Laboratory) database with 5 training images. The empirical results provide the accuracy of 94.50%, 93.00% and 90.25% for 20, 30 and 40 subjects respectively, with time delay of 0.0934 sec per image.

Keywords: Average filtering, Bicubic Interpolation, Neurons, vectorization.

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Authors: Yaoxin Zhang, Yafei Jia, Sam S.Y. Wang

Abstract:

A multi-block algorithm and its implementation in two-dimensional finite element numerical model CCHE2D are presented. In addition to a conventional Lagrangian Interpolation Method (LIM), a novel interpolation method, called Consistent Interpolation Method (CIM), is proposed for more accurate information transfer across the interfaces. The consistent interpolation solves the governing equations over the auxiliary elements constructed around the interpolation nodes using the same numerical scheme used for the internal computational nodes. With the CIM, the momentum conservation can be maintained as well as the mass conservation. An imbalance correction scheme is used to enforce the conservation laws (mass and momentum) across the interfaces. Comparisons of the LIM and the CIM are made using several flow simulation examples. It is shown that the proposed CIM is physically more accurate and produces satisfactory results efficiently.

Keywords: Multi-block algorithm, conservation, interpolation, numerical model, flow simulation.

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Authors: Mehdi Tatari

Abstract:

Finding the interpolation function of a given set of nodes is an important problem in scientific computing. In this work a kind of localization is introduced using the radial basis functions which finds a sufficiently smooth solution without consuming large amount of time and computer memory. Some examples will be presented to show the efficiency of the new method.

Keywords: Radial basis functions, local interpolation method, closed form solution.

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Authors: Wang liguo, Wang zongmin, Kong ying

Abstract:

A novel interpolation scheme to extend usable spectrum and upconvert in high performance D/A converters is addressed in this paper. By adjusting the pulse width of cycle and the production circuit of code, the expansion code is a null code or complementary code that is interpolation process. What the times and codes of interpolation decide DAC works in one of a normal mode or multi-mixer mode so that convert the input digital data signal into normal signal or a mixed analog signal having a mixer frequency that is higher than the data frequency. Simulation results show that the novel scheme and apparatus most extend the usable frequency spectrum into fifth to sixth Nyquist zone beyond conventional DACs.

Keywords: interpolation, upconversion, modulation, switching function, duty cycle.

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Authors: Hadi Taghvafard

Abstract:

We present a new quadrature rule based on the spline interpolation along with the error analysis. Moreover, some error estimates for the reminder when the integrand is either a Lipschitzian function, a function of bounded variation or a function whose derivative belongs to Lp are given. We also give some examples to show that, practically, the spline rule is better than the trapezoidal rule.

Keywords: Quadrature, Spline interpolation, Trapezoidal rule, Numericalintegration, Error analysis.

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Authors: Marzieh Dosti, Alireza Nazemi

Abstract:

In this paper, the telegraph equation is solved numerically by cubic B-spline quasi-interpolation .We obtain the numerical scheme, by using the derivative of the quasi-interpolation to approximate the spatial derivative of the dependent variable and a low order forward difference to approximate the temporal derivative of the dependent variable. The advantage of the resulting scheme is that the algorithm is very simple so it is very easy to implement. The results of numerical experiments are presented, and are compared with analytical solutions by calculating errors L2 and L∞ norms to confirm the good accuracy of the presented scheme.

Keywords: Cubic B-spline, quasi-interpolation, collocation method, second-order hyperbolic telegraph equation.

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Authors: Kazuo Komatsu, Hitoshi Takata

Abstract:

This paper discusses a design of nonlinear observer by a formal linearization method using an application of Chebyshev Interpolation in order to facilitate processes for synthesizing a nonlinear observer and to improve the precision of linearization. A dynamic nonlinear system is linearized with respect to a linearization function, and a measurement equation is transformed into an augmented linear one by the formal linearization method which is based on Chebyshev interpolation. To the linearized system, a linear estimation theory is applied and a nonlinear observer is derived. To show effectiveness of the observer design, numerical experiments are illustrated and they indicate that the design shows remarkable performances for nonlinear systems.

Keywords: nonlinear system, nonlinear observer, formal linearization, Chebyshev interpolation.

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Authors: Shujun Fu, Qiuqi Ruan, Wenqia Wang

Abstract:

Image interpolation is a common problem in imaging applications. However, most interpolation algorithms in existence suffer visually the effects of blurred edges and jagged artifacts in the image to some extent. This paper presents an adaptive feature preserving bidirectional flow process, where an inverse diffusion is performed to sharpen edges along the normal directions to the isophote lines (edges), while a normal diffusion is done to remove artifacts (“jaggies") along the tangent directions. In order to preserve image features such as edges, corners and textures, the nonlinear diffusion coefficients are locally adjusted according to the directional derivatives of the image. Experimental results on synthetic images and nature images demonstrate that our interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional interpolations.

Keywords: anisotropic diffusion, bidirectional flow, directional derivatives, edge enhancement, image interpolation, inverse flow, shock filter.

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Authors: T. P. Athira, Gibin Chacko George

Abstract:

This paper proposes a dual tree complex wavelet transform (DT-CWT) based directional interpolation scheme for noisy images. The problems of denoising and interpolation are modelled as to estimate the noiseless and missing samples under the same framework of optimal estimation. Initially, DT-CWT is used to decompose an input low-resolution noisy image into low and high frequency subbands. The high-frequency subband images are interpolated by linear minimum mean square estimation (LMMSE) based interpolation, which preserves the edges of the interpolated images. For each noisy LR image sample, we compute multiple estimates of it along different directions and then fuse those directional estimates for a more accurate denoised LR image. The estimation parameters calculated in the denoising processing can be readily used to interpolate the missing samples. The inverse DT-CWT is applied on the denoised input and interpolated high frequency subband images to obtain the high resolution image. Compared with the conventional schemes that perform denoising and interpolation in tandem, the proposed DT-CWT based noisy image interpolation method can reduce many noise-caused interpolation artifacts and preserve well the image edge structures. The visual and quantitative results show that the proposed technique outperforms many of the existing denoising and interpolation methods.

Keywords: Dual-tree complex wavelet transform (DT-CWT), denoising, interpolation, optimal estimation, super resolution.

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Authors: Shujun Fu, Qiuqi Ruan, Wenqia Wang

Abstract:

Image interpolation is a common problem in imaging applications. However, most interpolation algorithms in existence suffer visually to some extent the effects of blurred edges and jagged artifacts in the image. This paper presents an adaptive feature preserving bidirectional flow process, where an inverse diffusion is performed to enhance edges along the normal directions to the isophote lines (edges), while a normal diffusion is done to remove artifacts (''jaggies'') along the tangent directions. In order to preserve image features such as edges, angles and textures, the nonlinear diffusion coefficients are locally adjusted according to the first and second order directional derivatives of the image. Experimental results on synthetic images and nature images demonstrate that our interpolation algorithm substantially improves the subjective quality of the interpolated images over conventional interpolations.

Keywords: anisotropic diffusion, bidirectional flow, directionalderivatives, edge enhancement, image interpolation, inverse flow, shock filter.

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Authors: Nur Nadiah Abd Hamid, Ahmad Abd. Majid, Ahmad Izani Md. Ismail

Abstract:

Linear two-point boundary value problem of order two is solved using extended cubic B-spline interpolation method. There is one free parameters, λ, that control the tension of the solution curve. For some λ, this method produced better results than cubic B-spline interpolation method.

Keywords: two-point boundary value problem, B-spline, extendedcubic B-spline.

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Authors: Muhanned Alfarras

Abstract:

The recent growth of using multimedia transmission over wireless communication systems, have challenges to protect the data from lost due to wireless channel effect. Images are corrupted due to the noise and fading when transmitted over wireless channel, in wireless channel the image is transmitted block by block, Due to severe fading, entire image blocks can be damaged. The aim of this paper comes out from need to enhance the digital images at the wireless receiver side. Proposed Boundary Interpolation (BI) Algorithm using wavelet, have been adapted here used to reconstruction the lost block in the image at the receiver depend on the correlation between the lost block and its neighbors. New Proposed technique by using Boundary Interpolation (BI) Algorithm using wavelet with Pixel interleaver has been implemented. Pixel interleaver work on distribute the pixel to new pixel position of original image before transmitting the image. The block lost through wireless channel is only effects individual pixel. The lost pixels at the receiver side can be recovered by using Boundary Interpolation (BI) Algorithm using wavelet. The results showed that the New proposed algorithm boundary interpolation (BI) using wavelet with pixel interleaver is better in term of MSE and PSNR.

Keywords: Image Transmission, Wavelet, Pixel Interleaver, Boundary Interpolation Algorithm

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Authors: Dragos Nicolae VIZIREANU

Abstract:

One of the main image representations in Mathematical Morphology is the 3D Shape Decomposition Representation, useful for Image Compression and Representation,and Pattern Recognition. The 3D Morphological Shape Decomposition representation can be generalized a number of times,to extend the scope of its algebraic characteristics as much as possible. With these generalizations, the Morphological Shape Decomposition 's role to serve as an efficient image decomposition tool is extended to grayscale images.This work follows the above line, and further develops it. Anew evolutionary branch is added to the 3D Morphological Shape Decomposition's development, by the introduction of a 3D Multi Structuring Element Morphological Shape Decomposition, which permits 3D Morphological Shape Decomposition of 3D binary images (grayscale images) into "multiparameter" families of elements. At the beginning, 3D Morphological Shape Decomposition representations are based only on "1 parameter" families of elements for image decomposition.This paper addresses the gray scale inter frame interpolation by means of mathematical morphology. The new interframe interpolation method is based on generalized morphological 3D Shape Decomposition. This article will present the theoretical background of the morphological interframe interpolation, deduce the new representation and show some application examples.Computer simulations could illustrate results.

Keywords: 3D shape decomposition representation, mathematical morphology, gray scale interframe interpolation

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Authors: Nur Nadiah Abd Hamid , Ahmad Abd. Majid, Ahmad Izani Md. Ismail

Abstract:

Linear two-point boundary value problems of order two are solved using cubic trigonometric B-spline interpolation method (CTBIM). Cubic trigonometric B-spline is a piecewise function consisting of trigonometric equations. This method is tested on some problems and the results are compared with cubic B-spline interpolation method (CBIM) from the literature. CTBIM is found to approximate the solution slightly more accurately than CBIM if the problems are trigonometric.

Keywords: trigonometric B-spline, two-point boundary valueproblem, spline interpolation, cubic spline

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Authors: Saeed Mian Qaisar, Laurent Fesquet, Marc Renaudin

Abstract:

Mostly the systems are dealing with time varying signals. The Power efficiency can be achieved by adapting the system activity according to the input signal variations. In this context an adaptive rate filtering technique, based on the level crossing sampling is devised. It adapts the sampling frequency and the filter order by following the input signal local variations. Thus, it correlates the processing activity with the signal variations. Interpolation is required in the proposed technique. A drastic reduction in the interpolation error is achieved by employing the symmetry during the interpolation process. Processing error of the proposed technique is calculated. The computational complexity of the proposed filtering technique is deduced and compared to the classical one. Results promise a significant gain of the computational efficiency and hence of the power consumption.

Keywords: Level Crossing Sampling, Activity Selection, Rate Filtering, Computational Complexity, Interpolation Error.

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Authors: Tao Quang Bang, Insu Jeon

Abstract:

Most CT reconstruction system x-ray computed tomography (CT) is a well established visualization technique in medicine and nondestructive testing. However, since CT scanning requires sampling of radiographic projections from different viewing angles, common CT systems with mechanically moving parts are too slow for dynamic imaging, for instance of multiphase flows or live animals. A large number of X-ray projections are needed to reconstruct CT images, so the collection and calculation of the projection data consume too much time and harmful for patient. For the purpose of solving the problem, in this study, we proposed a method for tomographic reconstruction of a sample from a limited number of x-ray projections by using linear interpolation method. In simulation, we presented reconstruction from an experimental x-ray CT scan of a Aluminum phantom that follows to two steps: X-ray projections will be interpolated using linear interpolation method and using it for CT reconstruction based upon Ordered Subsets Expectation Maximization (OSEM) method.

Keywords: CT reconstruction, X-ray projections, Interpolation technique, OSEM

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Authors: Yakin Hajlaoui, Richard Labib, Jean-Franc¸ois Plante, Michel Gamache

Abstract:

This study presents the Multi-Layer Inverse Distance Weighting Model (ML-IDW), inspired by the mathematical formulation of both multi-layer neural networks (ML-NNs) and Inverse Distance Weighting model (IDW). ML-IDW leverages ML-NNs’ processing capabilities, characterized by compositions of learnable non-linear functions applied to input features, and incorporates IDW’s ability to learn anisotropic spatial dependencies, presenting a promising solution for nonlinear spatial interpolation and learning from complex spatial data. We employ gradient descent and backpropagation to train ML-IDW. The performance of the proposed model is compared against conventional spatial interpolation models such as Kriging and standard IDW on regression and classification tasks using simulated spatial datasets of varying complexity. Our results highlight the efficacy of ML-IDW, particularly in handling complex spatial dataset, exhibiting lower mean square error in regression and higher F1 score in classification.

Keywords: Deep Learning, Multi-Layer Neural Networks, Gradient Descent, Spatial Interpolation, Inverse Distance Weighting.

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Authors: Seung-Won Jung, Hye-Soo Kim, Le Thanh Ha, Seung-Jin Baek, Sung-Jea Ko

Abstract:

In this paper, a novel deinterlacing algorithm is proposed. The proposed algorithm approximates the distribution of the luminance into a polynomial function. Instead of using one polynomial function for all pixels, different polynomial functions are used for the uniform, texture, and directional edge regions. The function coefficients for each region are computed by matrix multiplications. Experimental results demonstrate that the proposed method performs better than the conventional algorithms.

Keywords: Deinterlacing, polynomial interpolation.

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Authors: Hanita Daud, Noorhana Yahya, Vijanth Sagayan, Muizuddin Talib

Abstract:

This paper discusses on the use of Spline Interpolation and Mean Square Error (MSE) as tools to process data acquired from the developed simulator that shall replicate sea bed logging environment. Sea bed logging (SBL) is a new technique that uses marine controlled source electromagnetic (CSEM) sounding technique and is proven to be very successful in detecting and characterizing hydrocarbon reservoirs in deep water area by using resistivity contrasts. It uses very low frequency of 0.1Hz to 10 Hz to obtain greater wavelength. In this work the in house built simulator was used and was provided with predefined parameters and the transmitted frequency was varied for sediment thickness of 1000m to 4000m for environment with and without hydrocarbon. From series of simulations, synthetics data were generated. These data were interpolated using Spline interpolation technique (degree of three) and mean square error (MSE) were calculated between original data and interpolated data. Comparisons were made by studying the trends and relationship between frequency and sediment thickness based on the MSE calculated. It was found that the MSE was on increasing trends in the set up that has the presence of hydrocarbon in the setting than the one without. The MSE was also on decreasing trends as sediment thickness was increased and with higher transmitted frequency.

Keywords: Spline Interpolation, Mean Square Error, Sea Bed Logging, Controlled Source Electromagnetic

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Authors: Mario Mastriani

Abstract:

Super-resolution is nowadays used for a high-resolution image produced from several low-resolution noisy frames. In this work, we consider the problem of high-quality interpolation of a single noise-free image. Such images may come from different sources, i.e., they may be frames of videos, individual pictures, etc. On the other hand, in the encoder we apply a downsampling via bidimen-sional interpolation of each frame, and in the decoder we apply a upsampling by which we restore the original size of the image. If the compression ratio is very high, then we use a convolutive mask that restores the edges, eliminating the blur. Finally, both, the encoder and the complete decoder are implemented on General-Purpose computation on Graphics Processing Units (GPGPU) cards. In fact, the mentioned mask is coded inside texture memory of a GPGPU.

Keywords: General-Purpose computation on Graphics ProcessingUnits, Image Compression, Interpolation, Super-resolution.

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Authors: Alina Trifan, António J. R. Neves

Abstract:

Although most digital cameras acquire images in a raw format, based on a Color Filter Array that arranges RGB color filters on a square grid of photosensors, most image compression techniques do not use the raw data; instead, they use the rgb result of an interpolation algorithm of the raw data. This approach is inefficient and by performing a lossless compression of the raw data, followed by pixel interpolation, digital cameras could be more power efficient and provide images with increased resolution given that the interpolation step could be shifted to an external processing unit. In this paper, we conduct a survey on the use of lossless compression algorithms with raw Bayer images. Moreover, in order to reduce the effect of the transition between colors that increase the entropy of the raw Bayer image, we split the image into three new images corresponding to each channel (red, green and blue) and we study the same compression algorithms applied to each one individually. This simple pre-processing stage allows an improvement of more than 15% in predictive based methods.

Keywords: Bayer images, CFA, losseless compression, image coding standards.

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Authors: Fakharuddin Ibrahim, Jamaludin Md. Ali, Ahmad Ramli

Abstract:

In this paper, we will discuss about the data interpolation by using the rational cubic Ball curve. To generate a curve with a better and satisfactory smoothness, the curve segments must be connected with a certain amount of continuity. The continuity that we will consider is of type G¹ continuity. The conditions considered are known as the G¹ Hermite condition. A simple application of the proposed method is to generate an Arabic font satisfying the required continuity.

Keywords: Continuity, data interpolation, Hermite condition, rational Ball curve.

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