Because of its simple approximate representation to nonlinear time variant frequency modulations, polynomial chirplet has been used in radar, gravity analysis, electronic warfare and acoustic signal processing. However, due to its high dimension parameter spaces, direct polynomial chirplet transform has extremely high computational cost. In addition, the discrete implementation of polynomial chirplet transform causes a limited parameter estimation accuracy which may not satisfy the requirement in real applications. In this paper, by combining a connected spectrogram graph fitting with random optimization, we develop a new technique to address these computational cost and parameter estimation accuracy issues. We first convert a high dimensional polynomial chirplet transform into a low dimensional spectrogram implementation which significantly reduces computational cost. We then introduce interpolation and random optimization methods to improve the parameter estimation accuracy.
K-Means is one of the most popular clustering techniques and has been successfully used in many data mining fields. K-Means is computationally expensive for high dimension and high quantity data and various techniques have been developed to reduce the computational cost. These techniques mainly involve improving implementation efficiency. In this paper, we proposed a Hausdorff distance and diffusion equation evolution combined technique to speed up K-Means through reducing number of data points in distance computation. Experiments show that for manifold data, the proposed method not only significantly reduces the computational cost, but also improve the clustering performance.
Polynomial chirplet representation of a frequency modulation radio frequency (FMRF) signal has many applications in radar and electronic support measurement (ESM). Fast and reliable estimation of polynomial chirplet parameters is valuable for FMRF signal processing. Traditionally, two types of techniques are used to estimate phase polynomial parameters, polynomial chirplet transform and multiple order phase difference approaches. The polynomial chirplet transform approach is robust to noises, however computationally expensive while the difference approach is computationally efficient, but sensitive noises. In this paper, a new multiple order difference approach is introduced to estimate parameters of a polynomial chirplet. In this new approach, we first compute the highest order coefficient of the polynomial chirplet, then remove the highest order monomial term by a monomial pursuit approach. Repeating these two steps, we develop a difference accumulation and monomial pursuit approach for estimating parameters of polynomial chirplet. Simulation tests show that the proposed method is robust to noises and has a low computational cost.
Handwritten word segmentation is one of the important components in the offline handwritten word recognition systems and has attracted many researchers. In this paper, we propose a Voronoi diagram approach to segment a handwritten word into stroke elements for handwritten word recognitions. Different from the conventional thinning techniques, the Voronoi diagram approach is proposed to create the segmentation path. Then the Chebyshev distance transform is suggested for implementing the Voronoi diagram. Test results and theoretical analysis show that the proposed method is capable to create similar quality handwritten word segmentation as the conventional thinning approach with much lower computational cost.
Convolutional neural networks (CNNs) provide the sensing and detection community with a discriminative approach for classifying images. However, one of the largest limitations of deep CNN image classifiers is the need for extensive training datasets containing a variety of image representations. While current methods, such as generative adversarial network data augmentation, additions of noise, rotations, and translations, can allow CNNs to better associate new images and their feature representations to ones of a learned image class, many fail to provide new contexts of ground truth feature information. To expand the association of critical class features within CNN image training datasets, an image pairing and training dataset augmentation paradigm via a multi-sensor domain image data fusion algorithm is proposed. This algorithm uses a mutual information (MI) and merit-based feature selection subroutine to pair highly correlated cross-domain images from multiple sensor domain image datasets. It then re-augments the corresponding cross-domain image pairs into the opposite sensor domain’s feature set via a highest MI, cross sensor domain, and image concatenation function. This augmented image set then acts to retrain the CNN to recognize greater generalizations of image class features via cross domain, mixed representations. Experimental results indicated an increased ability of CNNs to generalize and discriminate between image classes during testing of class images from synthetic aperture radar vehicle, solar cell device reliability screening, and lung cancer detection image datasets.
Deep convolutional neural networks (CNNs) provide the sensing and detection community with a discriminative, machine learning based approach for classifying images of objects. However, one of the largest limitations for deep CNN image classifiers is the need for extensive training data for a variety of appearances of class objects. While current methods such as GAN data augmentation, noise-perturbation, and rotation or translation of images can allow CNNs to better associate convolved features to ones similar to a learned image class, many fail to provide new context of ground truth information associated with each object class. To expand the association of new convolved feature examples with image classes within CNN training datasets, we propose a feature learning and training data enhancement paradigm via a multi-sensor domain data augmentation algorithm. This algorithm uses a mutual information, merit-based feature selection subroutine to iteratively select SAR object features that most correlate to each sensor domain’s class image objects. It then re-augments these features into the opposite sensor domain’s feature set via a highest mutual information, cross sensor domain image concatenation function. This augmented set then acts to retrain the CNN to recognize new cross domain class object features that each respective sensor domain’s network was not previously exposed to. Our experimental results using T60- class vs T70-class SAR object images from both the MSTAR and MGTD dataset repositories demonstrated an increase in classification accuracy from 88% and 61% to post-augmented cross-domain dataset training of 93.75% accuracy for the MSTAR, MGTD and subsequent fused datasets, respectively.
In high density communication and radar environments, radio frequency (RF) signal processing for receivers faces significant challenges. Receivers may receive the overlapped or pulse-on-pulse (POP) RF signals transmitted by cochannel and co-site RF emitters. Detecting, separating, and classifying these POP signals are valuable, but difficult due to the overlapping in time and frequency. In this paper, we propose a time-frequency manifold representation to solve these challenging problems. Using time frequency analysis, we show that a frequency modulation RF (FMRF) signal can be represented as a one dimensional manifold embedded in a two dimensional time frequency space. Using graph theory, we propose a path finding approach to extract this time frequency manifold. With both theoretical analysis and experiments, we show that the proposed approach can extract both simple single and complicated pulse-on-pulse FMRF signals
Graphical model, diffusion equation network dynamics, and manifold learning are different subareas in machine learning and network science. In this paper, combining the ideas from these subareas, we propose and implement graphical model and diffusion equation based manifold learning techniques for sensor data processing. We show that the graphical model and diffusion equation combined manifold learning can be used to perform data processing for one, two, and three-dimensional sensors. Experiments show that this manifold learning approach can solve many sensor data processing problems including radio frequency signal processing, image processing (computer vision), and three-dimensional lighting detection and ranging (LIDAR) processing problems better than some traditional methods.
Adaptive image filtering, removing noises without blurring the discontinuity of images, is important for many image processing, pattern recognition and computer vision applications. Many researches including anisotropic diffusion equation techniques have been conducted to address adaptive image filtering problems. Traditional techniques usually use differential characteristics of images to determine filtering coefficients for adaptively filtering images. As is well known, differential characteristics are difficult to estimate and the techniques to compute differential characteristics are usually sensitive to noises due to the intrinsic properties of derivatives. In this paper, we propose discrete Legendre polynomial based adaptive image filtering that effectively remove noises with preserving discontinuity of edges. We use polynomial fitting errors to choose masks to achieve the adaptivity. The fitting errors are computed by integrals (summation). This overcomes the derivative noise-sensitivity problems and allows us to achieve high performance.
Two-dimensional (2D) image processing and three-dimensional (3D) LIDAR point cloud data analytics are two important techniques of sensor data processing for many applications such as autonomous systems, auto driving cars, medical imaging and many other fields. However, 2D image data are the data that are distributed in regular 2D grids while 3D LIDAR data are represented in point cloud format that consist of points nonuniformly distributed in 3D spaces. Their different data representations lead to different data processing techniques. Usually, the irregular structures of 3D LIDAR data often cause challenges of 3D LIDAR analytics. Thus, very successful diffusion equation methods for image processing are not able to apply to 3D LIDAR processing. In this paper, applying network and network dynamics theory to 2D images and 3D LIDAR analytics, we propose graph-based data processing techniques that unify 2D image processing and 3D LIDAR data analytics. We demonstrate that both 2D images and 3D point cloud data can be processed in the same framework, and the only difference is the way to choose neighbor nodes. Thus, the diffusion equation techniques in 2D image processing can be used to process 3D point cloud data. With this general framework, we propose a new adaptive diffusion equation technique for data processing and show with experiments that this new technique can perform data processing with high performance.
Binary phase noise waveform is difficult to detect due to its thrum nail ambiguity function, has high transmission efficiency due to its constant magnitude, and is simple for implementation. Because of these advantages, binary phase noise waveform is widely used in noise radar. Many techniques have been developed to design binary phase noise waveform including Bernoulli trial, logistic mapping function, neural network optimization, genetic algorithm, kicked rotor chaos, polyphase perturbing, and particle swarm optimization techniques. In this paper, we first discuss the characteristics that good binary phase noise waveform needs to have. Then we introduce Kolmogorov-Smirnov (KS) two sample test in nonparametric statistics and derive balanced random walk. From balanced random walk theory, we propose balanced random based approach for noise waveform design and demonstrate that the noise waveforms generated by balanced random walk approach have zero direct components. We implement the proposed approach for noise waveform design and show that the proposed method is over 10% better on maximum sidelobes, 30% better on sidelobe energy than Bernoulli trial noise waveform design and selection.
In the era of big data, it appears imperative that future battle management systems be able to identify, decipher, and prioritize actionable information. This calls for fusing information and synchronizing operations across multiple domains and multiple sensor modalities. Fusing data from a diverse range of sensors across multiple domains is critical for improved situational awareness to enhance warfighters’ effectiveness. The basis for analyzing multiple field radar system data in real time remains a challenging yet promising threshold for military operational intelligence. Multiple domain sensor systems used to gather field intelligence requires gathering different types of information processing at required speeds that fall short of human reaction time and cognition. To press the advancement of field intelligence, the analysis, fusion and optimization of multi-domain systems, sensor data analysis is explored using probabilistic machine learning and supplemented heuristic signal processing to provide a basis for multi-system data integration, analysis and sensor suite selection.
Multidomain sensor data processing and fusion provide reliable ways for situational awareness in multidomain operations and receive great attention in both industry and academia. However, these data processing and fusion are complicated in implementation due to various modalities and high complexity of sensor data. In network dynamics, graph theory is used to represent complex data and extract information, and graph evolution is applied to analyze network dynamics. In this paper, combining the technologies of these two different domains, we propose using network dynamics to process and fuse multidomain sensor data for multidomain operations. First, we propose a graph-theory based framework for multidomain sensor data processing and fusion. Then we apply this general framework to multidomain sensor data processing. Using one-dimensional radio frequency (RF) signal processing, two-dimensional image processing and three-dimensional light detection and ranging (LIDAR) data analytics as examples, we demonstrate that with the proposed method, the same architecture can be used to extract critical features for these three types of sensor data. Furthermore, experiments also show that the proposed method creates higher performance than traditional methods.
KEYWORDS: Data modeling, Data processing, Distance measurement, Algorithm development, Lithium, Data mining, Quantization, Signal processing, Computer security, Data compression
KMeans is one of most popular algorithms in data mining (ranking number 2) and has be widely used in many fields. KMeans uses Euclidean distance to compare two data. However Euclidean distance is sensitive to linear transform in data collection process. Due to these linear transforms, the distance between two data points for the same class (intra-class distance) may larger than those for different classes (inter-class distance) that may cause low clustering performance for KMeans algorithm. In this paper, we propose simple linear regression approach for data clustering. Instead of using Euclidean distance to measure the difference, we recommend using the goodness of fitting (or normalized cross correlation) to measure the similarity and compare two data points. Using this new data comparison technique, we introduce linear regression approach for data clustering and demonstrate that the proposed method has higher performance and low computational cost than KMeans methods.
Noise radars have many advantages over conventional radars and receive great attentions recently. The waveforms and their characteristics of a noise radar are the crucial factors that impact the performance of this radar. Noise waveforms are usually generated by random number generators or logistic mapping. In this paper, we propose the kicked rotor quantum chaos perturbation (KRQCP) approach. First, we investigate the characteristics of noise waveforms generated by kicked rotor quantum chaos and demonstrate that like random number generation approach and logistic mapping approach, noise waveforms generated by kicked rotor quantum chaos (KRQC) also show the fractal relation between sidelobe values and code lengths. Then, we propose using KRQC noises to perturb quadratic phases and develop the KRQCP noise waveform generation method. We demonstrate that the performance for the noise waveforms generated by the new method is considerably higher than traditional random number generator and logistic mapping approaches.
KEYWORDS: Signal detection, Frequency modulation, Time-frequency analysis, Signal processing, Signal analysis, Receivers, RF communications, Radar, Signal analyzers, Image processing
Separation, detection and classification of multiple frequency modulation radio frequency (FMRF) signals are important in modern communication and radar environments. However, these problems are very challenging when multiple FMRF signals come from same locations, occupy in same frequency bands and transmit at same time. In this paper, we propose manifold representation and polynomial chirplet approach to separate, detect and classify multiple-FMRF-mixed signals. In the proposed approach, each FMRF signal is described by a one-dimensional manifold that is imbedded in a two-dimensional time-frequency space. This one-dimensional manifold is locally characterized by a vector that represents polynomial coefficients. Multiple FMRF signals, which may be inseparable in time and frequency, can be represented by multiple manifolds separable with polynomial coefficients. Through their separable polynomial coefficients, these inseparable FMRF signals can be separated and classified from the characteristics of their manifolds. In this paper, through simulation, we verify the manifold representation ideas and demonstrate that the proposed method can effectively separate, detect and classify multiple-FMRF-mixed signals even though these noisy FMRF signals are overlapped 100% in time and frequency band.
Binary phased codes have many applications in communication and radar systems. These applications, including spread spectrum communication and low probability of intercept radar, require low sidelobes and long code lengths. Many techniques for finding long binary phased codes with low sidelobes have been investigated in literatures. These techniques include exhaust search, neural network, and evolutionary methods, and they all have high computational cost. In this paper, we propose particle swarm optimization (PSO) to select long low sidelobe binary phased codes with reasonable computational cost. We investigate two techniques for initialization: random number approach and linear chirp approach and show that linear chirp initialization performs significantly better than random number approach. By implementing the proposed techniques, we demonstrate that PSO approach with linear chirp initialization can find binary codes with sidelobes equal to or lower than the neural network and genetic algorithm techniques in literatures.
Finding an object in images with orientation invariance has many applications in computer vision and pattern recognition. Template matching is a typical approach for finding objects. However, template matching requires pixel additions and multiplications on each pixel in a template and for each pixel in images, which involves a significant amount of pixel operations. Many techniques have been investigated to reduce the computational cost including FFT techniques, partial illumination approaches, and coarse to fine methods. These techniques may work for finding objects with the same orientation. However, when the object orientation in templates is different from the orientation in images, the computational cost is prohibitive even for the latest fast template matching techniques. In this paper, by combining the ideas of moment invariants in pattern recognition, Green theorem from physics and Bresenham line algorithm from computer graphics, we propose a mask size independent and orientation invariant object finding technique. From theoretical analysis and experiments, we demonstrate that this new technique significantly reduces computational cost for orientation free object finding from 0(ܰN3M2) of the direct implementation to 0(ܰN2)
Normalized cross correlation (NCC) based template matching is insensitive to intensity changes and it has many
applications in image processing, object detection, video tracking and pattern recognition. However, normalized
cross correlation implementation is computationally expensive since it involves both correlation computation and
normalization implementation. In this paper, we propose Legendre moment approach for fast normalized cross
correlation implementation and show that the computational cost of this proposed approach is independent of
template mask sizes which is significantly faster than traditional mask size dependent approaches, especially for
large mask templates. Legendre polynomials have been widely used in solving Laplace equation in electrodynamics
in spherical coordinate systems, and solving Schrodinger equation in quantum mechanics. In this paper, we extend
Legendre polynomials from physics to computer vision and pattern recognition fields, and demonstrate that
Legendre polynomials can help to reduce the computational cost of NCC based template matching significantly.
KEYWORDS: Telecommunications, Radar, Neural networks, Probability theory, Monte Carlo methods, Lithium, Fluctuations and noise, Tolerancing, Doppler effect, Frequency modulation
Binary phased codes have many applications in communication and radar systems. These applications require binary phased codes to have low sidelobes in order to reduce interferences and false detection. Barker codes are the ones that satisfy these requirements and they have lowest maximum sidelobes. However, Barker codes have very limited code lengths (equal or less than 13) while many applications including low probability of intercept radar, and spread spectrum communication, require much higher code lengths. The conventional techniques of finding binary phased codes in literatures include exhaust search, neural network, and evolutionary methods, and they all require very expensive computation for large code lengths. Therefore these techniques are limited to find binary phased codes with small code lengths (less than 100). In this paper, by analyzing Barker code, linear chirp, and P3 phases, we propose a new approach to find binary codes. Experiments show that the proposed method is able to find long low sidelobe binary phased codes (code length >500) with reasonable computational cost.
Sum of square difference (SSD) and normalized cross correlation (NCC) are two different template matching techniques and their fast implementations have been investigated independently. The SSD approach is known to be simple and fast, however it is variant to image intensity change that lead to low performance. On the other hand, the NCC method is invariant to intensity change and has high performance, but its computational cost is high. In this paper, we derive an equation that connects NCC and SSD. From this equation, we propose SSD based partial elimination for the fast implementation of NCC template matching. This new technique takes the advantages of both NCC’s high performance and SSD’s low computational cost. It is fast and has high performance. Then we propose a uniform smoothing approach that further reduces computational cost for NCC. Experiments show that the proposed method is significantly faster than the techniques reported in literature.
Noise radars have many advantages over conventional radars and receive great attentions recently. The performance of a noise radar is determined by its waveforms. Investigating characteristics of noise radar waveforms has significant value for evaluating noise radar performance. In this paper, we use binomial distribution theory to analyze general characteristics of binary phase coded (BPC) noise waveforms. Focusing on aperiodic autocorrelation function, we demonstrate that the probability distributions of sidelobes for a BPC noise waveform depend on the distances of these sidelobes to the mainlobe. The closer a sidelobe to the mainlobe, the higher the probability for this sidelobe to be a maximum sidelobe. We also develop Monte Carlo framework to explore the characteristics that are difficult to investigate analytically. Through Monte Carlo experiments, we reveal the Fractal relationship between the code length and the maximum sidelobe value for BPC waveforms, and propose using fractal dimension to measure noise waveform performance.
Morphological filtering, erosion and dilation on images, has been successfully applied to many image processing and pattern recognition fields. Landing site selection on the other hand, finding the best (safest) point to land with given obstacle distributions and aircraft shapes, has significant value for landing aircraft in tight environments. In this paper, we derive shape distance transform theory, and using this theory, we build a connection between morphological filtering and landing site selection which are traditionally treated as two different topics. From shape distance transform theory, we show that morphological filtering and land site selection can be implemented by shape distance transform. Thus, fast shape distance transform will provide fast implementation of morphological filtering and landing site selection. For convex polygon shape templates, we propose propagation techniques to compute shape distance transform. Then we introduce a new approach for faster morphological filtering and landing site selection. This new approach is independent of template sizes and its computational complexity is 0(1) which is much lower than 0(N) of the direct implementation, where N is the number of pixels in templates. For large templates, N is large, and the new approach is significantly faster than the direct implementation
Image feature extraction plays a significant role in image based pattern applications. In this paper, we propose a new
approach to generate hierarchical features. This new approach applies line fitting to adaptively divide regions based upon the amount of information and creates line fitting features for each subsequent region. It overcomes the feature
wasting drawback of the wavelet based approach and demonstrates high performance in real applications. For gray scale images, we propose a diffusion equation approach to map information-rich pixels (pixels near edges and ridge
pixels) into high values, and pixels in homogeneous regions into small values near zero that form energy map
images. After the energy map images are generated, we propose a line fitting approach to divide regions recursively
and create features for each region simultaneously. This new feature extraction approach is similar to wavelet based
hierarchical feature extraction in which high layer features represent global characteristics and low layer features represent local characteristics. However, the new approach uses line fitting to adaptively focus on information-rich regions so that we avoid the feature waste problems of the wavelet approach in homogeneous regions. Finally, the
experiments for handwriting word recognition show that the new method provides higher performance than the
regular handwriting word recognition approach.
Image registration is a process of transforming a data set from one coordinate system into another. There are two
typical approaches for image registration: Feature point match based and Area similarity comparison based. The
feature point match based approach, using points to establish the correspondence between two images, is relatively
fast, but it involves feature extractions and parameter selection to create feature points. Feature extractions involve
derivatives which are ill-posed problems and may lead to robustness issues. The area similarity comparison based
approach compares intensity patterns using a correlation metric such as normalized cross correlation (NCC). Since
it does not require feature extraction, is simple and not sensitive to noise. However its computational cost is high.
Even when some fast techniques like FFT are used to reduce the computational cost, the implementation is still time
consuming.
In this paper, we propose a diffusion equation and normalized cross correlation (NCC) combined method to perform
robust image registration with low computational cost. We first apply the diffusion equation to two images received
from two sensors (or the same sensor) and allow these two images to evolve by this diffusion equation. Based on
the characteristics of evolutions, we select a very small percentage of stable points in the first image and perform the
normalized cross correlation to the second image at each transformation point. The highest NCC point provides the
transformation parameters for registering these two images. This new method is resistant to noise since the
evolution of the diffusion equation reduces noise and it chooses only stable points for the NCC computation.
Furthermore, the new method is computationally efficient since only a small percentage of pixels involve in the
transformation estimation. Finally, the experiments for video motion estimation and image registration are provided
to demonstrate that the new method is able to estimate the registration transformation reliably in real time.
Digital Image Correlation (DIC) of time-sequenced-imagery (TSI) is a very popular method in the study of medical,
material deformation, and electronic packaging. Its use in processing the before-and-after images provides critical
information about the scene deformation and structural differences between the imagery.
Several correlation methods for implementing DIC have been developed and will be compared in this study. Each of
these methods offer distinct trades offs with respect to processing complexity and lock-in accuracy.
There are several factors that influence the effectiveness of these methods to provide robust operation and strongly
localized correlation peaks. These factors include; camera positional stability during the time of image acquisitions,
deformation of the object under study, and measurement noise. In addition, the signatures that are captured during DIC
can often times be amplified through preprocessing and thus potentially enhancing DIC performance.
This paper examines the impacts on two of these factors (measurement noise and image digital sharpening) using four
popular correlation methods that are often implemented in DIC analyses.
Line fitting and moments are two different problems and most articles discuss these two problems separately. In this paper, using the constraint optimization, we relate the line fitting to moments. We show that the eigen vectors of the second order central moments are fitted line directional vectors, and the eigen value is the fitting error. Then, we further show that the line fitting errors can be computed directly from the first and second moment invariants. From the relation between line fitting and moments, we propose a mask-size independent approach to implement the line fitting for curves or object contours. The computational cost of the new approach is independent of the mask size. It is computationally efficient if compared to the conventional approach whose computational cost is proportional to the fitting mask size.
Erosion and dilation are two basic morphological filters and have been widely used in both academic and industrial fields. When they are used in industry, such as automated visual inspection, their implementation cost especially for large masks is a challenging issue. In this paper, we propose a FDT (form distance transform) method for implementing erosion and dilation for some regular shapes. In this proposed method, the implementation of erosion and dilation is first converted into the computation of its FDT. Then a propagation technique is used to compute the FDT. The computational cost of the new method is independent of mask sizes. In contrast of the direct implementation, if the pixel number in a morphological mask is N, the proposed method reduces the implementation cost from O(N) to O(1).
The diffusion equation has received increasing attention in the fields of image analysis and computer vision for tasks such as image smoothing, image enhancement, feature extraction as well as dominant point detection. In this paper, the diffusion equation is applied to the inspection of surface mounted devices. It is shown experimentally that diffusion-equation-based methods could render very good discriminating features. The inspection experiments show that the correct inspection rate of the diffusion-equation- based method is very high for both training boards and test boards.
KEYWORDS: Correlation function, Signal processing, Signal to noise ratio, Fourier transforms, Electroluminescence, Phase only filters, Radon, Osmium, Detection theory, Nickel
In this paper, a robust-statistic-based method is proposed to implement template matching. The proposed method has two novel advantages. First, it is computationally efficient because only a very small fraction of the template pixels is used to do template matching. Second, it generates very high mainlobes and very low sidelobes because it only accumulates the gradient magnitudes of edges when the template is moved to the object center (signal focusing (SF) accumulation), and summarizes the gradient magnitudes in homogeneous regions when the template goes to other positions (interference avoiding (IA) accumulation). It is shown experimentally that compared with the normalized correlation method, the SFIA method increases the DSNR for about 11 approximately 30 db and decrease computational cost about 20 approximately 50 times.
In this paper, a generalized zero crossing (GZC) theorem is proposed. The GZC theorem has much less constraints on filters so that the design of filters can be flexible. Then, it is shown that ramp models can be effectively approximated by step models. Based on the GZC theorem, a difference-of-exponential (DoE) operator is proposed. It is shown both theoretically and experimentally that the new operator is computationally efficient, and its edge detection performance is higher than that of the Laplacian-of-Gaussian (LOG) operator.
In this paper, we propose a multiscale filtering method to compute derivatives with any orders. As a special case, we consider the computation of the second derivatives, and show that the difference of two smoothers with the same kernel, but different scales constructs a Laplacian operator and has a zero crossing at a step edge. Selecting a Gaussian function as the smoother, we show the DOG (difference of Gaussian) itself is a zero crossing edge extractor, and it needn't approximate to LoG (Laplacian of Gaussian). At the same time, we show that even though DOG for bandwidth ratio 0.625 (1:1.6) is the optimal approximation to LoG, it is not optimal for edge detection. Finally, selecting an exponential function as the smoothing kernel, we obtain a Laplacian of exponential (LoE) operator, and it is shown theoretically and experimentally that the LoE has a high edge detection performance, furthermore its computation is efficient and its computational complexity is independent of the filter kernel bandwidths.
Fractal Brownian random (FBR) field is an extension of fractional Brownian motion (FBM) and has been successfully used in image analysis, the generation of natural scenes, fractal geometry, and other areas. But its implementation is difficult and limits its applications. In this paper, by extending the AR model, we propose a new approach, the pyramid-AR-model approach, to implement FBR fields. The new method has the same computational complexity as A. Fournier's, but it can generate FBR fields much more accurately.
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