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Scale-Space And Edge Detection Using Anisotropic Diffusion
Abstract
A new definition of scale-space is suggested, and a class of
algorithms used to realize a diffusion process is introduced. The
diffusion coefficient is chosen to vary spatially in such a way as to
encourage intraregion smoothing rather than interregion smoothing. It is
shown that the `no new maxima should be generated at coarse scales'
property of conventional scale space is preserved. As the region
boundaries in the approach remain sharp, a high-quality edge detector
which successfully exploits global information is obtained. Experimental
results are shown on a number of images. Parallel hardware
implementations are made feasible because the algorithm involves
elementary, local operations replicated over the image
... PDE-based image algorithms have been well used in recent years for image processing. Perona and Malik proposed the anisotropic diffusion model, also named the PM model [13], and the total variation (TV) model [14] applies anisotropic diffusion to image denoising, and their improved models [15] have great performance. However, due to its property of step image, the second-order PDE diffusion model has blocky effects in the processing results. ...
... The improvement of nonlinear diffusion models often starts from the perspective of the diffusion function. For example, improving the detection operator, using the image gradient magnitude [13], the local variance [24], the local residual energy [25], or changing the diffusion function curve [26]. The principle of all the above methods is to detect the effective information in the image more accurately. ...
... . (13) The local variance in the proposed model is slightly different from the normalization method used in the literature [24], which is related to the maximum gradient magnitude of the image: ...
Low-dose computed tomography (LDCT) reduces radiation damage to patients, however, it adds noise and artifacts to the reconstructed images, which deteriorates the quality of CT images. To address the problem of easy edge blurring in the denoising process of LDCT images, this paper proposes a fourth-order partial differential diffusion model with adaptive Laplacian kernel that can protect edge and detail information. The model incorporates the guided filter with edge preserving as a fidelity term in the energy function. Then, using gradient magnitude and the local variance to construct edge and detail detectors in the diffusion function, which can protect the edge and detail information of the LDCT images during the diffusion process. Finally, using the adaptive Laplacian kernel replaces the conventional Laplacian kernel, which has stronger edge preserving. The experimental results show that the proposed model achieves excellent performance in edge preserving and noise suppression in actual thoracic phantom and AAPM clinical LDCT images. In terms of both visual effects and quantitative indexes, the proposed model has good processing performance compared with other excellent algorithms.
... . . . [35]. This architecture is useful for edge preservation and segmentation but not convolution. ...
... In image processing literature, diffusion refers to a class of algorithms that aim to smooth out the image -similar to how heat pattern diffuses in a 2D material. The equation for diffusion in image processing (as stated in the early vision literature [35]) is typically given by the diffusion equation, ...
... Anisotropic diffusion, on the other hand, is a type of diffusion where the diffusion coefficient varies based on the local image structure, i.e., c(x, y, t) is a function of the local image structure. In this vein, the most relevant implementation is Perona-Malik diffusion [35], where they modified the diffusion coefficient in (1) to be a nonlinear function of the gradient magnitude, i.e., ...
In this article, we introduce a groundbreaking approach for ultra-low-power hybrid analog-digital processing of multimodal physiological data at multiple locations, emphasizing EEG signals. We propose an innovative analog Convolutional Processing Unit (CvPU) that uniquely harnesses the properties of anisotropic diffusion in electrical circuits for convolution. This novel use of anisotropic diffusion-driven convolution sets our work apart. Additionally, we present a controller architecture that allows for the sequential execution of multiple consecutive convolutional layers using the same CvPU array. The proposed neural network architecture to detect seizures using EEG signals is evaluated on a publicly available clinical dataset. Our CvPU array-based convolution’s performance and feasibility metrics have been assessed using SPICE simulation software. Furthermore, we have delved deep into studying the scalability of our approach in terms of power and space and its feasibility for battery-less and implantable applications and have compared it with both digital and hybrid analog-digital methods.
... Contrast is a unitless quantity measured by the ratio of brightness values. There are different ways to calculate it [26], for example: ...
... Anisotropic diffusion is a method aimed at reducing interference in the image without masking significant parts of the image [26]. ...
This study explores the English language learning challenges faced by first-year economy and finance stu
dents at Nakhchivan State University. Through surveys, interviews, and academic performance analysis, the re
search identifies key difficulties in vocabulary, grammar, pronunciation, and listening comprehension, and their
impact on academic and professional proficiency. The findings suggest the need for specialized English teaching
strategies tailored to these students' specific needs.
... Perona et al [27] proposed an anisotropic diffusion algorithm, which has since been improved by numerous research works [28]. The PMD filter reduces speckle by preserving image edges, lines, and regions. ...
... Diffusion filtrations, such as the anisotropic dissemination that reduces speckle filtration and more sophisticated variance, the information-retaining anisotropic dissemination filter [71] and the oriented additive noise-decreasing anisotropic dissemination filter [72], detach unwanted elements from such an image by resolving a differential equation with partials [27]. Sharp-transition detail is significantly lost throughout the entire filter. ...
Congenital heart defects (CHD) are one of the serious problems that arise during pregnancy. Early CHD detection reduces death rates and morbidity but is hampered by the relatively low detection rates (i.e., 60%) of current screening technology. The detection rate could be increased by supplementing ultrasound imaging with fetal ultrasound image evaluation (FUSI) using deep learning techniques. As a result, the non-invasive foetal ultrasound image has clear potential in the diagnosis of CHD and should be considered in addition to foetal echocardiography. This review paper highlights cutting-edge technologies for detecting CHD using ultrasound images, which involve pre-processing, localization, segmentation, and classification. Existing technique of preprocessing includes spatial domain filter, non-linear mean filter, transform domain filter, and denoising methods based on Convolutional Neural Network (CNN); segmentation includes thresholding-based techniques, region growing-based techniques, edge detection techniques, Artificial Neural Network (ANN) based segmentation methods, non-deep learning approaches and deep learning approaches. The paper also suggests future research directions for improving current methodologies.
... Nonlinear diffusion is a method to simplify and reduce noise reduction of the image. This method was invented by Perona et al. [28] for edge detection. Gerig et al. [29] developed other versions of this method for imaging such as texture and color images. ...
... In this research for evaluating image retrieval system, precision (Pr) and retrieval (Re) are used. These two criteria are defined as below [28]: ...
... These methods have progressed to accommodate the particularities of MRI data as more sophisticated noise models for MRI have been developed. Numerous examples, such as the Conventional Approach (CA), linear estimators, Maximum Likelihood (ML), and modified Non-Local Mean (NLM) methods, may be found in the current literature [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Single-coil acquisitions present the simplest case, and the complicated spatial MR data is usually represented as a complex Gaussian process. ...
... The experiments involve conducting tests with two types of noise distributions: Gaussian and Rician. When considering the Gaussian distribution, the observations z, which are affected by noise, follow the distribution described in Equation (1). The noisy observations : → + , in the case of Rician distributed noise, obey the definition. ...
Denoising volumetric data is essential for improving data quality and facilitating accurate analysis, interpretation, and diagnostic capabilities in various fields, including medical imaging, scientific visualization, and engineering simulations. This article proposes a BM4D, an expansion of the BM3D filter designed specifically for volumetric data. This unique approach combines grouping and Collaborative Filtering (CF) principles. It entails grouping dimensionally similar patches into a (d + 1)-dimensional array, then processing them collectively in the transform domain. Unlike BM3D, where basic data patches are pixel blocks, BM4D uses voxel cubes combined into a Four-Dimensional (4-D) "Group." This 4-D transformation takes advantage of local correlations inside each voxel cube as well as non-local correlations between matching voxels from other cubes. This produces a highly sparse spectrum within this group, allowing for effective signal-to-noise distinction via coefficient reduction. Estimates for each grouped cube are received after applying the inverse transformation, which is then adaptively mixed at their original positions. The proposed algorithm's performance is assessed in the context of denoising volumetric data corrupted by Gaussian and Rician noise. Experimental results showcase BM4D's outstanding denoising capabilities, establishing its effectiveness in the domain of volumetric data denoising and positioning it as a cutting-edge solution.
... In this work, we reinterpret images as parametrised surfaces on the unit plane Ω and explore them from a metric perspective. This perspective had gained popularity prior to the rise of deep learning [42,43,34,7,12,48,47,5]. Metric geometry focuses on curved spaces, or manifolds, denoted as X, each with well-defined tangent planes T x X at every point x ∈ X. Manifolds are equipped with metrics, positive functions on the tangent bundle X × T x X → R + guiding local distance calculations. ...
... This metric yields symmetric distances, making traversal direction irrelevant. If M is everywhere a scaled identity matrix, then the metric is isotropic, and it is non-uniform yet isotropic if the scale differs between points, although it is sometimes mistakenly called anisotropic [42]. ...
Standard convolutions are prevalent in image processing and deep learning, but their fixed kernel design limits adaptability. Several deformation strategies of the reference kernel grid have been proposed. Yet, they lack a unified theoretical framework. By returning to a metric perspective for images, now seen as two-dimensional manifolds equipped with notions of local and geodesic distances, either symmetric (Riemannian metrics) or not (Finsler metrics), we provide a unifying principle: the kernel positions are samples of unit balls of implicit metrics. With this new perspective, we also propose metric convolutions, a novel approach that samples unit balls from explicit signal-dependent metrics, providing interpretable operators with geometric regularisation. This framework, compatible with gradient-based optimisation, can directly replace existing convolutions applied to either input images or deep features of neural networks. Metric convolutions typically require fewer parameters and provide better generalisation. Our approach shows competitive performance in standard denoising and classification tasks.
... Diffusion has a long tradition in image processing [26,74,75] and it has also been used for image inpainting; see e.g. [29,35]. ...
... In our experiments, we found the rational Perona-Malik diffusivity [74] g s 2 = 1 1 + s 2 λ 2 (19) to be a suitable choice. It attenuates the variance σ 2 at image locations with dominant structures where the edge detector |∇f | exceeds some contrast parameter λ. ...
Inpainting-based codecs store sparse selected pixel data and decode by reconstructing the discarded image parts by inpainting. Successful codecs (coders and decoders) traditionally use inpainting operators that solve partial differential equations. This requires some numerical expertise if efficient implementations are necessary. Our goal is to investigate variants of Shepard inpainting as simple alternatives for inpainting-based compression. They can be implemented efficiently when we localise their weighting function. To turn them into viable codecs, we have to introduce novel extensions of classical Shepard interpolation that adapt successful ideas from previous codecs: Anisotropy allows direction-dependent inpainting, which improves reconstruction quality. Additionally, we incorporate data selection by subdivision as an efficient way to tailor the stored information to the image structure. On the encoding side, we introduce the novel concept of joint inpainting and prediction for isotropic Shepard codecs, where storage cost can be reduced based on intermediate inpainting results. In an ablation study, we show the usefulness of these individual contributions and demonstrate that they offer synergies which elevate the performance of Shepard inpainting to surprising levels. Our resulting approaches offer a more favourable trade-off between simplicity and quality than traditional inpainting-based codecs. Experiments show that they can outperform JPEG and JPEG2000 at high compression ratios.
... The diffusion equation has been a useful tool for image processing for a long time. Perona and Malik [33] introduced a nonlinear diffusion equation model for attenuating additive Gaussian noise (PM) in 1990, though its solutions were criticized for being ill-posed. Cattéet al. [4] refined the PM model's diffusion coefficient function using a Gaussian con-volution kernel, achieving unique solutions and enhanced experimental performance (RPM). ...
... The PSNR value of the image result with the speckle noise level L = 4 processed by HTNet is 33.74db, while the PSNR value of the adversarial attack image result processed by HTNet is 27.79db. Since the gradient information usually appears dark, we use edge indicator function [33] Edge(u) = 255 1+k|∇u| 2 to distinguish the gradient features of the image. From Fig. 4, we can see that the texture information of these three images gradually decreases, and the difference is large. ...
Speckle removal aims to smooth noise while preserving image boundaries and texture information. In recent years, speckle removal models based on deep learning methods have attracted a lot of attention. However, it was found that these models are less robust to adversarial attacks. The adversarial attack makes the image recovery of deep learning methods significantly less effective when the speckle noise distribution is almost unchanged. In purpose of addressing the above problem, we propose a diffusion equation-based speckle removal model that can improve the robustness of deep learning algorithms in this paper. The model utilizes a deep learning image prior and an image grayscale detection operator together to construct the coefficient function of the diffusion equation. Among them, there is a high possibility that the deep learning image prior is inaccurate or even incorrect, but it will not affect the performance and the properties of the proposed diffusion equation model for noise removal. Moreover, we analyze the robustness of the proposed diffusion equation model in terms of theoretical and numerical properties. Experiments show that our proposed diffusion equation speckle removal model is not affected by adversarial attacks in any way and has stronger robustness.
... • We demonstrate that our operator̃( , ) gives rise to a method that gives better results than both the Perona-Malik method [23] and total variation methods [5] in image denoising. ...
... Proposition 2. The scheme defined in (27) is consistent with the equation for the boundary problem (23).Proof. For given ∈ ∞ ([0, ] ×̄) and arbitrary ( 0 , 0 ) ∈ [0, ] ×̄. ...
... Change-point smoothing arises in image processing when filters are designed to detect boundaries or "edges" in an image, and smooth between those boundaries. Anisotropic diffusion filtering, also known as Perona-Malik diffusion, is a partial differential equation methododology for smoothing between boundaries while preserving edges, and can be framed as an edge-preserving extension to (isotropic) Gaussian filtering [21,22]. Anisotropic diffusion is a convolution of the image and an isotropic kernel function, using an edge-stopping function to preserve boundaries between regions without supervision [21]. ...
... Anisotropic diffusion filtering, also known as Perona-Malik diffusion, is a partial differential equation methododology for smoothing between boundaries while preserving edges, and can be framed as an edge-preserving extension to (isotropic) Gaussian filtering [21,22]. Anisotropic diffusion is a convolution of the image and an isotropic kernel function, using an edge-stopping function to preserve boundaries between regions without supervision [21]. ...
Understanding wildfire spread in Canada is critical to promoting forest health and protecting human life and infrastructure. Quantifying fire spread from noisy images, where change-point boundaries separate regions of fire, is critical to accurately estimating fire spread rates. The challenge lies in denoising the fire images and accurately identifying highly non-linear fire lines without smoothing over boundaries. In this paper, we develop an iterative smoothing algorithm for change-point data that utilizes oversmoothed estimates of the underlying data generating process to inform re-smoothing. We demonstrate its effectiveness on simulated one- and two-dimensional change-point data, and robustness to response outliers. Then, we apply the methodology to fire spread images from laboratory micro-fire experiments and show that the regions fuel, burning and burnt-out are smoothed while boundaries are preserved.
... given degraded image. Traditional restoration techniques [1][2][3][4][5][6][7] have relied on explicitly defined image priors, handcrafted based on empirical observations. Such an approach, however, is inherently limited due to the difficulty of designing these priors and their lack of generalizability across different scenarios. ...
RainDNet is an advanced image deraining model that refines the “Multi-Stage Progressive Image Restoration Network” (MPRNet) for superior computational efficiency and perceptual fidelity. RainDNet’s innovative architecture employs depthwise separable convolutions instead of MPRNet’s traditional ones, reducing model complexity and improving computational efficiency while preserving the feature extraction ability. RainDNet’s performance is enhanced by a multi-objective loss function combining perceptual loss for visual quality and Structural Similarity Index Measure (SSIM) loss for structural integrity. Experimental evaluations demonstrate RainDNet’s superior performance over MPRNet in terms of Peak Signal-to-Noise Ratio (PSNR), SSIM, and BRISQUE (Blind Referenceless Image Spatial Quality Evaluator) scores across multiple benchmark datasets, underscoring its aptitude for maintaining image fidelity while restoring structural and textural details. Our findings invite further explorations into more efficient architectures for image restoration tasks, contributing significantly to the field of computer vision. Ultimately, RainDNet lays the foundation for future, resource-efficient image restoration models capable of superior performance under diverse real-world scenarios.
... In (5), the parameter K controls the sensitivity to the edges and is a hyper-parameter and * represents the convolution operator. The algorithm presented in (5) is in fact the nonlinear diffusion method, also known as modified Perona-Malik diffusion technique [25], which is used for image de-noising . We will implement (5) directly in Python to achieve the de-noised image. ...
In this paper, we present a unique 2D high resolution , compact, low-cost, low-weight, and highly accurate millimeter wave (mm-Wave) imagery system capable of operating in all weather conditions. We describe mm-Wave imaging process in detail and present several novel signal processing methods with their applications. To create the array, we utilize the Synthetic Aperture Radar (SAR) concept. The imagery system presented in this work, can strongly compete with Lidar systems as the resolution limit is at the same level. Furthermore, in contrast to the Lidar systems, our imagery system can operate in heavy rain and dense fog and produce high quality images. We use our custom-made Frequency Modulated Continuous Wave (FMCW) radar operating at W-band with 33 GHz band-width for data collection and present the results.
... In the last few years, there has been a growing interest among researchers in anisotropic partial differential equations due to their applicability in various scientific fields. For instance, in the early 1990s, Perona and Malik [43] introduced the initial anisotropic partial differential equation (PDE) model, utilized for image enhancement and denoising by preserving significant image features while employing anisotropic PDEs, as detailed in Tschumperlé and Deriche [51]. Anisotropic problems also arise in physics models describing fluid dynamics with different conductivities in different directions. ...
... We can observe from the penalty functions shown in Fig. 3a that our function would cause more losses when the gradients (x) are smaller, and cause fewer losses for larger gradients. The better edge-preserving capability of our function can be validated form the edge stopping function ( (x) = � (x)∕x ) [33] shown in Fig. 3b, where we see our function imposes larger penalties when the gradients are smaller, while imposing milder penalties when the gradients are larger. ...
Edge-aware image decomposition is an essential topic in the field of multimedia signal processing. In this paper, we propose a novel non-convex penalty function, which we name the generalized Welsch function. We show that the proposed penalty function is more than a generalization of most existing penalty functions for edge-aware regularization, thus, it better facilitates edge-awareness. We embed the proposed penalty function into a novel optimization model for edge-aware image decomposition. To solve the optimization model with non-convex penalty function, we propose an efficient algorithm based on the additive quadratic minimization and Fourier domain optimization. We have experimented with the proposed method in a variety of tasks, including image smoothing, detail enhancement, HDR tone mapping, and JPEG compression artifact removal. Experiment results show that our method outperforms the state-of-the-art image decomposition methods. Furthermore, our method is highly efficient, it is able to render real-time processing of 720P color images on a modern GPU.
... where D = diag(A1) is the normalization. As derived in [35], when applying the filter in (4) multiple times, i.e., with the initial state x 0 = y, the iterations x t = D −1 Ax t−1 = (D −1 A) t y are essentially a discrete version of anisotropic diffusion [36]. More importantly, it is shown that optimizing A through diffusion can make the filter spectrum closer to those of an ideal Wiener filter that minimizes the reconstruction error. ...
Depth images captured by low-cost three-dimensional (3D) cameras are subject to low spatial density, requiring depth completion to improve 3D imaging quality. Image-guided depth completion aims at predicting dense depth images from extremely sparse depth measurements captured by depth sensors with the guidance of aligned Red–Green–Blue (RGB) images. Recent approaches have achieved a remarkable improvement, but the performance will degrade severely due to the corruption in input sparse depth. To enhance robustness to input corruption, we propose a novel depth completion scheme based on a normalized spatial-variant diffusion network incorporating measurement uncertainty, which introduces the following contributions. First, we design a normalized spatial-variant diffusion (NSVD) scheme to apply spatially varying filters iteratively on the sparse depth conditioned on its certainty measure for excluding depth corruption in the diffusion. In addition, we integrate the NSVD module into the network design to enable end-to-end training of filter kernels and depth reliability, which further improves the structural detail preservation via the guidance of RGB semantic features. Furthermore, we apply the NSVD module hierarchically at multiple scales, which ensures global smoothness while preserving visually salient details. The experimental results validate the advantages of the proposed network over existing approaches with enhanced performance and noise robustness for depth completion in real-use scenarios.
... Conversely, a lower diffusion coefficient restricts the spread of information, preserving edges and fine details in the image. This method follows prior work (Perona and Malik, 1990), which defines c(g p , g n ) = K 2 K 2 + || g p − g n || 2 2 (2) ...
A low-resolution digital surface model (DSM) features distinctive attributes impacted by noise, sensor limitations and data acquisition conditions, which failed to be replicated using simple interpolation methods like bicubic. This causes super-resolution models trained on synthetic data does not perform effectively on real ones. Training a model on real low and high resolution DSMs pairs is also a challenge because of the lack of information. On the other hand, the existence of other imaging modalities of the same scene can be used to enrich the information needed for large-scale super-resolution. In this work, we introduce a novel methodology to address the intricacies of real-world DSM super-resolution, named REAL-GDSR, breaking down this ill-posed problem into two steps. The first step involves the utilization of a residual local refinement network. This strategic approach departs from conventional methods that trained to directly predict height values instead of the differences (residuals) and utilize large receptive fields in their networks. The second step introduces a diffusion-based technique that enhances the results on a global scale, with a primary focus on smoothing and edge preservation. Our experiments underscore the effectiveness of the proposed method. We conduct a comprehensive evaluation, comparing it to recent state-of-the-art techniques in the domain of real-world DSM super-resolution (SR). Our approach consistently outperforms these existing methods, as evidenced through qualitative and quantitative assessments.
... The anisotropic [36] algorithm that Perona and Malik developed has been enhanced by numerous researchers [37]. By keeping image borders, lines, and areas intact, the Perona-Malik Diffusion (PMD) filter attempts to reduce speckle noise. ...
Medical ultrasound imaging involves the use of high-frequency sound waves to produce images of various body parts. A transducer generates these sound waves, which traverse through bodily tissues, providing measurements of soft tissue and organ dimensions, shapes, and consistencies. The quality of an ultrasound image depends on the frequency of the transducer used. Higher-frequency transducers yield better resolution, but they are limited in their ability to penetrate deeply into the body due to their shorter wavelengths, making them more susceptible to absorption. Lower-frequency transducers can penetrate deeper because they have longer wavelengths, although their resolution isn’t as sharp as that of high-frequency transducers. The primary drawback associated with ultrasound imaging is the introduction of noise during the signal processing stage, which can lead to images that are challenging to interpret. Efficient medical image processing plays a pivotal role in improving the quality and utility of ultrasound imaging for medical applications, enhancing image comprehension, and aiding in accurate diagnoses. One critical aspect of the pre-processing stage in medical image analysis, particularly in ultrasound images, is speckle noise reduction. To improve the analysis and diagnosis in various applications, it has become essential to employ software tools for speckle noise removal. In this paper we propose hybrid filter, which is combination of filters to be used to remove noise along with Convolutional Neural Networks. Experiments were performed at different speckle variances and the results showed that hybrid filter performed well when the speckle variance is between 0.1 and 0.3, while DnCNNL5 performed well when the speckle variance was high (between 0.5 and 0.9).
... where div and ∇ are respectively the divergence and gradient operators, and c is a conductivity function. Perona and Malik (1990) proposed a gradient dependent conductivity function c. The conductivity function c reduces the diffusion at the location of edge within a region and smooths a region preserving boundaries. ...
... Existing studies [36] put forward the application of anisotropic diffusion equation to the image field, which is a classical filtering method. The essence of anisotropy is to control the intensity of the filtering process with a differential coefficient decreasing with the gray gradient of the image. ...
In the ground-based infrared optical remote sensing system, the target signal is very weak due to the dynamic strong light background and the movement of small and small targets. In order to improve the limit detection capability, the background suppression and target enhancement methods are required to be more suitable for this scene. To solve this problem, we first analyze the image features in the current scene, and propose a more perfect point target and noise model. Then, we propose a new WY distribution function based on the Fermi Dirac distribution function, and propose an anisotropic filtering method based on this function, and further suppress the back-ground through the difference results of two steps. Then, based on the distribution function, we further designed an energy concentration accumulation strategy in 9 scale directions, through which the SNR of the target is effectively improved and the suppression ability of the background is enhanced. Through quantitative and qualitative experimental analysis, compared with the same type of algorithms, the proposed method has better robustness against extremely weak targets and dynamic backgrounds.
... Concretely, we obtain two eigenvalues µ 1 = g(z 1 ), µ 2 = g(z 2 ) and one (Perona & Malik, 1990), where the free parameter λ is learned during training. In contrast to the formulation in Equation 6, we apply the diffusivity to both eigenvalues. ...
Deep learning has revolutionized the field of computer vision by introducing large scale neural networks with millions of parameters. Training these networks requires massive datasets and leads to intransparent models that can fail to generalize. At the other extreme, models designed from partial differential equations (PDEs) embed specialized domain knowledge into mathematical equations and usually rely on few manually chosen hyperparameters. This makes them transparent by construction and if designed and calibrated carefully, they can generalize well to unseen scenarios. In this paper, we show how to bring model- and data-driven approaches together by combining the explicit PDE-based approaches with convolutional neural networks to obtain the best of both worlds. We illustrate a joint architecture for the task of inpainting optical flow fields and show that the combination of model- and data-driven modeling leads to an effective architecture. Our model outperforms both fully explicit and fully data-driven baselines in terms of reconstruction quality, robustness and amount of required training data. Averaging the endpoint error across different mask densities, our method outperforms the explicit baselines by 11-27%, the GAN baseline by 47% and the Probabilisitic Diffusion baseline by 42%. With that, our method sets a new state of the art for inpainting of optical flow fields from random masks.
... Conventional suppression methods for random noise in seismic data are mainly divided into space domain and transform domain methods (Necati, 1986;Joachim, 1997). Noise suppression methods in the space domain can be divided into median filtering , diffusion filtering (Perona and Malik, 1990), etc.,; The methods in transform domain noise suppression can be divided into frequency domain denoising (Necati, 1986), wavelet transform denoising (Morlet et al., 1982), meander transform denoising (Cao et al., 2015), and empirical mode decomposition-based denoising methods (Mirko and Maiza, 2009). ...
The signal-to-noise ratio (SNR) of seismic data is the key to seismic data processing, and it also directly affects interpretation of seismic data results. The conventional denoising method, independent variable analysis, uses adjacent traces for processing. However, this method has problems, such as the destruction of effective signals. The widespread use of velocity and acceleration geophones in seismic exploration makes it possible to obtain different types of signals from the same geological target, which is fundamental to the joint denoising of these two types of signals. In this study, we propose a joint denoising method using seismic velocity and acceleration signals. This method selects the same trace of velocity and acceleration signal for Independent Component Analysis (ICA) to obtain the independent initial effective signal and separation noise. Subsequently, the obtained effective signal and noise are used as the prior information for a Kalman filter, and the final joint denoising results are obtained. This method combines the advantages of low-frequency seismic velocity signals and high-frequency and high-resolution acceleration signals. Simultaneously, this method overcomes the problem of inconsistent stratigraphic reflection caused by the large spacing between adjacent traces, and improves the SNR of the seismic data. In a model data test and in field data from a work area in the Shengli Oilfield, the method increases the dominate frequency of the signal from 20 to 40 Hz. The time resolution was increased from 8.5 to 6.8 ms. The test results showed that the joint denoising method based on seismic velocity and acceleration signals can better improve the dominate frequency and time resolution of actual seismic data.
... In Eq. 12, P t indicates the photo diffusion, Δ = gradient operator, Δ = Laplacian operator, R(x, y, t) = diffusion rate, and t is the number of iterations. It is vital to keep the stability of diffusion, and this can be achieved using forward time central space (FTCS), which is accomplished by maximum principle [39]. It can be computed by Eq. 13: ...
The fusion of multimodal medical images has garnered painstaking attention for clinical diagnosis and surgical planning. Although various scholars have designed numerous fusion methods, the challenges of extracting substantial features without introducing noise and non-uniform contrast hindered the overall quality of fused photos. This paper presents a multimodal medical image fusion (MMIF) using a novel deep convolutional neural network (D-CNN) along with preprocessing schemes to circumvent the mentioned issues. A non-linear average median filtering (NL-AMF) and multiscale improved top-hat (MI-TH) approach are utilized at the preprocessing stage to remove noise and improve the contrast of images. The non-linear anisotropic diffusion (NL-AD) scheme is employed to split the photos into base and detailed parts. The fusion of base parts is accomplished by a dimension reduction method to retain the energy information. In contrast, the detailed parts are fused by novel D-CNN to preserve the enriched detailed features effectively. The simulation results demonstrate that the proposed method produces better brightness contrast and more image details than existing methods by acquiring 0.7649 to 0.8986, 0.3520 to 0.4783, 0.7639 to 0.9056, 68.8932 to 81.0487 gain for quality transfer ratio from source photo to a generated photo (QGAB\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_{G}^{AB}$$\end{document}), feature mutual information (FMI), structural similarity index (SSIM), and average pixel intensity (API) respectively.
We investigate whether geomorphic signatures of permafrost are embedded in planforms of river
meanders, and we inquire as to how physical factors unique to permafrost environments are able to affect their
dynamics. By exploiting satellite imagery, a data set of 19 freely‐meandering Arctic rivers is compared against
an independent data set of 23 freely‐meandering streams flowing through temperate and tropical regions.
Suitable dimensionless metrics are defined to characterize morphometric properties of meanders in terms of the
spatio‐temporal distribution of curvature and channel width. Results show the absence of marked contrasts in
the amplitude of bend‐curvature between the two data set. Differently, we find a permafrost signature in the
channel width response, which manifests itself through larger values of the average bend‐width and by peaks of
width fluctuations. Field data suggest that permafrost meanders tend to widen for increasing bend sinuosity,
likely promoting a shift of their morphodynamic regime as final cutoff is approached.
This chapter presents a multi-objective approach to obtain the anisotropic diffusion (AD) parameters for effective filter results. In the proposed scheme, the appropriate parameters for the anisotropic diffusion methodology are calculated as the solution which involves the best trade-of between two conflictive scenarios: minimize the noise content in an image as at the same time maximize the image contrast.
IT SEEMS THAT ACCORDING TO THE CLINICAL TRIALS REGISTERED IN 2021 in clinicalTrials.gov Identifier: NCT04850703.
Double blind with control group, group treated with tES and group treated with Dapoxetine.
Only the group treated with tES improved their sexual dysfunction, improved emotionally and their relationship.
The research carried out first identifies 2 groups of patients, those who only specifically have LPE and those who have LPE plus something else, although a true classification method such as SVM would have to be applied, etc.
In the LPE or 1 group, the 3 therapeutic cards for tES are identified:
1.- CCA
2.- Tonsils
3.- the best the right orbitofrontal cortex
The ERP P300 curve is not an adequate marker, but the CNV presented a p<0.005. at F4, F3, T3, T4 and Fz
Applying 24 days, 2 times a week for 35 minutes of 1000 to 1500 mA in the described areas.
Cone-beam computed laminography (CL) is still a very challenging problem for the inspection of thin-plate objects. Since CL projections are incomplete, the reconstructed images always suffer from severe aliasing and blurring in the z direction. To mitigate this problem, we propose an anisotropic adaptive weighted total variation (AAwTV) reconstruction model, which takes the edge properties between adjacent voxels into account and introduces different weights in different directions. In addition, we solved the proposed AAwTV using the Chambolle-Pock (CP) framework, since it has good computational efficiency and stable convergence, and is often easy to get a satisfactory reconstruction result. Experiments on simulated PCB phantom and simulated workpiece phantom show that the proposed algorithm can preserve the detailed features of the object well, and can effectively suppress inter-slice aliasing and blurring.
Thyroid ultrasound video provides significant value for thyroid diseases diagnosis, but the ultrasound imaging process is often affected by the speckle noise, resulting in poor quality of the ultrasound video. Numerous video denoising methods have been proposed to remove noise while preserving texture details. However, existing methods still suffer from the following problems: (1) relevant temporal features in the low-contrast ultrasound video cannot be accurately aligned and effectively aggregated by simple optical flow or motion estimation, resulting in the artifacts and motion blur in the video; (2) fixed receptive field in spatial features integration lacks the flexibility of aggregating features in the global region of interest and is susceptible to interference from irrelevant noisy regions. In this work, we propose a deformable spatial-temporal attention denoising network to remove speckle noise in thyroid ultrasound video. The entire network follows the bidirectional feature propagation mechanism to efficiently exploit the spatial-temporal information of the whole video sequence. In this process, two modules are proposed to address the above problems: (1) a deformable temporal attention module (DTAM) is designed after optical flow pre-alignment to further capture and aggregate relevant temporal features according to the learned offsets between frames, so that inter-frame information can be better exploited even with the imprecise flow estimation under the low contrast of ultrasound video; (2) a deformable spatial attention module (DSAM) is proposed to flexibly integrate spatial features in the global region of interest through the learned intra-frame offsets, so that irrelevant noisy information can be ignored and essential information can be precisely exploited. Finally, all these refined features are rectified and merged through residual convolution blocks to recover the clean video frames. Experimental results on our thyroid ultrasound video (US-V) dataset and the DDTI dataset demonstrate that our proposed method exceeds 1.2\(\sim\)1.3 dB on PSNR and has clearer texture detail compared to other state-of-the-art methods. In the meantime, the proposed model can also assist thyroid nodule segmentation methods to achieve more accurate segmentation effect, which provides an important basis for thyroid diagnosis. In the future, the proposed model can be improved and extended to other medical image sequence datasets, including CT and MRI slice denoising. The code and datasets are provided at https://github.com/Meta-MJ/DSTAN.
Magnetic resonance electrical impedance tomography (MREIT) is a high-resolution imaging modality that aims to reconstruct the objects' conductivity distributions at low frequency using the measurable $z$-th component of the magnetic flux density obtained from an MRI scanner. Traditional reconstruction algorithms in MREIT use two data subject to two linearly independent current densities. However, the temporal resolution of such a MREIT image is relatively low. Recently, a single current MREIT has been proposed to improve the temporal resolution. Even though a series of reconstruction algorithms have been proposed in the last two decades, the theoretical studies of MREIT are still quite limited. This paper presents the stability theorems for two datum and a single data-based isotropic conductivity reconstruction using MREIT. Using the regularity theory of elliptic partial differential equations, we prove that the only instability in the inverse problem of MREIT comes from taking the second-order derivative of the measured data $B_z$, the $z$-th component of the magnetic flux density. To get a stable reconstruction from the noisy $B_z$ data, we note that the edge structure of $\nabla B_z$ reveals the edge features in the unknown conductivity and provides an edge-preserving denoising approach for the $\nabla B_z$ data. We use a modified Shepp-Logan phantom model to validate the proposed theory and the denoising approach.
Neurodegenerative disorders are one of the most insidious disorders that affect millions around the world. Presently, these disorders do not have any remedy, however, if detected at an early stage, therapy can prevent further degeneration. This study aims to detect the early onset of one such neurodegenerative disorder called Alzheimer’s Disease, which is the most prevalent neurological disorder using the proposed Convolutional Neural Network (CNN). These MRI scans are pre-processed by applying various filters, namely, High-Pass Filter, Contrast Stretching, Sharpening Filter, and Anisotropic Diffusion Filter to enhance the Biomarkers in MRI images. A total of 21 models are proposed using different preprocessing and enhancement techniques on transverse and sagittal MRI images. The comparative analysis of the proposed five-layer Convolutional Neural Network (CNN) model with Alex Net is presented. The proposed CNN model outperforms AlexNet and achieves an accuracy of 99.40%, with a precision of 0.988, and recall of 1.00, by using an edge enhanced, contrast stretched, anisotropic diffusion filter. The proposed method may be used to implement automated diagnosis of neurodegenerative disorders.
In this paper we deal with a reaction–diffusion equation in a bounded interval of the real line with a nonlinear diffusion of Perona–Malik’s type and a balanced bistable reaction term. Under very general assumptions, we study the persistence of layered solutions, showing that it strongly depends on the behavior of the reaction term close to the stable equilibria \(\pm 1\), described by a parameter \(\theta >1\). If \(\theta \in (1,2)\), we prove existence of steady states oscillating (and touching) \(\pm 1\), called compactons, while in the case \(\theta =2\) we prove the presence of metastable solutions, namely solutions with a transition layer structure which is maintained for an exponentially long time. Finally, for \(\theta >2\), solutions with an unstable transition layer structure persist only for an algebraically long time.
The most critical part of a neutron computed tomography (NCT) system is the image processing algorithm, which directly affects the quality and speed of the reconstructed images. Various types of noise in the system can degrade the quality of the reconstructed images. Therefore, to improve the quality of the reconstructed images of NCT systems, efficient image processing algorithms must be used. The anisotropic diffusion filtering (ADF) algorithm can not only effectively suppress the noise in the projection data, but also preserve the image edge structure information by reducing the diffusion at the image edges. Therefore, we propose the application of the ADF algorithm for NCT image reconstruction. To compare the performance of different algorithms in NCT systems, we reconstructed images using the ordered subset simultaneous algebraic reconstruction technique (OS-SART) algorithm with different regular terms as image processing algorithms. In the iterative reconstruction, we selected two image processing algorithms, the Total Variation and split Bregman solved total variation algorithms, for comparison with the performance of the ADF algorithm. Additionally, the filtered back-projection algorithm was used for comparison with an iterative algorithm. By reconstructing the projection data of the numerical and clock models, we compared and analyzed the effects of each algorithm applied in the NCT system. Based on the reconstruction results, OS-SART-ADF outperformed the other algorithms in terms of denoising, preserving the edge structure, and suppressing artifacts. For example, when the 3D Shepp–Logan was reconstructed at 25 views, the root mean square error of OS-SART-ADF was the smallest among the four iterative algorithms, at only 0.0292. The universal quality index, mean structural similarity, and correlation coefficient of the reconstructed image were the largest among all algorithms, with values of 0.9877, 0.9878, and 0.9887, respectively.
In the field of image restoration, denoising is considered one of the most important techniques. It is a pre
processing approach aims to refine image clarity and enhance its overall quality by effectively reducing noise
present in the image. The aim is to obtain good-quality images from a version degraded by additive noise
or convolutional noise that introduces blur. As a result, more advanced treatments can be performed on the
resulting image. In order to remove Gaussian noise from input images, we propose the following methodology
using a fractional differential equation in time-space based on Gaussian convolution, where the integer and
fractional order derivatives of Caputo can be discretized using finite difference and 𝐿1−approximations. Once
the equation is solved numerically, the scheme is applied to grayscale digital images using the presented
algorithm. The parameters must be optimized and adjusted. As a result of testing with natural images, we
are able to successfully suppress the noise present in the images. Aside from that. Our model demonstrates
strong visual quality, as verified by the calculation of indexes such as Peak Signal-To-Noise Ratio (PSNR)
and Structural Similarity Index Measure (SSIM). Our denoising technique demonstrates its effectiveness in
mitigating noise present in both MRI and X-ray images, resulting in improved image quality and diagnostic
accuracy for these vital medical imaging modalities.
This paper describes a computational approach to edge detection. The success of the approach depends on the definition of a comprehensive set of goals for the computation of edge points. These goals must be precise enough to delimit the desired behavior of the detector while making minimal assumptions about the form of the solution. We define detection and localization criteria for a class of edges, and present mathematical forms for these criteria as functionals on the operator impulse response. A third criterion is then added to ensure that the detector has only one response to a single edge. We use the criteria in numerical optimization to derive detectors for several common image features, including step edges. On specializing the analysis to step edges, we find that there is a natural uncertainty principle between detection and localization performance, which are the two main goals. With this principle we derive a single operator shape which is optimal at any scale. The optimal detector has a simple approximate implementation in which edges are marked at maxima in gradient magnitude of a Gaussian-smoothed image. We extend this simple detector using operators of several widths to cope with different signal-to-noise ratios in the image. We present a general method, called feature synthesis, for the fine-to-coarse integration of information from operators at different scales. Finally we show that step edge detector performance improves considerably as the operator point spread function is extended along the edge.
Scale-space filtering constructs hierarchic symbolic signal descriptions by transforming the signal into a continuum of versions of the original signal convolved with a kernel containing a scale or bandwidth parameter. It is shown that the Gaussian probability density function is the only kernel in a broad class for which first-order maxima and minima, respectively, increase and decrease when the bandwidth of the filter is increased. The consequences of this result are explored when the signal¿or its image by a linear differential operator¿is analyzed in terms of zero-crossing contours of the transform in scale-space.
We characterize some properties of the zero crossings of the Laplacian of signals¿in particular images¿filtered with linear filters, as a function of the scale of the filter (extending recent work by Witkin [16]). We prove that in any dimension the only filter that does not create generic zero crossings as the scale increases is the Gaussian. This result can be generalized to apply to level crossings of any linear differential operator: it applies in particular to ridges and ravines in the image intensity. In the case of the second derivative along the gradient, there is no filter that avoids creation of zero crossings, unless the filtering is performed after the derivative is applied.
Mathematics Version of Record
Visual Reconstruction presents a unified and highly original approach to the treatment of continuity in vision. It introduces, analyzes, and illustrates two new concepts. The first—the weak continuity constraint—is a concise, computational formalization of piecewise continuity. It is a mechanism for expressing the expectation that visual quantities such as intensity, surface color, and surface depth vary continuously almost everywhere, but with occasional abrupt changes. The second concept—the graduated nonconvexity algorithm—arises naturally from the first. It is an efficient, deterministic (nonrandom) algorithm for fitting piecewise continuous functions to visual data. The book first illustrates the breadth of application of reconstruction processes in vision with results that the authors' theory and program yield for a variety of problems. The mathematics of weak continuity and the graduated nonconvexity (GNC) algorithm are then developed carefully and progressively.
Contents Modeling Piecewise Continuity • Applications of Piecewise Continuous Reconstruction • Introducing Weak Continuity Constraints • Properties of the Weak String and Membrane • Properties of Weak Rod and Plate • The Discrete Problem • The Graduated Nonconvexity (GNC) Algorithm • Appendixes: Energy Calculations for the String and Membrane • Noise Performance of the Weak Elastic String • Energy Calculations for the Rod and Plate • Establishing Convexity • Analysis of the GNC Algorithm
Visual Reconstruction is included in the Artificial Intelligence series, edited by Michael Brady and Patrick Winston.
We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.
Pyramid data structures for image processing are usually defined using discrete grids and discrete levels. It has proven useful to formulate pyramids in terms of continuous variable. When the level of the pyramid is changed to be a continuous variables, we talk of the resulting domain as 'scale-space.' When both the image domain and level are treated as continuous, the resulting pyramid structures are most naturally viewed in terms of partial differential equations governing their formation. This viewpoint allows one to generalize to new kinds of pyramid data structures, analyze their information content, and develop rational methods for treating borders and other problems in the discrete construction of pyramids. Keywords: Gaussian and Laplacian pyramids; Zero crossings; Heat equation. (Author)
Using the Heat Equation to formulate the notion of scale-space filtering, we show that the evolution property of level-crossings in scale-space is equivalent to the maximum principle. We briefly discuss filtering over bounded domains. We then consider the completeness of the representation of data by zero-crossings, and observe that for polynomial data, the issue is solved by standard results in algebraic geometry. For more general data, we argue that gradient information along the zero-crossings is needed, and that although such information more than suffices, the representation is still not stable. We give a simple linear procedure for reconstruction of data from zero-crossings and gradient data along zero-crossings in both continuous and discrete scale-space domains. Keywords: Signal and image data; Convolution.
The extrema in a signal and its first few derivatives provide a useful general purpose qualitative description for many kinds of signals. A fundamental problem in computing such descriptions is scale: a derivative must be taken over some neighborhood, but there is seldom a principled basis for choosing its size. Scale-space filtering is a method that describes signals qualitatively, managing the ambiguity of scale in an organized and natural way. The signal is first expanded by convolution with gaussian masks over a continuum of sizes. This "scale-space" image is then collapsed, using its qualitative structure, into a tree providing a concise but complete qualitative description covering all scales of observation. The description is further refined by applying a stability criterion, to identify events that persist of large changes in scale.
Gaussian blur, or convolution against a Gaussian kernel, is a common model for image and signal degradation. In general, the process of reversing Gaussian blur is unstable, and cannot be represented as a convolution filter in the spatial domain. If we restrict the space of allowable functions to polynomials of fixed finite degree, then a convolution inverse does exist. We give constructive formulas for the deblurring kernels in terms of Hermite polynomials, and observe that their use yields optimal approximate deblurring solutions among the space of bounded degree polynomials. The more common methods of achieving stable approximate deblurring include restrictions to band-limited functions or functions of bounded norm.
We make an analogy between images and statistical mechanics systems. Pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice-like physical system. The assignment of an energy function in the physical system determines its Gibbs distribution. Because of the Gibbs distribution, Markov random field (MRF) equivalence, this assignment also determines an MRF image model. The energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the MRF. For a range of degradation mechanisms, including blurring, nonlinear deformations, and multiplicative or additive noise, the posterior distribution is an MRF with a structure akin to the image model. By the analogy, the posterior distribution defines another (imaginary) physical system. Gradual temperature reduction in the physical system isolates low energy states (``annealing''), or what is the same thing, the most probable states under the Gibbs distribution. The analogous operation under the posterior distribution yields the maximum a posteriori (MAP) estimate of the image given the degraded observations. The result is a highly parallel ``relaxation'' algorithm for MAP estimation. We establish convergence properties of the algorithm and we experiment with some simple pictures, for which good restorations are obtained at low signal-to-noise ratios.
In this paper, we study the problem of interpreting line drawings of scenes composed of opaque regular solid objects bounded by piecewise smooth surfaces with no markings or texture on them. It is assumed that the line drawing has been formed by orthographic projection of such a scene under general viewpoint, that the line drawing is error free, and that there are no lines due to shadows or specularities. Our definition implicitly excludes laminae, wires, and the apices of cones.
A major component of the interpretation of line drawings is line labelling. By line labelling we mean (a) classification of each image curve as corresponding to either a depth or orientation discontinuity in the scene, and (b) further subclassification of each kind of discontinuity. For a depth discontinuity we determine whether it is a limb—a locus of points on the surface where the line of sight is tangent to the surface—or an occluding edge—a tangent plane discontinuity of the surface. For an orientation discontinuity, we determine whether it corresponds to a convex or concave edge. This paper presents the first mathematically rigorous scheme for labelling line drawings of the class of scenes described. Previous schemes for labelling line drawings of scenes containing curved objects were heuristic, incomplete, and lacked proper mathematical justification.
By analyzing the projection of the neighborhoods of different kinds of points on a piecewise smooth surface, we are able to catalog all local labelling possibilities for the different types of junctions in a line drawing. An algorithm is developed which utilizes this catalog to determine all legal labellings of the line drawing. A local minimum complexity rule—at each vertex select those labellings which correspond to the minimum number of faces meeting at the vertex—is used in order to prune highly counter-intuitive interpretations. The labelling scheme was implemented and tested on a number of line drawings. The labellings obtained are few and by and large in accordance with human interpretations.
In practice the relevant details of images exist only over a restricted range of scale. Hence it is important to study the dependence of image structure on the level of resolution. It seems clear enough that visual perception treats images on several levels of resolution simultaneously and that this fact must be important for the study of perception. However, no applicable mathematically formulated theory to deal with such problems appears to exist. In this paper it is shown that any image can be embedded in a one-parameter family of derived images (with resolution as the parameter) in essentially only one unique way if the constraint that no spurious detail should be generated when the resolution is diminished, is applied. The structure of this family is governed by the well known diffusion equation (a parabolic, linear, partial differential equation of the second order). As such the structure fits into existing theories that treat the front end of the visual system as a continuous stack of homogeneous layers, characterized by iterated local processing schemes. When resolution is decreased the images becomes less articulated because the extrem ("light and dark blobs") disappear one after the other. This erosion of structure is a simple process that is similar in every case. As a result any image can be described as a juxtaposed and nested set of light and dark blobs, wherein each blob has a limited range of resolution in which it manifests itself. The structure of the family of derived images permits a derivation of the sampling density required to sample the image at multiple scales of resolution.(ABSTRACT TRUNCATED AT 250 WORDS)
The process of detecting edges in a one-dimensional signal by
finding the zeros of the second derivative of the signal can be
interpreted as the process of detecting the critical points of a general
class of contrast functions that are applied to the signal. It is shown
that the second derivative of the contrast function at the critical
point is related to the classification of the associated edge as being
phantom or authentic. The contrast of authentic edges decreases with
filter scale, while the contrast of phantom edges are shown to increase
with scale. As the filter scale increases, an authentic edge must either
turn into a phantom edge or join with a phantom edge and vanish. The
points in the scale space at which these events occur are seen to be
singular points of the contrast function. Using ideas from singularity,
or catastrophy theory, the scale map contours near these singular points
are found to be either vertical or parabolic
Simple sets of parallel operations are described which can be used to detect texture edges, "spots," and "streaks" in digitized pictures. It is shown that, by comparing the outputs of the operations corresponding to (e.g.,) edges of different sizes, one can construct a composite output in which edges between differently textured regions are detected, and isolated objects are also detected, but the objects composing the textures are ignored. Relationships between this class of picture processing operations and the Gestalt psychologists' laws of pictorial pattern organization are also discussed.
Stochastic relaxation. Gibbs distribu-tions, and the Bayesian restoration of images I€€€ TrtrrisRepresentations based on zero-crossings in scale-space
- Perona And
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- S Geman
- D Gernan
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A network for edge detection and scale spaceEdge and curve detection for v~sual scene analysis
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- M Thurston
[I91 -, "A network for edge detection and scale space," in Proc. I€€E Inr. Symp. Circuits and Sysrmis, Helsinki. June 1988. pp. 2565-2568. 1201 A. Rosenfeld and M. Thurston. "Edge and curve detection for v~sual scene analysis.'' /€E€ Trans. Compur., vol. C-20, pp. 562-569. May
A network for edge detection and scale space
- P Pcrona
- J Malik
The scale-spacc formulation of pyramid data structures" in Parallel Computer Vision
- L Uhrin