Isotropic Gaussian Filter Research Articles

Expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain

Expressions for the calculation of isotropic Gaussian filter kernels in the spherical harmonic domain

Examining the Influence of Spatial Smoothing on Spatiotemporal Features of Intrinsic Connectivity Networks at Low ICA Model Order.

Using a relatively high model order of independent component analysis (ICA with 75 ICs) of functional magnetic resonance imaging (fMRI) data, we have reported a clear effect of spatial smoothing Gaussian kernel size on spatiotemporal properties of intrinsic connectivity networks (ICNs). However, many if not the majority of ICA fMRI studies are usually performed at low model order, e.g., 20-IC decomposition, as such low order is generally enough to extract the few networks of interest such as the default-mode network (DMN). The aim of this study is to investigate if we can replicate the spatial smoothing effects on spatiotemporal features of ICNs at low ICA model order. Same resting state fMRI data that we used with 75-IC analysis were used here. Spatial smoothing using an isotropic Gaussian filter kernel with full width at half maximum (FWHM) of 4, 8, and 12 mm was applied during preprocessing. ICNs were identified from 20-IC decomposition and evaluated in terms of three primary features: spatial map intensity, functional network connectivity (FNC), and power spectra. The results identified similar effects of spatial smoothing on spatial map intensities and power spectra at p < 0.01, false discovery rate (FDR) corrected for multiple comparisons. Reduced spatial smoothing kernel size resulted in decreased spatial map intensities as well as a generally decreased low-frequency power (0.01 - 0.10 Hz) but increased high-frequency power (0.15 - 0.25 Hz). FNC, however, did not show a uniform change in correlation values with the size of smoothing kernel. Notably, FNC between DMNs decreased but FNC between central executive and visual networks increased with an increase in smoothing kernel size. These preliminary findings confirm spatial smoothing influences ICN features regardless of model order. The discussion focuses on differences between observed changes at low and high ICA model orders.

Diffusion-informed spatial smoothing of fMRI data in white matter using spectral graph filters

Brain activation mapping using functional magnetic resonance imaging (fMRI) has been extensively studied in brain gray matter (GM), whereas in large disregarded for probing white matter (WM). This unbalanced treatment has been in part due to controversies in relation to the nature of the blood oxygenation level-dependent (BOLD) contrast in WM and its detectability. However, an accumulating body of studies has provided solid evidence of the functional significance of the BOLD signal in WM and has revealed that it exhibits anisotropic spatio-temporal correlations and structure-specific fluctuations concomitant with those of the cortical BOLD signal. In this work, we present an anisotropic spatial filtering scheme for smoothing fMRI data in WM that accounts for known spatial constraints on the BOLD signal in WM. In particular, the spatial correlation structure of the BOLD signal in WM is highly anisotropic and closely linked to local axonal structure in terms of shape and orientation, suggesting that isotropic Gaussian filters conventionally used for smoothing fMRI data are inadequate for denoising the BOLD signal in WM. The fundamental element in the proposed method is a graph-based description of WM that encodes the underlying anisotropy observed across WM, derived from diffusion-weighted MRI data. Based on this representation, and leveraging graph signal processing principles, we design subject-specific spatial filters that adapt to a subject’s unique WM structure at each position in the WM that they are applied at. We use the proposed filters to spatially smooth fMRI data in WM, as an alternative to the conventional practice of using isotropic Gaussian filters. We test the proposed filtering approach on two sets of simulated phantoms, showcasing its greater sensitivity and specificity for the detection of slender anisotropic activations, compared to that achieved with isotropic Gaussian filters. We also present WM activation mapping results on the Human Connectome Project’s 100-unrelated subject dataset, across seven functional tasks, showing that the proposed method enables the detection of streamline-like activations within axonal bundles.

Corner Detection Using Second-Order Generalized Gaussian Directional Derivative Representations.

Corner detection is a critical component of many image analysis and image understanding tasks, such as object recognition and image matching. Our research indicates that existing corner detection algorithms cannot properly depict the difference between edges and corners and this results in wrong corner detections. In this paper, the capability of second-order generalized (isotropic and anisotropic) Gaussian directional derivative filters to suppress Gaussian noise is evaluated. The second-order generalized Gaussian directional derivative representations of step edge, L-type corner, Y- or T-type corner, X-type corner, and star-type corner are investigated and obtained. A number of properties for edges and corners are discovered which enable us to propose a new image corner detection method. Finally, the criteria on detection accuracy and average repeatability under affine image transformation, JPEG compression, and noise degradation, and the criteria on region repeatability are used to evaluate the proposed detector against nine state-of-the-art methods. The experimental results show that our proposed detector outperforms all the other tested detectors.

Additional methods for the stable calculation of isotropic Gaussian filter coefficients: The case of a truncated filter kernel

The isotropic Gaussian filter is frequently used as a post-processing method for mitigating errors in Gravity Recovery and Climate Experiment (GRACE) time-variable gravity field solutions. It is known that the recurrent calculation of isotropic Gaussian filter coefficients in the spherical harmonic domain results in numerical instabilities. This issue has been recently resolved by Piretzidis & Sideris (Computers & Geosciences 133:104303) for the case of a global Gaussian filter. In this paper, we extend this work for the case of a regional Gaussian filter (i.e., a Gaussian filter with a truncated kernel). We show that instabilities also occur when the coefficients of a regional Gaussian filter are calculated using a conventional recurrence relation, and we provide methods for their proper treatment. We also briefly present some theoretical preliminaries related to recurrence relations and examine algorithms (i.e., initial value and boundary value algorithms) for their stable evaluation. Our analysis indicates that the initial value algorithms are ineffective for the calculation of regional Gaussian filter coefficients, contrary to the global Gaussian filter. The comparison of the boundary value algorithms shows that the Thomas and lower-upper (LU) decomposition algorithms are the most suitable for this case. Their main advantage is their insusceptibility to numerical limitations, such as overflow errors. The Thomas algorithm is also the most efficient in terms of computational speed.

Diagnosis of Cataract using a Robust Algorithm for Telemedicine Applications

This project proposes and evaluates an algorithm to automatically detect cataract from colored images of adult human subjects. The methods currently available make use of DSLR (Digital Single Lens Reflex) cameras which are very expensive thereby, making the whole process expensive. The main objective of this project is to provide a robust and affordable alternative to the classic treatment method which adopts the above methods. In this project via an algorithm we aim to diagnose the presence of cataract from the true colour images of an eye. The algorithm proposed makes use of OpenCV for cataract screening based on textural features such as uniformity, intensity and standard deviation. These features are first computed and mapped using Data Mining techniques after consultation with an eye expert to define the basic threshold of screening system and later tested on real subjects in an eye clinic. Pre-processing includes conservative smoothing followed by image de-noising. The isotropic Gaussian filter is widely used as a low pass filter for image de-noising.Feature extraction is done after pre-processing to extract all the information for cataract detection from the eye’s pupil region. The extracted parameters were compared with the values obtained from an ophthalmologist to determine the presence of Cataract in the eye of the patient. Finally, a tele-ophthalmology model using our proposed system has been suggested, which confirms the telemedicine application of the proposed system.

Diagnosis of Cataract using a Robust Algorithm for Telemedicine Applications

This project proposes and evaluates an algorithm to automatically detect cataract from colored images of adult human subjects. The methods currently available make use of DSLR (Digital Single Lens Reflex) cameras which are very expensive thereby, making the whole process expensive. The main objective of this project is to provide a robust and affordable alternative to the classic treatment method which adopts the above methods. In this project via an algorithm we aim to diagnose the presence of cataract from the true colour images of an eye. The algorithm proposed makes use of OpenCV for cataract screening based on textural features such as uniformity, intensity and standard deviation. These features are first computed and mapped using Data Mining techniques after consultation with an eye expert to define the basic threshold of screening system and later tested on real subjects in an eye clinic. Pre-processing includes conservative smoothing followed by image de-noising. The isotropic Gaussian filter is widely used as a low pass filter for image de-noising.Feature extraction is done after pre-processing to extract all the information for cataract detection from the eye’s pupil region. The extracted parameters were compared with the values obtained from an ophthalmologist to determine the presence of Cataract in the eye of the patient. Finally, a tele-ophthalmology model using our proposed system has been suggested, which confirms the telemedicine application of the proposed system.

Stable recurrent calculation of isotropic Gaussian filter coefficients

Filtering in the spherical harmonic domain has become a common practice for the post-processing of temporal gravity field solutions derived from the Gravity Recovery and Climate Experiment (GRACE) satellite mission. The most frequently used filtering technique to suppress high-frequency noise in GRACE data is the isotropic Gaussian averaging filter. The implementation of the isotropic Gaussian filter in the spherical harmonic domain requires the calculation of filter coefficients, which is performed using either a forward recurrence relation, or numerical integration. Both methods show limitations, namely, the forward recurrence relation is numerically unstable and the numerical integration suffers from low accuracy. The objective of this study is to investigate these open problems more thoroughly and to provide alternative expressions for the calculation of isotropic Gaussian filter coefficients. We test three methods (i.e., backward recurrence, continued fraction and sweep algorithms) for the stable evaluation of three-term recurrence relations and examine their properties. All methods successfully solve the instability issues of the forward recurrence relation. They are also more accurate and more computationally efficient than numerical integration.

Designing an anisotropic noise filter for measuring critical dimension and line edge roughness from scanning electron microscope images

The scanning electron microscope (SEM) is often employed in inspecting patterns transferred through a lithographic process. A typical inspection is to measure the critical dimension (CD) and line edge roughness (LER) of each feature in a transferred pattern. Such inspection may be done by utilizing image processing techniques to detect the boundaries of a feature. Since SEM images tend to include a substantial level of noise, a proper reduction of noise is essential before the subsequent process of edge detection. In a previous study, a method of designing an isotropic Gaussian filter adaptive to the noise level was developed. However, its performance for relatively small features was not so good as for large features, especially in the case of LER. The main objective of this study is to improve the design method such that the accuracy of the measured CD and LER is not deteriorated substantially as the feature size decreases. The new design method allows a Gaussian filter to be anisotropic for the better adaptability to the signal and noise, both of which show a substantial level of directional correlation. The cutoff frequency for the direction normal to features is determined to include most of the signal components, and the cutoff frequency in the other direction is set to balance the signal and noise components to be included. This procedure enables a systematic and easy design of the filter. Also, the method of estimating the noise has been modified for higher accuracy. The performance of the new design method has been thoroughly analyzed using the reference images for which feature boundaries are known. It has been shown that the anisotropic filter designed by the proposed method performs better than the isotropic filter from the viewpoint of CD, LER, and power spectral density accuracy.

SU‐E‐J‐261: The Importance of Appropriate Image Preprocessing to Augment the Information of Radiomics Image Features

Purpose:To investigate how different image preprocessing techniques, their parameters, and the different boundary handling techniques can augment the information of features and improve feature's differentiating capability.Methods:Twenty‐seven NSCLC patients with a solid tumor volume and no visually obvious necrotic regions in the simulation CT images were identified. Fourteen of these patients had a necrotic region visible in their pre‐treatment PET images (necrosis group), and thirteen had no visible necrotic region in the pre‐treatment PET images (non‐necrosis group). We investigated how image preprocessing can impact the ability of radiomics image features extracted from the CT to differentiate between two groups. It is expected the histogram in the necrosis group is more negatively skewed, and the uniformity from the necrosis group is less. Therefore, we analyzed two first order features, skewness and uniformity, on the image inside the GTV in the intensity range [−20HU, 180HU] under the combination of several image preprocessing techniques: (1) applying the isotropic Gaussian or anisotropic diffusion smoothing filter with a range of parameter(Gaussian smoothing: size=11, sigma=0:0.1:2.3; anisotropic smoothing: iteration=4, kappa=0:10:110); (2) applying the boundaryadapted Laplacian filter; and (3) applying the adaptive upper threshold for the intensity range. A 2‐tailed T‐test was used to evaluate the differentiating capability of CT features on pre‐treatment PT necrosis.Result:Without any preprocessing, no differences in either skewness or uniformity were observed between two groups. After applying appropriate Gaussian filters (sigma&gt;=1.3) or anisotropic filters(kappa &gt;=60) with the adaptive upper threshold, skewness was significantly more negative in the necrosis group(p&lt;0.05). By applying the boundary‐adapted Laplacian filtering after the appropriate Gaussian filters (0.5 &lt;=sigma&lt;=1.1) or anisotropic filters(20&lt;=kappa &lt;=50), the uniformity was significantly lower in the necrosis group (p&lt;0.05).Conclusion:Appropriate selection of image preprocessing techniques allows radiomics features to extract more useful information and thereby improve prediction models based on these features.

Improving Surface Geostrophic Current from a GOCE-Derived Mean Dynamic Topography Using Edge-Enhancing Diffusion Filtering

With increased geoid resolution provided by the gravity and steady-state ocean circulation explorer (GOCE) mission, the ocean’s mean dynamic topography (MDT) can be now estimated with an accuracy not available prior to using geodetic methods. However, an altimetric-derived MDT still needs filtering in order to remove short wavelength noise unless integrated methods are used in which the three quantities are determined simultaneously using appropriate covariance functions. We studied nonlinear anisotropic diffusive filtering applied to the ocean´s MDT and a new approach based on edge-enhancing diffusion (EED) filtering is presented. EED filters enable controlling the direction and magnitude of the filtering, with subsequent enhancement of computations of the associated surface geostrophic currents (SGCs). Applying this method to a smooth MDT and to a noisy MDT, both for a region in the Northwestern Pacific Ocean, we found that EED filtering provides similar estimation of the current velocities in both cases, whereas a non-linear isotropic filter (the Perona and Malik filter) returns results influenced by local residual noise when a difficult case is tested. We found that EED filtering preserves all the advantages that the Perona and Malik filter have over the standard linear isotropic Gaussian filters. Moreover, EED is shown to be more stable and less influenced by outliers. This suggests that the EED filtering strategy would be preferred given its capabilities in controlling/preserving the SGCs.

MRI-guided brain PET image filtering and partial volume correction

Positron emission tomography (PET) image quantification is a challenging problem due to limited spatial resolution of acquired data and the resulting partial volume effects (PVE), which depend on the size of the structure studied in relation to the spatial resolution and which may lead to over or underestimation of the true tissue tracer concentration. In addition, it is usually necessary to perform image smoothing either during image reconstruction or afterwards to achieve a reasonable signal-to-noise ratio. Typically, an isotropic Gaussian filtering (GF) is used for this purpose. However, the noise suppression is at the cost of deteriorating spatial resolution. As hybrid imaging devices such as PET/MRI have become available, the complementary information derived from high definition morphologic images could be used to improve the quality of PET images. In this study, first of all, we propose an MRI-guided PET filtering method by adapting a recently proposed local linear model and then incorporate PVE into the model to get a new partial volume correction (PVC) method without parcellation of MRI. In addition, both the new filtering and PVC are voxel-wise non-iterative methods. The performance of the proposed methods were investigated with simulated dynamic FDG brain dataset and 18F-FDG brain data of a cervical cancer patient acquired with a simultaneous hybrid PET/MR scanner. The initial simulation results demonstrated that MRI-guided PET image filtering can produce less noisy images than traditional GF and bias and coefficient of variation can be further reduced by MRI-guided PET PVC. Moreover, structures can be much better delineated in MRI-guided PET PVC for real brain data.

Structure tensor based image interpolation method

Feature preserving image interpolation is an active area in image processing field. In this paper a new direct edge directed image super-resolution algorithm based on structure tensors is proposed. Using an isotropic Gaussian filter, the structure tensor at each pixel of the input image is computed and the pixels are classified to three distinct classes; uniform region, corners and edges, according to the eigenvalues of the structure tensor. Due to application of the isotropic Gaussian filter, the classification is robust to noise presented in image. Based on the tangent eigenvector of the structure tensor, the edge direction is determined and used for interpolation along the edges. In comparison to some previous edge directed image interpolation methods, the proposed method achieves higher quality in both subjective and objective aspects. Also the proposed method outperforms previous methods in case of noisy and JPEG compressed images. Furthermore, without the need for optimization in the process, the algorithm can achieve higher speed.

Two-dimensional multi-pixel anisotropic Gaussian filter for edge-line segment (ELS) detection

In order to further improve the performance of the existing anisotropic Gaussian filters and more fully take advantage of structural information of a boundary, we heuristically develop a new multi-pixel anisotropic Gaussian filter to detect edges or edge-line segments directly from low signal-to-noise ratio images. To significantly increase computational efficiency, the classical isotropic Gaussian filters are first used for quickly estimating an approximate direction along an edge; then our filter is applied to more accurately search edge-line segment direction by a few directional filter masks only near such approximate direction. By comparing the proposed filter with the isotropic Gaussian filters, we analyze two improvement factors associated with the localization and SNR of the proposed filter. Experimental results show that the proposed detector can achieve better performance than several existing edge-detection methods in the sense of noise reduction, good localization, and high edge continuity.

Non-isotropic Gaussian smoothing and leakage reduction for determining mass changes over land and ocean using GRACE data

SUMMARY We present a new non-isotropic Gaussian filter for smoothing mass changes computed from Gravity Recovery and Climate Experiment (GRACE) L2 products and a new method to reduce land–ocean signal leakage caused by Gaussian smoothing. The kernel of our non-isotropic filter is the product of two Gaussian functions with distinct latitudinal and longitudinal smoothing radii. When expressed as number of kilometres at the Earth’s surface, the longitudinal smoothing radius, defined as a fixed longitude interval, is longer at the equator, and shorter at higher latitude. This is principally in accordance with the resolution of the GRACE data, and permits us to produce homogeneously smoothed results without excessive smoothing in latitudinal direction. This filter is not applicable in polar regions, where we choose to use the classical isotropic Gaussian filter. A smoothing radius choice scheme is proposed for the two filters to mesh seamlessly. In our leakage reduction method, the inputs are the mass change data after smoothing using a Gaussian filter. Along coasts where mass change signal on land is far larger than that over ocean (or signal over ocean is reduced to a very small magnitude by removing a model beforehand, and adding the model back afterwards), our method approximately recovers a smoothed mass change signal over both land and ocean sides as if a regional Gaussian filter with the same smoothing radius were applied over land and ocean separately, in which no signal leakages appear. The side lobe problem does not appear in our approach. Our leakage reduction method could also be used to study mass changes within a region where signal is far larger than that in surrounding regions, or where signal in surrounding regions could be reduced to very low magnitude by removing a model.

Improved inverse and probabilistic methods for geophysical applications of GRACE gravity data

SUMMARY Mapping time-varying gravity via satellite-to-satellite tracking systems holds great potential as a new way to monitor the Earth's global climate system. Measurement noises and systematic deficiencies in sampling, both in time and space, cause global geoid or surface mass solutions to have a structured spherical harmonic error spectrum, with strong degree and order dependences and cross-correlations. To extract average values of geoid or surface mass variations around global gridpoints on Earth's surface and over various geographic regions, both the shape of the averaging kernel and the resulting average uncertainties must be considered quantitatively and statistically. We investigate two methods of the Backus and Gilbert continuous geophysical inverse formalism for optimal averages around points on Earth's surface. The first averaging kernel optimally approximates the Dirac-δ function. With an equivalent measure of deviation from the Dirac-δ function, the optimal average has greater (up to 2.6 times) accuracy than does the most widely used isotropic Gaussian filter for GRACE analysis. The second method was crafted to decrease the kernel weight as the distance from the point of interest increases. A new method is presented to use a modified Gaussian averaging kernel that reduces average uncertainties with minimum loss of resolution. The modified method has some advantages over using the kernel that optimally approximates the Dirac-δ function. Both methods are computationally efficient and are applied to simulated and real GRACE data to compute improved averages around fine-resolution global gridpoints and used with non-diagonal covariance matrices to intelligently reduce effects of correlated errors. The optimal probabilistic method of least squares with a priori information is discussed in the spherical harmonic domain. The property of optimality will be preserved when the estimates are mapped to the geographic domain for spatial averages. A regionally-bounded Gaussian a priori function is derived in the spherical harmonic domain to better represent different change regimes separated by major geographic boundaries. We also introduce an algorithm to derive the optimal regional average incorporating a constraint such that the average weight over the region is unity. Applications of such more realistic a priori information (and/or constraint) can produce improved average estimates using satellite gravity data.

Effects of spatial smoothing on fMRI group inferences

The analysis of functional magnetic resonance imaging (fMRI) data involves multiple stages of data pre-processing before the activation can be statistically detected. Spatial smoothing is a very common pre-processing step in the analysis of functional brain imaging data. This study presents a broad perspective on the influence of spatial smoothing on fMRI group activation results. The data obtained from 20 volunteers during a visual oddball task were used for this study. Spatial smoothing using an isotropic gaussian filter kernel with full width at half maximum (FWHM) sizes 2 to 30 mm with a step of 2 mm was applied in two levels - smoothing of fMRI data and/or smoothing of single-subject contrast files prior to general linear model random-effects group analysis generating statistical parametric maps. Five regions of interest were defined, and several parameters (coordinates of nearest local maxima, t value, corrected threshold, effect size, residual values, etc.) were evaluated to examine the effects of spatial smoothing. The optimal filter size for group analysis is discussed according to various criteria. For our experiment, the optimal FWHM is about 8 mm. We can conclude that for robust experiments and an adequate number of subjects in the study, the optimal FWHM for single-subject inference is similar to that for group inference (about 8 mm, according to spatial resolution). For less robust experiments and fewer subjects in the study, a higher FWHM would be optimal for group inference than for single-subject inferences.

Using signal processing techniques to model worm propagation over wireless sensor networks

This paper describes how a topologically aware worm propagation model (TWPM) for wireless sensor networks is formulated and derived. The paper defines worm propagation characteristics that are specific to sensor networks. It also parameterize the effects of physical channel conditions, medium access control (MAC) layer contention, network layer routing, and transport layer protocol on worm propagation in sensor networks. The basic model formulation results in a partial differential equation, which is solved in the frequency domain to yield a closed-form solution for the TWPM. It is shown that in the spatial domain, the TWPM spread function is low-pass filtered by a two-dimensional isotropic Gaussian filter, thereby providing an intuitive feel for the dependence of the model on its underlying parameters. Simulation results demonstrate that the TWPM provides an effective and accurate worm propagation model for sensor networks

Computational Aspects of Three-Term Recurrence Relations

Computational Aspects of Three-Term Recurrence Relations