Physics-Informed Neural Networks (PINNs) have gained increasing attention for solving partial differential equations, including the Helmholtz equation, due to their flexibility and mesh-free formulation. However, their low-frequency bias limits their accuracy and convergence speed for high-frequency wavefield simulations. To alleviate these problems, we propose a simplified PINN framework that incorporates Gabor functions, designed to capture the oscillatory and localized nature of wavefields more effectively. Unlike previous attempts that rely on auxiliary networks to learn Gabor parameters, we redefine the network's task to map input coordinates to a custom Gabor coordinate system, simplifying the training process without increasing the number of trainable parameters compared to a simple PINN. We validate the proposed method across multiple velocity models, including the complex Marmousi and Overthrust models, and demonstrate its superior accuracy, faster convergence, and better robustness features compared to both traditional PINNs and earlier Gabor-based PINNs. Additionally, we propose an efficient integration of a Perfectly Matched Layer (PML) to enhance wavefield behavior near the boundaries. These results suggest that our approach offers an efficient and accurate alternative for scattered wavefield modeling and lays the groundwork for future improvements in PINN-based seismic applications.

Recently, Physics-Informed Neural Networks (PINNs) have gained significant attention for their versatile interpolation capabilities in solving partial differential equations (PDEs). Despite their potential, the training can be computationally demanding, especially for intricate functions like wavefields. This is primarily due to the neural-based (learned) basis functions, biased toward low frequencies, as they are dominated by polynomial calculations, which are not inherently wavefield-friendly. In response, we propose an approach to enhance the efficiency and accuracy of neural network wavefield solutions by modeling them as linear combinations of Gabor basis functions that satisfy the wave equation. Specifically, for the Helmholtz equation, we augment the fully connected neural network model with an adaptable Gabor layer constituting the final hidden layer, employing a weighted summation of these Gabor neurons to compute the predictions (output). These weights/coefficients of the Gabor functions are learned from the previous hidden layers that include nonlinear activation functions. To ensure the Gabor layer's utilization across the model space, we incorporate a smaller auxiliary network to forecast the center of each Gabor function based on input coordinates. Realistic assessments showcase the efficacy of this novel implementation compared to the vanilla PINN, particularly in scenarios involving high-frequencies and realistic models that are often challenging for PINNs.

Leaf vein forms the basis of leaf characterization and classification. Different species have different leaf vein patterns. It is seen that leaf vein segmentation will help in maintaining a record of all the leaves according to their specific pattern of veins thus provide an effective way to retrieve and store information regarding various plant species in database as well as provide an effective means to characterize plants on the basis of leaf vein structure which is unique for every species. The algorithm proposes a new way of segmentation of leaf veins with the use of Odd Gabor filters and the use of morphological operations for producing a better output. The Odd Gabor filter gives an efficient output and is robust and scalable as compared with the existing techniques as it detects the fine fiber like veins present in leaves much more efficiently.

We study the application of a strongly non-linear generative model to image patches. As in standard approaches such as Sparse Coding or Independent Component Analysis, the model assumes a sparse prior with independent hidden variables. However, in the place where standard approaches use the sum to combine basis functions we use the maximum. To derive tractable approximations for parameter estimation we apply a novel approach based on variational Expectation Maximization. The derived learning algorithm can be applied to large-scale problems with hundreds of observed and hidden variables. Furthermore, we can infer all model parameters including observation noise and the degree of sparseness. In applications to image patches we find that Gabor-like basis functions are obtained. Gabor-like functions are thus not a feature exclusive to approaches assuming linear superposition. Quantitatively, the inferred basis functions show a large diversity of shapes with many strongly elongated and many circular symmetric functions. The distribution of basis function shapes reflects properties of simple cell receptive fields that are not reproduced by standard linear approaches. In the study of natural image statistics, the implications of using different superposition assumptions have so far not been investigated systematically because models with strong non-linearities have been found analytically and computationally challenging. The presented algorithm represents the first large-scale application of such an approach.

A crucial part of developing mathematical models of how the brain works is the quantification of their success. One of the most widely-used metrics yields the percentage of the variance in the data that is explained by the model. Unfortunately, this metric is biased due to the intrinsic variability in the data. This variability is in principle unexplainable by the model. We derive a simple analytical modification of the traditional formula that significantly improves its accuracy (as measured by bias) with similar or better precision (as measured by mean-square error) in estimating the true underlying Variance Explained by the model class. Our estimator advances on previous work by a) accounting for the uncertainty in the noise estimate, b) accounting for overfitting due to free model parameters mitigating the need for a separate validation data set and c) adding a conditioning term. We apply our new estimator to binocular disparity tuning curves of a set of macaque V1 neurons and find that on a population level almost all of the variance unexplained by Gabor functions is attributable to noise.

Maximizing the efficiency of data collection is important in any experimental setting.

Linear implementations of the efficient coding hypothesis, such as independent component analysis (ICA) and sparse coding models, have provided functional explanations for properties of simple cells in V1 [1, 2]. These models, however, ignore the nonlinear behavior of neurons and fail to match individual and population properties of neural receptive fields in subtle but important ways. Hierarchical models, including Gaussian Scale Mixtures [3, 4] and other generative statistical models [5, 6], can capture higher-order regularities in natural images and explain nonlinear aspects of neural processing such as normalization and context effects [6,7]. Previously, it had been assumed that the lower level representation is independent of the hierarchy, and had been fixed when training these models. Here we examine the optimal lower-level representations derived in the context of a hierarchical model and find that the resulting representations are strikingly different from those based on linear models. Unlike the the basis functions and filters learned by ICA or sparse coding, these functions individually more closely resemble simple cell receptive fields and collectively span a broad range of spatial scales. Our work unifies several related approaches and observations about natural image structure and suggests that hierarchical models might yield better representations of image structure throughout the hierarchy.

Linear implementations of the efficient coding hypothesis, such as independent componentanalysis (ICA) and sparse coding models, have provided functional explanations for properties of simple cells in V1 [1, 2]. These models, however, ignore the nonlinear behavior of neurons and fail to match individual and population properties of neural receptive fields in subtle but important ways. Hierarchical models, including Gaussian ScaleMixtures [3, 4] and other generative statistical models [5, 6], can capture higher-order regularities in natural images and explain nonlinear aspectsof neural processing such as normalization and context effects [6,7]. Previously, it had been assumed that the lower level representation isindependent of the hierarchy, and had been fixed when training these models. Here we examine the optimal lower-level representations derived in the context of a hierarchical model and find that the resulting representations are strikingly different from those based on linear models. Unlikethe the basis functions and filters learned by ICA or sparse coding, these functions individually more closely resemble simple cell receptive fields and collectively span a broad range of spatial scales. Our work unifies several related approaches and observations about natural image structure and suggests that hierarchical models might yield better representations of image structure throughout the hierarchy.