Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Improved bearing estimation in ocean by nonlinear wavelet denoising under non-Gaussian noise conditions

Bearing estimation of underwater acoustic sources is an important aspect of passive localization in the ocean. The performance of all bearing estimation techniques degrades under conditions of low signal-to-noise ratio (SNR) at the sensor array. The degradation may be arrested by denoising the array data before performing the task of bearing estimation. In the last few years, there has been considerable progress in the use of the wavelet transform for denoising signals. It is known that the traditional wavelet transform, which is a linear transformation, can be used for denoising signals in Gaussian noise; but this method is not suitable if the noise is strongly non-Gaussian. Statistical measurements of ocean acoustic ambient noise data indicate that the noise may have a significantly non-Gaussian heavy-tailed distribution in some environments. In this work, we have explored the possibility of employing nonlinear wavelet denoising [1, 2], a robust technique based on median interpolation, to improve the performance of bearing estimation techniques in ocean in a strongly non-Gaussian noise environment. We propose the application of nonlinear wavelet denoising to the noisy signal at each sensor in the array to boost the SNR before performing bearing estimation by known techniques such as MUSIC and Subspace Intersection Method [3]. Simulation results are presented to show that denoising leads to a significant reduction in the mean square errors (MSE) of the estimators, and enhancement of resolution of closely spaced sources

Improved localization of underwater acoustic sources by nonlinear wavelet denoising under non-Gaussian noise conditions

Bearing estimation of underwater acoustic sources is an important aspect of passive localization in the ocean. The performance of all bearing estimation techniques degrades under conditions of low signal-to-noise ratio (SNR) at the sensor array. The degradation may be arrested by denoising the array data before performing the task of bearing estimation. In the last few years, there has been considerable progress in the use of the wavelet transform for denoising signals. It is known that the traditionalwavelet transform, which is a linear transformation, can be used for denoising signals in Gaussian noise; but this method is not suitable if the noise is strongly non-Gaussian. Statistical measurements of ocean acoustic ambient noise data indicate that the noise may have a significantly non-Gaussian heavy-tailed distribution in some environments. In this work, we have explored the possibility of employing nonlinear wavelet denoising [1, 2], a robust technique based on median interpolation, to improve the performance of bearing estimation techniques in ocean in a strongly non-Gaussian noise environment. We propose the application of nonlinear wavelet denoising to the noisy signal at each sensor in the array to boost the SNR before performing bearing estimation by known techniques such as MUSIC and Subspace Intersection Method [3]. Simulation results are presented to show that denoising leads to a significant reduction in the mean square errors (MSE) of the estimators, and enhancement of resolution of closely spaced sources

Improved Stroke Detection at Early Stages Using Haar Wavelets and Laplacian Pyramid

Stroke merupakan pembunuh nomor tiga di dunia, namun hanya sedikit metode tentang deteksi dini. Oleh karena itu dibutuhkan metode untuk mendeteksi hal tersebut. Penelitian ini mengusulkan sebuah metode gabungan untuk mendeteksi dua jenis stroke secara simultan. Haar wavelets untuk mendeteksi stroke hemoragik dan Laplacian pyramid untuk mendeteksi stroke iskemik. Tahapan dalam penelitian ini terdiri dari pra proses tahap 1 dan 2, Haar wavelets, Laplacian pyramid, dan perbaikan kualitas citra. Pra proses adalah menghilangkan bagian tulang tengkorak, reduksi derau, perbaikan kontras, dan menghilangkan bagian selain citra otak. Kemudian dilakukan perbaikan citra. Selanjutnya Haar wavelet digunakan untuk ekstraksi daerah hemoragik sedangkan Laplacian pyramid untuk ekstraksi daerah iskemik. Tahapan terakhir adalah menghitung fitur Grey Level Cooccurrence Matrix (GLCM) sebagai fitur untuk proses klasifikasi. Hasil visualisasi diproses lanjut untuk ekstrasi fitur menggunakan GLCM dengan 12 fitur dan kemudian GLCM dengan 4 fitur. Untuk proses klasifikasi digunakan SVM dan KNN, sedangkan pengukuran performa menggunakan akurasi. Jumlah data hemoragik dan iskemik adalah 45 citra yang dibagi menjadi 2 bagian, 28 citra untuk pengujian dan 17 citra untuk pelatihan. Hasil akhir menunjukkan akurasi tertinggi yang dicapai menggunakan SVM adalah 82% dan KNN adalah 88%

De-Noising via Wavelet Transforms Using Steerable Filters

Feature extraction remains an important part of low-level vision. Traditional oriented filters have been effective tools to identify features, such as lines and edges. Steerable filters, which can be adjusted at arbitrary orientation, have made decisions of feature orientations more precise. Combined with a pyramid structure of a multiscale representation, these filters can provide a reliable and efficient tool for image analysis. This paper takes advantage of multiscale steerable filters in the context of de-noising. First a set of novel filters are designed, that decompose the frequency plane into distinct directional bands. Next, we identify the dominant direction and strength at each point of an image from quadrature pairs of steerable filters. A nonlinear threshold function is then applied to the filtered coefficients to suppress noise. The denoised image is restored from coefficients modified at each level of transform space. We demonstrate the benefits of multiscale steerable filters for de-noising and show that it can greatly reduce noise while preserving image features. Two examples are presented to verify the efficacy of the technique

A New Formalism for Nonlinear and Non-Separable Multi-scale Representation

In this paper, we present a new formalism for nonlinear and non-separable multi-scale representations. We first show that most of the one-dimensional nonlinear multi-scale representations described in the literature are based on prediction operators which are the sum of a linear prediction operator and a perturbation defined using finite differences. We then extend this point of view to the multi-dimensional case where the scaling factor is replaced by a non-diagonal dilation matrix $M$. The new formalism we propose brings about similarities between existing nonlinear multi-scale representations and also enables us to alleviate the classical hypotheses made to prove the convergence of the multi-scale representations