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Register a new userThe Signed Cumulative Distribution Transform for 1-D Signal Analysis and Classification
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Sumati Thareja
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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This paper presents a new mathematical signal transform that is especially suitable for decoding information related to non-rigid signal displacements. We provide a measure theoretic framework to extend the existing Cumulative Distribution Transform [ACHA 45 (2018), no. 3, 616-641] to arbitrary (signed) signals on $overline{mathbb{R}}$. We present both forward (analysis) and inverse (synthesis) formulas for the transform, and describe several of its properties including translation, scaling, convexity, linear separability and others. Finally, we describe a metric in transform space, and demonstrate the application of the transform in classifying (detecting) signals under random displacements.
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We present a new supervised image classification method applicable to a broad class of image deformation models. The method makes use of the previously described Radon Cumulative Distribution Transform (R-CDT) for image data, whose mathematical properties are exploited to express the image data in a form that is more suitable for machine learning. While certain operations such as translation, scaling, and higher-order transformations are challenging to model in native image space, we show the R-CDT can capture some of these variations and thus render the associated image classification problems easier to solve. The method -- utilizing a nearest-subspace algorithm in R-CDT space -- is simple to implement, non-iterative, has no hyper-parameters to tune, is computationally efficient, label efficient, and provides competitive accuracies to state-of-the-art neural networks for many types of classification problems. In addition to the test accuracy performances, we show improvements (with respect to neural network-based methods) in terms of computational efficiency (it can be implemented without the use of GPUs), number of training samples needed for training, as well as out-of-distribution generalization. The Python code for reproducing our results is available at https://github.com/rohdelab/rcdt_ns_classifier.
Invertible image representation methods (transforms) are routinely employed as low-level image processing operations based on which feature extraction and recognition algorithms are developed. Most transforms in current use (e.g. Fourier, Wavelet, etc.) are linear transforms, and, by themselves, are unable to substantially simplify the representation of image classes for classification. Here we describe a nonlinear, invertible, low-level image processing transform based on combining the well known Radon transform for image data, and the 1D Cumulative Distribution Transform proposed earlier. We describe a few of the properties of this new transform, and with both theoretical and experimental results show that it can often render certain problems linearly separable in transform space.
Electroencephalogram (EEG) has shown a useful approach to produce a brain-computer interface (BCI). One-dimensional (1-D) EEG signal is yet easily disturbed by certain artifacts (a.k.a. noise) due to the high temporal resolution. Thus, it is crucial to remove the noise in received EEG signal. Recently, deep learning-based EEG signal denoising approaches have achieved impressive performance compared with traditional ones. It is well known that the characteristics of self-similarity (including non-local and local ones) of data (e.g., natural images and time-domain signals) are widely leveraged for denoising. However, existing deep learning-based EEG signal denoising methods ignore either the non-local self-similarity (e.g., 1-D convolutional neural network) or local one (e.g., fully connected network and recurrent neural network). To address this issue, we propose a novel 1-D EEG signal denoising network with 2-D transformer, namely EEGDnet. Specifically, we comprehensively take into account the non-local and local self-similarity of EEG signal through the transformer module. By fusing non-local self-similarity in self-attention blocks and local self-similarity in feed forward blocks, the negative impact caused by noises and outliers can be reduced significantly. Extensive experiments show that, compared with other state-of-the-art models, EEGDnet achieves much better performance in terms of both quantitative and qualitative metrics.
Convolutional neural networks (CNNs) have achieved state-of-the-art results on many visual recognition tasks. However, current CNN models still exhibit a poor ability to be invariant to spatial transformations of images. Intuitively, with sufficient layers and parameters, hierarchical combinations of convolution (matrix multiplication and non-linear activation) and pooling operations should be able to learn a robust mapping from transformed input images to transform-invariant representations. In this paper, we propose randomly transforming (rotation, scale, and translation) feature maps of CNNs during the training stage. This prevents complex dependencies of specific rotation, scale, and translation levels of training images in CNN models. Rather, each convolutional kernel learns to detect a feature that is generally helpful for producing the transform-invariant answer given the combinatorially large variety of transform levels of its input feature maps. In this way, we do not require any extra training supervision or modification to the optimization process and training images. We show that random transformation provides significant improvements of CNNs on many benchmark tasks, including small-scale image recognition, large-scale image recognition, and image retrieval. The code is available at https://github.com/jasonustc/caffe-multigpu/tree/TICNN.
Approximate message passing (AMP) is an efficient iterative signal recovery algorithm for compressed sensing (CS). For sensing matrices with independent and identically distributed (i.i.d.) Gaussian entries, the behavior of AMP can be asymptotically described by a scaler recursion called state evolution. Orthogonal AMP (OAMP) is a variant of AMP that imposes a divergence-free constraint on the denoiser. In this paper, we extend OAMP to incorporate generic denoisers, hence the name D-OAMP. Our numerical results show that state evolution predicts the performance of D-OAMP well for generic denoisers when i.i.d. Gaussian or partial orthogonal sensing matrices are involved. We compare the performances of denosing-AMP (D-AMP) and D-OAMP for recovering natural images from CS measurements. Simulation results show that D-OAMP outperforms D-AMP in both convergence speed and recovery accuracy for partial orthogonal sensing matrices.
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