Blind spectral unmixing is the problem of decomposing the spectrum of a mixed signal or image into a collection of source spectra and their corresponding activations indicating the proportion of each source present in the mixed spectrum. To perform this task, nonnegative matrix factorization (NMF) based on the $\beta$-divergence, referred to as $\beta$-NMF, is a standard and state-of-the art technique. Many NMF-based methods factorize a data matrix that is the result of a resolution trade-off between two adversarial dimensions. Two instrumental examples are (1)~audio spectral unmixing for which the frequency-by-time data matrix is computed with the short-time Fourier transform and is the result of a trade-off between the frequency resolution and the temporal resolution, and (2)~blind hyperspectral unmixing for which the wavelength-by-location data matrix is a trade-off between the number of wavelengths measured and the spatial resolution. In this paper, we propose a new NMF-based method, dubbed multi-resolution $\beta$-NMF (MR-$\beta$-NMF), to address this issue by fusing the information coming from multiple data with different resolutions in order to produce a factorization with high resolutions for all the dimensions. MR-$\beta$-NMF performs a form of nonnegative joint factorization based on the $\beta$-divergence. In order to solve this problem, we propose multiplicative updates based on a majorization-minimization algorithm. We show on numerical experiments that MR-$\beta$-NMF is able to obtain high resolutions in both dimensions for two applications: the joint-factorization of two audio spectrograms, and the hyperspectral and multispectral data fusion problem.

中文翻译：

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用于盲谱解混的多分辨率 Beta-Divergence NMF

盲光谱解混是将混合信号或图像的光谱分解为源光谱集合及其相应激活的问题，该集合指示混合光谱中存在的每个源的比例。为了执行此任务，基于 $\beta$-divergence 的非负矩阵分解 (NMF)，称为 $\beta$-NMF，是一种标准且最先进的技术。许多基于 NMF 的方法分解数据矩阵，该矩阵是两个对抗维度之间的分辨率权衡的结果。两个工具示例是 (1)~音频频谱解混，其中使用短时傅立叶变换计算频率-时间数据矩阵，并且是频率分辨率和时间分辨率之间权衡的结果，(2)~盲高光谱解混，其中波长-位置数据矩阵是测量的波长数量和空间分辨率之间的权衡。在本文中，我们提出了一种新的基于 NMF 的方法，称为多分辨率 $\beta$-NMF（MR-$\beta$-NMF），通过融合来自不同分辨率的多个数据的信息来解决这个问题。为了对所有维度产生高分辨率的分解。MR-$\beta$-NMF 基于 $\beta$-divergence 执行一种形式的非负联合分解。为了解决这个问题，我们提出了基于majorization-minimization算法的乘法更新。我们在数值实验中表明 MR-$\beta$-NMF 能够在两个应用中获得两个维度的高分辨率：两个音频频谱图的联合分解，