Due to the many applications in Magnetic Resonance Imaging (MRI), Nuclear Magnetic Resonance (NMR), radio interferometry, helium atom scattering etc., the theory of compressed sensing with Fourier transform measurements has reached a mature level. However, for binary measurements via the Walsh transform, the theory has long been merely non-existent, despite the large number of applications such as fluorescence microscopy, single pixel cameras, lensless cameras, compressive holography, laser-based failure-analysis etc. Binary measurements are a mainstay in signal and image processing and can be modelled by the Walsh transform and Walsh series that are binary cousins of the respective Fourier counterparts. We help bridging the theoretical gap by providing non-uniform recovery guarantees for infinite-dimensional compressed sensing with Walsh samples and wavelet reconstruction. The theoretical results demonstrate that compressed sensing with Walsh samples, as long as the sampling strategy is highly structured and follows the structured sparsity of the signal, is as effective as in the Fourier case. However, there is a fundamental difference in the asymptotic results when the smoothness and vanishing moments of the wavelet increase. In the Fourier case, this changes the optimal sampling patterns, whereas this is not the case in the Walsh setting.

由于在磁共振成像（MRI），核磁共振（NMR），无线电干涉仪，氦原子散射等方面的许多应用，利用傅立叶变换测量进行压缩感测的理论已经达到了成熟的水平。但是，对于通过Walsh变换进行二进制测量，尽管有大量应用，例如荧光显微镜，单像素相机，无透镜相机，压缩全息术，基于激光的故障分析等，但该理论长期以来一直不存在。测量是信号和图像处理的中流tay柱，可以通过Walsh变换和Walsh序列建模，它们是相应Fourier副本的二元表亲。通过为Walsh样本和小波重构的无限维压缩感测提供非均匀的恢复保证，我们有助于弥合理论差距。理论结果表明，只要采样策略高度结构化并且遵循信号的结构稀疏性，使用Walsh采样进行压缩感测与傅立叶情况一样有效。但是，当小波的平滑度和消失矩增加时，渐近结果存在根本差异。在傅立叶情况下，这会更改最佳采样模式，而在沃尔什设置中则不是这种情况。只要采样策略是高度结构化的并且遵循信号的结构化稀疏性，就与傅立叶情况一样有效。但是，当小波的平滑度和消失力矩增加时，渐近结果存在根本差异。在傅立叶情况下，这会更改最佳采样模式，而在沃尔什设置中则不是这种情况。只要采样策略是高度结构化的并且遵循信号的结构化稀疏性，就与傅立叶情况一样有效。但是，当小波的平滑度和消失力矩增加时，渐近结果存在根本差异。在傅立叶情况下，这会更改最佳采样模式，而在沃尔什设置中则不是这种情况。