Much recent activity is aimed at reconstructing images from a few projections. Images in any application area are not random samples of all possible images, but have some common attributes. If these attributes are reflected in the smallness of an objective function, then the aim of satisfying the projections can be complemented with the aim of having a small objective value. One widely investigated objective function is total variation (TV), it leads to quite good reconstructions from a few mathematically ideal projections. However, when applied to measured projections that only approximate the mathematical ideal, TV-based reconstructions from a few projections may fail to recover important features in the original images. It has been suggested that this may be due to TV not being the appropriate objective function and that one should use the ℓ(1)-norm of the Haar transform instead. The investigation reported in this paper contradicts this. In experiments simulating computerized tomography (CT) data collection of the head, reconstructions whose Haar transform has a small ℓ(1)-norm are not more efficacious than reconstructions that have a small TV value. The search for an objective function that provides diagnostically efficacious reconstructions from a few CT projections remains open.

中文翻译：

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通过 Haar 变换的 ℓ1 最小化从一些投影中重建

最近的许多活动旨在从一些投影中重建图像。任何应用领域的图像都不是所有可能图像的随机样本，而是具有一些共同的属性。如果这些属性反映在目标函数的小值上，那么满足预测的目标可以与具有小的目标值的目标相补充。一种广泛研究的目标函数是全变分（TV），它可以从一些数学上理想的投影中得到相当好的重建结果。然而，当应用于仅近似数学理想的测量投影时，基于电视的一些投影重建可能无法恢复原始图像中的重要特征。有人建议，这可能是由于 TV 不是合适的目标函数，应该使用 Haar 变换的 ℓ(1)-范数。本文报道的调查与此相矛盾。在模拟头部计算机断层扫描 (CT) 数据采集的实验中，Haar 变换具有较小 ℓ(1) 范数的重建并不比具有较小 TV 值的重建更有效。寻找一种能够根据一些 CT 投影提供诊断有效的重建的目标函数仍然处于开放状态。