Gradient Descent Provably Solves Nonlinear Tomographic Reconstruction.

In computed tomography (CT), the forward model consists of a linear Radon transform followed by an exponential nonlinearity based on the attenuation of light according to the Beer-Lambert Law. Conventional reconstruction often involves inverting this nonlinearity as a preprocessing step and then solving a convex inverse problem. However, this nonlinear measurement preprocessing required to use the Radon transform is poorly conditioned in the vicinity of high-density materials, such as metal. This preprocessing makes CT reconstruction methods numerically sensitive and susceptible to artifacts near high-density regions. In this paper, we study a technique where the signal is directly reconstructed from raw measurements through the nonlinear forward model. Though this optimization is nonconvex, we show that gradient descent provably converges to the global optimum at a geometric rate, perfectly reconstructing the underlying signal with a near minimal number of random measurements. We also prove similar results in the under-determined setting where the number of measurements is significantly smaller than the dimension of the signal. This is achieved by enforcing prior structural information about the signal through constraints on the optimization variables. We illustrate the benefits of direct nonlinear CT reconstruction with cone-beam CT experiments on synthetic and real 3D volumes. We show that this approach reduces metal artifacts compared to a commercial reconstruction of a human skull with metal dental crowns.

DAB-quant: An open-source digital system for quantifying immunohistochemical staining with 3,3'-diaminobenzidine (DAB).

Here, we describe DAB-quant, a novel, open-source program designed to facilitate objective quantitation of immunohistochemical (IHC) signal in large numbers of tissue slides stained with 3,3'-diaminobenzidine (DAB). Scanned slides are arranged into separate folders for negative controls and test slides, respectively. Otsu's method is applied to the negative control slides to define a threshold distinguishing tissue from empty space, and all pixels deemed tissue are scored for normalized red minus blue (NRMB) color intensity. Next, a user-defined tolerance for error is applied to the negative control slides to set a NRMB threshold distinguishing stained from unstained tissue and this threshold is applied to calculate the fraction of stained tissue pixels on each test slide. Results are recorded in a spreadsheet and pseudocolor images are presented to document how each pixel was categorized. Slides can be analyzed in full, or sampled using small boxes scattered randomly and automatically across the tissue area. Quantitation of sampling boxes enables faster processing, reveals the degree of heterogeneity of signal, and enables exclusion of problem areas on a slide, if needed. This system should prove useful for a broad range of applications. The code, usage instructions, and sample data are freely and publicly available on GitHub (https://github.com/sarafridov/DAB-quant) and at protocols.io (dx.doi.org/10.17504/protocols.io.dm6gpb578lzp/v1).