# Noise- and Outlier-Resistant Tomographic Reconstruction under Unknown Viewing Parameters

**Ritwick Chaudhry ^{1}^{†}, Arunabh Ghosh^{2}^{†} and Ajit Rajwade^{3}**

^{†}Equal Contribution

^{1}Adobe Research

^{2}Department of Electrical Engineering, Indian Institute of Technology Bombay

^{3}Department of Computer Science & Engineering, Indian Institute of Technology Bombay

**[Full Text]**,

**[Code]**

# Abstract

This paper presents an algorithm for effectively reconstructing an object from a set of its tomographic projections without any knowledge of the viewing directions or any prior structural information, in the presence of pathological amounts of noise, unknown shifts in the projections, and outliers among the projections. The outliers are mainly in the form of a number of projections of a completely different object, as compared to the object of interest. We introduce a novel approach of first processing the projections, then obtaining an initial estimate for the orientations and the shifts, and then define a refinement procedure to obtain the final reconstruction. Even in the presence of high noise variance (up to 50% of the average value of the (noiseless) projections) and presence of outliers, we are able to successfully reconstruct the object. We also provide interesting empirical comparisons of our method with the sparsity-based optimization procedures that have been used earlier for image reconstruction tasks.

# Results

The images used for our experiments were taken from the Yale and ORL face databases and the image sizes used were 192×192 and 112 × 112 respectively. A total of Q = 2 × 104 projections per image were simulated using angles from Uniform(0, π). A fraction of these projections were outliers of class 1, i.e. they were projections of non-face images taken from the CIFAR-10 dataset. For another fraction f2 of projections, we deliberately generated them from a copy of the same image, but with a small number of pixel values (at randomly selected locations) set to 0. We term the corresponding projections ‘Outliers of Class 2’. These simulate projections of biological specimens corrupted by overlapping ice particles or minor structural changes.

Below are some of the reconstructions produced by our algorithm -

*0% Noise and no outliers of class 1 or class 2, RMSE - 4.58%*

*10% Noise, 10% outliers of class 1, 10% outliers ofclass 2, RMSE - 8.95%*

*50% Noise, 0% outliers of class 1, 0% outliers of class 2, RMSE - 11.99%*

*50% Noise, 5% outliers of class 1, 5% outliers of class 2, RMSE - 18.39%*

*20% Noise, 10% outliers of class 1, 0% outliers of class 2, Non-Uniform distribution of angles, RMSE - 17.69%*