The ill-posed problem of aerosol distribution determination from a small number of backscatter and extinction lidar measurements was solved successfully via a hybrid method by a variable dimension of projection with B-Splines. Numerical simulation results with noisy data at different measurement situations show that it is possible to derive a reconstruction of the aerosol distribution only with 4 measurements.

In this age, there are many applications of inverse problems to lots of areas ranging from astronomy, geoscience and so on. For example, image reconstruction and deblurring require the use of methods to solve inverse problems. Since the problems are subject to many factors and noise, we can't simply apply general inversion methods. Furthermore in the problems of interest, the number of unknown variables is huge, and some may depend nonlinearly on the data, such that we must solve nonlinear problems. It is quite different and significantly more challenging to solve nonlinear problems than linear inverse problems, and we need to use more sophisticated methods to solve these kinds of problems. / Master of Science / In various research areas, there are many required measurements which can't be observed due to physical and economical reasons. Instead, these unknown measurements can be recovered by known measurements. This phenomenon can be modeled and be solved by mathematics.

Planning is a central research area in artificial intelligence, and a lot of effort has gone into constructing more and more efficient planning algorithms. In real-world examples, many problem instances do not have a solution. Hence, there is an obvious need for methods that are capable of identifying unsolvable instances efficiently. It is not possible to efficiently identify all unsolvable instances due to the inherent high complexity of planning, but many unsolvable instances can be identified in polynomial time. We present a number of novel methods for doing this. We adapt the notion of k-consistency (a well-studied concept from constraint satisfaction) for testing unsolvability of planning instances. The idea is to decompose a given problem instance into a number of smaller instances which can be solved in polynomial time. If any of the smaller instances are unsolvable, then the original instance is unsolvable. If all the smaller instances are solvable, then it is possible to extract information which can be used to guide the search. For instance, we introduce the notion of forbidden state patterns that are partial states that must be avoided by any solution to the problem instance. This can be viewed as the opposite of pattern databases which give information about states which can lead to a solution.  We also introduce the notion of critical sets and show how to identify them. Critical sets describe operators or values which must be used or achieved in any solution. It is a variation on the landmark concept, i.e., operators or values which must be used in every solution. With the help of critical sets we can identify superfluous operators and values. These operators and values can be removed by preprocessing the problem instance to decrease planning time.

Bundle adjustment is the process of simultaneously optimizing camera poses and 3D structure given image point tracks. In structure-from-motion, it is typically used as the final refinement step due to the nonlinearity of the problem, meaning that it requires sufficiently good initialization. Contrary to this belief, recent literature showed that useful solutions can be obtained even from arbitrary initialization for fixed-rank matrix factorization problems, including bundle adjustment with affine cameras. This property of wide convergence basin of high quality optima is desirable for any nonlinear optimization algorithm since obtaining good initial values can often be non-trivial. The aim of this thesis is to find the key factor behind the success of these recent matrix factorization algorithms and explore the potential applicability of the findings to bundle adjustment, which is closely related to matrix factorization. The thesis begins by unifying a handful of matrix factorization algorithms and comparing similarities and differences between them. The theoretical analysis shows that the set of successful algorithms actually stems from the same root of the optimization method called variable projection (VarPro). The investigation then extends to address why VarPro outperforms the joint optimization technique, which is widely used in computer vision. This algorithmic comparison of these methods yields a larger unification, leading to a conclusion that VarPro benefits from an unequal trust region assumption between two matrix factors. The thesis then explores ways to incorporate VarPro to bundle adjustment problems using projective and perspective cameras. Unfortunately, the added nonlinearity causes a substantial decrease in the convergence basin of VarPro, and therefore a bootstrapping strategy is proposed to bypass this issue. Experimental results show that it is possible to yield feasible metric reconstructions and pose estimations from arbitrary initialization given relatively clean point tracks, taking one step towards initialization-free structure-from-motion.