Erdem Altuntac

Erdem Altuntac

This work studies the usage of well-known smoothed total variation regularization for solving an atmospheric tomography problem named as {\em GPS-tomography} in some quasi-Newton methods. That is we solve an unconstrained, convex, smooth minimization problem associated with a general type Tikhonov functional containing smoothed type total variation penalty term by quasi-Newton methods. As a result of the conducted experiments, it is concluded that limited memory BFGS algorithm with trust region is the effective algorithm in terms obtaining a reasonably optimum solution.

Erdem Altuntac

We introduce a variational dictionary-based learning algorithm with hybrid penalization for single-cell microscopy signals. The cost functional couples a least-squares data fidelity term with total-variation (TV) regularization and a non-negativity constraint, promoting edge-preserving, physically meaningful reconstructions under heterogeneous backgrounds and imaging artifacts. We formulate the learning task with an explicit unitary (orthonormal) constraint on the dictionary operator, ensuring well-conditioned representations and predictable numerical behavior. The resulting optimization problem is solved by an alternating proximal-gradient scheme that combines smooth updates with closed-form proximal steps for non-smooth penalties. We prove that the PDHG iterates converge to the regularized minimizer under an explicit step-size condition ($τσ< 1/8$), and that when the true solution satisfies a variational source condition (VSC), the regularized solution converges to the true solution at the optimal $O(δ)$ rate under a noise-proportional regularization parameter choice $λ\propto δ$. Beyond reconstruction, we address the problem of multi-channel cell feature unification: given five imaging channels of the BSCCM dataset (DPC Left, Right, Top, Bottom, and Brightfield), we propose a \emph{deterministic} approach to synthesize a unified single-cell representation. Rather than probabilistic latent encodings, we pursue a joint dictionary learning framework in which all five channels share a common dictionary, and the sparse codes across channels are combined to form a channel-agnostic cell descriptor. This deterministic unification strategy is mathematically transparent, reproducible, and directly compatible with the clinical requirement that AI systems for diagnostics must be interpretable and auditable.

Erdem Altuntac

Primal-dual splitting involving proximity operators in order to be able to find some approximation to the minimizer for a general form of Tikhonov type functional is in the focus of this work. This approximation is produced by a pair of iterative variational regularization procedures. Under the assumption of some variational source condition (VSC), total error estimation both in the iterative sense and in the continuous sense has been analysed separately. Rates of convergence will be obtained in terms some concave and positive definite index function. Of the choice of the penalty term, we are interested in Bregman distance penalization associated with the non-smooth total variation (TV) functional. Furthermore, following up the lower and bounds defined for the regularization parameter, some deterministic choice of the regularization parameter is given explicitly. It is in the emphasis of this work that the regularization parameter obeys Morozov`s discrepancy principle (MDP) in order for the stability analysis of regularized solution. In the computerized environment, the algorithms are verified as iterative regularization methods by applying it to an atmospheric tomography problem named as GPS-Tomography. Apart from this 3-D tomographic inverse problem, we also apply the algorithms to some 2-D conventional tomographic image reconstruction problems in order to be able test algorithms` capability of capturing the details and observe that algorithms behave as iterative regularization procedures.

Erdem Altuntac

A general deterministic analysis to state the necessary conditions with a coefficient determination for the variational source condition to hold is provided. Of particular interest in terms of the choice of the regularization parameter, it is revealed that Morozov's discrepancy principle can be used both for determining new stable lower and upper bounds for the regularization parameter. With these bounds, it is also possible to establish quantitative estimations for the index function as well as for the different definitions of the Bregman distance. Inclusion of the variational source condition into the stability analysis enables one to re-establish convergence and convergence rate results in terms of the index function. The coefficient in the variational source condition is explicitly defined as a multivariable function of constants in Morozov's discrepancy principle. As expected, the results here are applicable when any strictly convex, smooth/non-smooth objective functional is considered.

Erdem Altuntac

The problem of minimization of the least squares functional with a smooth, lower semi-continuous, convex regularizer $J(\cdot)$ is considered to be solved. Over some compact and convex subset $Ω$ of the Hilbert space $\mathcal{H},$ the regularizer is implicitly defined as $ J(\cdot) : \mathcal{C}^{k}(Ω, \mathcal{H}) \rightarrow \mathbb{R}_{+}$ where $k \in \{1,2\}.$ So the cost functional associated with some given linear, compact and injective forward operator $\mathcal{T} :Ω\subset \mathcal{H} \rightarrow \mathcal{H},$ \begin{align} F_α(\cdot , f^δ) := \frac{1}{2} \Vert \mathcal{T}( \cdot ) - f^δ\Vert_{\mathcal{H}}^2 + αJ(\cdot) , \nonumber \end{align} where $f^δ$ is the given perturbed data with its perturbation amount $δ$ in it. Convergence of the regularized optimum solution $φ_{α(δ)} \in \mbox{argmin} F_α(φ, f^δ)$ to the true solution $φ^{\dagger}$ is analysed depending on the smoothness degree of the regularizer, \textit{i.e.} the cases $k \in \{1,2\}$ in $ J(\cdot) : \mathcal{C}^{k}(Ω, \mathcal{H}) \rightarrow \mathbb{R}_{+}.$ In both cases, we define such a regularization parameter that is in cooperation with the condition \begin{align} α(δ, f^δ) \in \{ α> 0 \mbox{ }\vert \mbox{ }\Vert\mathcal{T}φ_α^δ - f^δ\Vert \leq τδ\} , \nonumber \end{align} for some fixed $τ\geq 1.$ In the case of $k = 2,$ we are able to evaluate the discrepancy $\Vert\mathcal{T}φ_{α(δ)} - f^δ\Vert\leq τδ$ with the Hessian Lipschitz constant $L_H$ of the functional $F_α(\cdot , f^δ).$