In this talk we consider the identifiability of metabolic parameters from multi-compartment measurement data in quantitative positron emission tomography (PET) imaging, a non-invasive clinical technique that images the distribution of a radio tracer in-vivo.

We discuss how, for the frequently used two-tissue compartment model and under reasonable assumptions, it is possible to uniquely identify metabolic tissue parameters from standard PET measurements, without the need of additional concentration measurements from blood samples. The core assumption requirements for this result are that PET measurements are available at sufficiently many time points, and that the arterial tracer concentration is parametrized by a polyexponential, an approach that is commonly used in practice. Our analytic identifiability result, which holds in the idealized, noiseless scenario, indicates that costly concentration measurements from blood samples in quantitative PET imaging can be avoided in principle. The connection to noisy measurement data is made via a consistency result in Tikhonov regularization theory, showing that exact reconstruction is maintained in the vanishing noise limit.

We further present numerical experiments with a regularization approach based on the Iteratively Regularized Gauss-Newton Method (IRGNM) supporting these analytic results in an application example.

The random coefficients regression model $Y_i={\beta_0}_i+{\beta_1}_i {X_1}_i+{\beta_2}_i {X_2}_i+\ldots+{\beta_d}_i {X_d}_i$, with $\boldsymbol{X}_i$, $Y_i$, $\boldsymbol{\beta}_i$ i.i.d random variables, and $\boldsymbol{\beta}_i$ independent of $\boldsymbol{X}_i$ is often used to capture unobserved heterogeneity in a population. Reconstructing the joint density of random coefficients $\boldsymbol{\beta}_i=({\beta_0}_i,\ldots, {\beta_d}_i)$ implicitly involves the inversion of a Radon transformation. We propose a regularized maximum likelihood method with non-negativity and $\|\cdot\|_{L^1}=1$ constraint to estimate the density. We analyse the convergence of the method under general assumptions and illustrate the performance in a real data application and in simulations comparing it to the method of approximate inverse.

In this talk we study an adaptive two-step estimation method for statistical linear inverse problems with repeated measurements for smoothness classes expressed as $\alpha-$generalized random fields [1]. In a first step, the minimum fractional singularity order $\alpha$ is estimated, and in the second step the penalized least squares estimator with smoothness estimated in the first step is studied [2]. Rates of convergence for both the process smoothness and the penalized estimator are proven and illustrated through numerical simulations.

[1] M. D. Ruiz-Medina, J. M. Angulo, V. V. Anh. Fractional generalized random fields on bounded domains. Stochastic Anal. Appl. 21: 465--492, 2005.

[2] S. Golovkine, N. Klutchnikoff, V. Patilea. Learning the smoothness of noisy curves with application to online curve estimation. Electron. J. Stat. 16: 1485--1560, 2022.

Tomographic techniques have become a vital tool in medicine, allowing doctors to observe patients’ interior features. Modeling the measurement process (and the underlying physics) are projection operators, the most well-known one being the classical Radon transform. Identifying the range of projection operators has proven itself useful in various tomography-related applications [1-3], such as geometric calibration, motion detection, or more general projection model parameter identification. Projection operators feature the integration of density functions along certain curves (typically straight lines representing paths of radiation), and are subdivided into individual projections -- data obtained during a single step of the measurement process.

Mathematically, given a bounded open set $\Omega\subset \mathbb{R}^2$ and bounded open sets $R,T\subset \mathbb{R}$, a function $\gamma\colon R\times T \to \mathbb{R}^2$ that diffeomorphically covers $\Omega$ and a function $\rho \colon R\times T \to \mathbb{R}^+$, an individual projection is an operator $p\colon L^2(\Omega) \to L^2(R)$ with $$ [p{f}](r) = \int_{T} f\big(\gamma(r,t)\big) \rho\big(r,t\big) \,\mathrm{d}{t} \qquad \text{for all } r\in R $$ for $f \in \mathcal{C}^\infty_c(\Omega)$ (the unknown density). In other words, $r$ determines an integration curve $\gamma(r,\cdot)$, and $[pf](r)$ is the associated line integral weighted by $\rho$ (representing physical effects such as attenuation). Note that we do not allow projection truncation as $\Omega$ is covered by $\gamma$. More general projection operators are $P\colon L^2(\Omega)\to L^2(R_{1})\times \cdots \times L^2(R_{N})$ with $Pf=(p_{1}f,\dots,p_{N}f)$ with $N$ projections (with associated $\gamma_n,\rho_n,R_n,T_n$ for $n\in \{1,\dots,N\}$). In this work, we are concerned with characterizing the range of what we call projection pair operators, i.e., projection operators $P=(p_1,p_2)$ consisting of only two projections $(N=2)$. Conditions on the range of projection pair operators naturally impose properties on larger projection operators' ranges. These pairwise range conditions are particularly convenient for applications.

A natural approach for identifying the range is determining the range's orthogonal complement. The orthogonal complement being small would facilitate determining whether a projection pair is inside the range. We find that such normal vectors naturally consist of two functions $G_1$ and $G_2$ -- one per projection -- that need to satisfy $$ -\frac{\rho_{1}\big(\gamma_{1}^{-1 }(x)\big) \left |\det\left( \frac{\,\mathrm{d}{ { \gamma_{1}^{-1 }}}}{\,\mathrm{d}{ x} }(x)\right)\right|}{\rho_{2}\big(\gamma_{2}^{-1 }(x)\big) \left|\det\left( \frac{ \mathrm{d}{\gamma_{2}^{-1 }}}{\mathrm{d}{ x}}(x)\right)\right|} = \frac{G_2(r_2(x))}{G_1(r_1(x))} \qquad \text{for a.e. }x\in \Omega, $$ where $r_1(x)$ is such that $x\in \gamma_{1}(r_1(x),\cdot)$ and analogously for $r_2(x)$. This uniquely determines the orthogonal direction; therefore, the orthogonal complement's dimension is at most one. Thus, two projections' information can only overlap in a single way. Due to this equation's specific structure -- the right-hand side is a ratio of functions depending only on $r_1$ and $r_2$, respectively -- it is easy to imagine that this equation is not always solvable. While it is solvable for some standard examples like the conventional and exponential Radon transforms (whose ranges were already characterized [4,5]), we find that no solution exists for the exponential fanbeam transform and for the Radon transform with specific depth-effects. The fact that no solution exists implies that the operator's range is dense. Range conditions of this type can only precisely characterize the range when it is closed (otherwise, only the closure is characterized). In this regard, we find that the question of the range's closedness is equivalent for all projection pair operators whose $\gamma$ and $\rho$ functions are suitably related.

Acknowledgment: This work was supported by the ANR grant ANR-21-CE45-0026 `SPECT-Motion-eDCC'.

[1] F. Natterer. “Computerized Tomography with Unknown Sources”. SIAM Journal on Applied Mathematics 43.5,DOI : 10.1137/0143079:1201–1212 1983.

[2] J. Xu, K. Taguchi, B. Tsui. “Statistical Projection Completion in X-ray CT Using Consistency Conditions”. IEEE Trans. Med. Imaging 29, DOI : 10.1109/TMI.2010. 2048335: 1528–1540 2010.

[3] R. Clackdoyle, L. Desbat. “Data consistency conditions for truncated fanbeam and parallel projections.” Medical physics 42 2:831–45, 2015.

[4] V. Aguilar, P. Kuchment. “Range conditions for the multidimensional exponential X-ray transform”. Inverse Problems 11.5 ,DOI : 10.1088/0266-5611/11/5/002: 977, 1995.

[5] F. Natterer. The Mathematics of Computerized Tomography. Philadelphia: Society for Industrial and Applied Mathematics, Chap. II.4, 2001.