# Multidimensional Analysis of Time-Resolved Charged Particle Imaging Experiments
## 1 Introduction  
A broad range of experimental scientific fields have moved over the last few years from the study of scalar or vectorial quantities (i.e., measurement of a magnitude or a string of data as a function of a relevant variable) to the acquisition of images (i.e., 2D sets of data). The broad availability of cheap charge-coupled devices (CCD), the larger storage capabilities and the faster communication protocols have contributed to this remarkable change. In particular, in the field of Atomic and Molecular Physics, traditional methods, such as time-of-flight detection of ions or photoelectrons, are increasingly being replaced by techniques like ion or photoelectron imaging, or even ion-photoelectron coincidence imaging, for the study of photoionization, photodissociation, reaction dynamics or molecular alignment [1; 2; 3; 4; 5; 6; 7; 8]. The acquisition of spatially-resolved data on the final position of these particles can yield a wealth of information of the process under study that was unimaginable with previous methods. In the analysis of these data, it has been common practice to analyze either integrated sections or cuts of the images to obtain 1D information that is then analyzed with standard 0D or 1D fitting procedures [9; 10; 11; 12]. In many cases, the richness of the information that can be extracted from the data is lost. Multidimensional analysis can separate certain contributions that are often the key to unravel the dynamical processes.  
In this work, we present a multidimensional fitting solution dedicated to obtain the best fit to the complete set of data contained in a time sequence of a sequence of velocity map images measured in femtosecond pump-probe experiments, through parameterized functionals that describeradial and angular properties of the particle distribution as a function of time. As will be shown, the multidimensional fit becomes essential for instance when contributions with different time behavior are overlapping. Additionally, for those cases where the initial guesses for the parameters or functional forms of the fit are misguided (on the number or nature of the contributions to the image, on the time behavior of anisotropy, etc...), discrepancies can be detected easily through the use of the analysis of the residuals. It is important to note that the multidimensional nature of the fit allows the discrimination of the different contributions to the images, in a manner that a reduced-dimensionality analysis cannot achieve. This method has revealed its value in a number of contributions where the need for discrimination of overlapping contributions in charged particle images was crucial [13; 14; 15; 16; 17; 18]. The method is analogous to the global 2D fit approach used in [PERSON]'s group [19; 20; 21; 22] but can easily be extended to extra dimensions such as the anisotropy, the temperature, the intensity dependence, measurements in coincidence, etc...With some modifications, the method can be equally applied to other problems, such as the detection of spectrally and spatially resolved X-rays from high-harmonic generation [23], or temporal-spectral-spatial ultrashort pulse characterization [24; 25; 26] and time-resolved time-of-flight measurements [27]. For each particular application, functional shapes and dependences have to be adapted, but the strategy presented here is of general applicability to the study of 2D or higher dimensionality data. We describe the procedure in a pedagogical way to adapt any algorithms of optimization to a multidimensional analysis.  
The paper is organized as follows. In Section 2, the method to construct velocity map images and to describe their time evolution is presented, including both 3D and 2D versions. The multidimensional numerical fitting procedure is explained in Section 3, and Section 4 is dedicated to the application of the method to a case example: the femtosecond pump-probe velocity map imaging (VMI) experiment on CH\({}_{3}\)I predissociation on the \(B\)-band. Section 5 closes the paper with the main conclusions.