4 Femtosecond Photodissociation Dynamics by Velocity Map Imaging
67
4.2.2 The Multidimensional Analysis
This section is devoted to details concerning image analysis for the particular case
of velocity map charged particle images. It is common that the analysis of such
data is carried out using methods that involve cuts or partial integrations through the
multidimensional data. As a consequence, in many instances, the full information
that can be extracted from the data is not totally and accurately recovered. A homemade procedure developed in our group [9] for the complete multidimensional fit
of this type of data will be described here. This procedure has proven to be crucial
for the extraction of all the relevant information from the images if, in addition,
the temporal dimension is included, as it is the case in time-resolved velocity map
imaging experiments. Some examples can be seen in [14, 15]. The key advantage of
the method consists of its capability to distinguish the different overlapped contributions present in the set of images corresponding to different reaction channels of
interest from secondary signals arising from other pathways.
Briefly, the method consists of an application of the well-known Levenberg–
Marquardt nonlinear regression method [16–18] to n-dimensional data, but adapted
to the particular case of velocity map images to find a balance between calculation
speed, accuracy, and human-guided input. The procedure assumes that each image
contains the sum of a number of “contributions” (related to each of the mechanisms
producing a given species with a certain speed distribution). Each contribution is parameterized as a function of all variables (radius and angle for each image, but also
time, for instance, to fit a time-dependent series of images) with a test functional
form with physical meaning using a sufficient number of adjustable parameters.
The first test functions are chosen guided by the known physical properties of the
system. The least-squares procedure is then applied to the complete data collection.
Inspection of the residuals (typically, also in image format), guides the choice of the
second set of functional forms. An iterative procedure of this kind allows the complete parameterization of the data, and from this, quantities such as decay times,
anisotropy parameters, etc. can be obtained for each contribution, with estimates of
error bars. For those cases where the initial guesses for the parameters or functional
forms 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. In addition, there is no conceptual problem to extend the fitting procedure to n dimensions,
the only limitation being computational time restrictions to analyze large quantities
of data. Once the procedure has yielded an analytical expression for the complete
set of data, the behavior of each “contribution” can be analyzed separately.
A typical image acquired in this type of experiments, either raw (through slice
imaging), or, equivalently, mathematically inverted (through velocity-map imaging),
contains, in general, a set of “contributions”, by which we mean each of the possible
processes or channels associated with a given type of charged particle (ion or photoelectron). Typically, a “channel” is characterized by a given kinetic energy, which,
67
4.2.2 The Multidimensional Analysis
This section is devoted to details concerning image analysis for the particular case
of velocity map charged particle images. It is common that the analysis of such
data is carried out using methods that involve cuts or partial integrations through the
multidimensional data. As a consequence, in many instances, the full information
that can be extracted from the data is not totally and accurately recovered. A homemade procedure developed in our group [9] for the complete multidimensional fit
of this type of data will be described here. This procedure has proven to be crucial
for the extraction of all the relevant information from the images if, in addition,
the temporal dimension is included, as it is the case in time-resolved velocity map
imaging experiments. Some examples can be seen in [14, 15]. The key advantage of
the method consists of its capability to distinguish the different overlapped contributions present in the set of images corresponding to different reaction channels of
interest from secondary signals arising from other pathways.
Briefly, the method consists of an application of the well-known Levenberg–
Marquardt nonlinear regression method [16–18] to n-dimensional data, but adapted
to the particular case of velocity map images to find a balance between calculation
speed, accuracy, and human-guided input. The procedure assumes that each image
contains the sum of a number of “contributions” (related to each of the mechanisms
producing a given species with a certain speed distribution). Each contribution is parameterized as a function of all variables (radius and angle for each image, but also
time, for instance, to fit a time-dependent series of images) with a test functional
form with physical meaning using a sufficient number of adjustable parameters.
The first test functions are chosen guided by the known physical properties of the
system. The least-squares procedure is then applied to the complete data collection.
Inspection of the residuals (typically, also in image format), guides the choice of the
second set of functional forms. An iterative procedure of this kind allows the complete parameterization of the data, and from this, quantities such as decay times,
anisotropy parameters, etc. can be obtained for each contribution, with estimates of
error bars. For those cases where the initial guesses for the parameters or functional
forms 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. In addition, there is no conceptual problem to extend the fitting procedure to n dimensions,
the only limitation being computational time restrictions to analyze large quantities
of data. Once the procedure has yielded an analytical expression for the complete
set of data, the behavior of each “contribution” can be analyzed separately.
A typical image acquired in this type of experiments, either raw (through slice
imaging), or, equivalently, mathematically inverted (through velocity-map imaging),
contains, in general, a set of “contributions”, by which we mean each of the possible
processes or channels associated with a given type of charged particle (ion or photoelectron). Typically, a “channel” is characterized by a given kinetic energy, which,