Statistical analysis of ICA decomposition results. The signal components for the ICA routine
1.1. Anatomical connectivity mapping
“Why should we bother about connectivity in the age of functional imaging, at a time when magnets of ever increasing strength promise to detect the location of even the faintest thought? Isn’t it enough to locate cortical areas engaged in deception, introspection, empathy? Do we really have to worry about their connections? The answer is “yes”. In the case of the nervous system, the unit of relational architecture that allows the whole to exceed the sum of the parts is known as large-scale network. Its elucidation requires an elaborate understanding of connectivity patterns” . Despite considerable advances in experimental techniques and in our understanding of animal anatomy over the last decades, the real connectivity of the human brain has essentially remained a mystery. It is the human brain’s multiscale topology that poses a particular challenge to any neuroimaging technique and prevented the neuroscientists from unraveling the connectome so far.
However, it is also the brain’s architecture that allows different morphological entities to be defined at different scales depending on the spatial resolution provided by the available neuroimaging techniques and the scientific objectives. Consequently, a comprehensive description of neuronal networks and their intricate fiber connections requires a multimodal approach based on complementary imaging techniques targeting different levels of organization (microscale, mesoscale, and macroscale) [2,3].
MR-based diffusion imaging is the most frequently used method to visualize fiber pathways in both the living and the postmortem human brain (for a comprehensive introduction to the field cf. [4,5]). Diffusion imaging contributes to the understanding of the macroscopic connectivity (i.e., at the millimeter scale) in the living brain, while postmortem studies already explore the upper mesoscale (i.e., the sub-millimeter scale). Hence, not surprisingly, diffusion imaging is of great appeal to neuroscientists as a method for the visualization of connectivity patterns in both clinical and basic research. However, complex fiber networks and small fiber tracts are difficult to be disentangled reliably at present. Furthermore, the termination of fields of pathways emanating from cortical areas no larger than a few millimeters in size cannot be demonstrated with the required precision.
Conversely, microscopic techniques generate data sets of impressing neuroanatomical detail, but they are limited to small sample sizes (i.e., small areas of interest in a small number of subjects) of postmortem tissue. This substantially restricts their predictive power. In the recent years, anatomical connections in the human postmortem brains were studied with dissection techniques [6,7], in myelin stained sections of adult human brains , or of immature brains taking advantage of heterochronic myelination of different fiber tracts during pre- and early postnatal development , in lesioned brains using various techniques for staining degenerating fibers [10,11], and using tract-tracing methods for discovering local connections [12,13]. These studies have enriched our knowledge about human brain fiber tracts, but all of them suffer from severe restrictions if the 3D courses of fiber pathways are to be mapped in the adult human brain. In contrast to studies in animals, the tight packing of different fiber tracts in the white substance, and the lack of specific tracers for in vitro tracking of long distance fibers made comprehensive fiber tract mapping impossible in the adult human brain .
1.2. 3D-polarized light imaging (3D-PLI)
Recently, a neuroimaging technique referred to as
The birefringence of brain tissue is measured by passing linearly polarized light through histological brain sections and by detecting local changes in the polarization state of light using a polarimeter setup (Figure 1a-b). The polarimeter is equipped with a pair of crossed polarizers, a tilting specimen stage, a quarter-wave retarder, an LED light source (with a narrow-band green wavelength spectrum), and a charge-coupled device (CCD) camera. By rotating the optical devices simultaneously around the stationary brain section and by imaging the sample at discrete rotation angles
The amplitude of the profile quantifies the phase retardation
For estimating the sinusoidal profile of the 3D-PLI signal Discrete Fourier Analysis (DFA) can be used to deduce
1.3. Relevance and restoration of the light intensity profile
In 3D-PLI, the light intensity profile (cf. Figure 1) represents a crucial data set in terms of fiber orientation determination, since peak position and signal amplitude of the profile are directly related to the in-section direction and out-of-section inclination of the fiber orientation, respectively. A precise and undisturbed recording of the light intensity passing through a microtome section is therefore mandatory for the reliable reconstruction of high-resolution fiber tracts. The signal quality is however influenced by several conditions. Thermal effects and electrical noise in both the light source and the CCD-electronics deteriorate the PLI signal characteristics at each image pixel. Filter inhomogeneities of the polarizers or retarder may also manipulate the intensity profile. Therefore, a standardized image calibration technique using flat fields is usually applied to all raw images prior to analysis to compensate pixel-wise for inhomogeneities across the field-of-view . Depending on the section thickness the pixel-by-pixel intensity is also influenced by the scatter properties of the investigated object. Another possible source of artifacts is dust on the polarizer. Although the polarimeter should be operated in a shielded construction to prevent for external light and dust particles, dust cannot completely be avoided. As a result of the rotation of the polarizer dust particles will deteriorate the measured light intensity only once within a half circle, if and only if they are located on the rotating system (cf. Figure 1). Hence, the measured intensity at the CCD array is a linear mixture of different light sources.
Recently, signal enhancement and restoration techniques for PLI images utilizing independent component analysis (ICA) have been introduced [22,23]. As a result, component maps corresponding to gray and white matter structures as well as noise and artifacts can be identified automatically using statistical analysis tools . Remarkably, even in the presence of dust on the polarimeter ICA can effectively restore the original sinusoidal signal (Figure 2). After ICA filtering the noise and artifacts are removed and the sinusoidal nature of the PLI signal is restored (Figure 2).
2. A new concept for optimal signal decomposition in polarized light imaging
Although blind source separation methods are successfully applied for signal separation in all kinds of neuroimaging modalities [24–29] two major problems remain: i) the method applied must be selected carefully and the separation strategy (i.e., the internal cost function) of the applied method should be optimal for the type of data. This however, is one of the reasons why many different ICA algorithms co-exist. ii) Assuming the measured signals are adequately separated into the underlying source signals (i.e., signal of interest and non-interest), identification of the signal of interest should be performed user independently; preferably in an automatic fashion. For the latter, significant effort has to be invested in order to identify components of interest automatically from data recorded utilizing different neuroimaging techniques [29–33]. In contrast to many other ICA applications, the great advantage in PLI signal decomposition is that all profiles of the basis vectors that correspond to the signal of interest must show a sinusoidal waveform [16,22]. Since these waveforms can only vary in amplitude and phase, an automatic extraction of the signal components can be achieved utilizing this property as demonstrated in .
Alternatively, it has been demonstrated that by incorporating prior knowledge, i.e., by imposing temporal or spatial constraints for the source separation task, decomposition can be effectively improved [34–37]. Hesse and James , for example, showed that using different types of spatial constraints ICA can be trained to
2.1. A short introduction to independent component analysis (ICA)
For 3D-PLI a linear superposition of light at the CCD camera is assumed, where each elementary signal component refers to a distinct region in space. By applying independent component analysis (ICA) to a set of polarized light images (here a stack of 18 images at different rotation angles is used) the decomposition of the data results in spatially independent components (often called feature or basis vectors) yielding maximally (i.e., statistically) independent spatial component maps.
Let be the
Within ICA the
Signal restoration, i.e. the cleaning process, is performed by zeroing columns in which reflect signal contributions from unwanted (e.g. artifact) sources. This is identical to zeroing rows in
2.2. Constrained ICA for polarized light imaging (
Signal separation in constrained ICA is based on incorporating prior knowledge about the underlying signals. In the study of Barriga and colleagues ,
In Equation (6)
where refers to the
where is the expected value of the deviation function of the
The basic idea here is that a very small
For the weight update described in Equation (10) the choice of the confidence value
2.2.1. Evaluation of the signal enhancement
For measuring the noise reduction and signal enhancement in a set of 3D-PLI data after ICA application we use the weighted reduced chi-squared statistic as introduced in . The reduced chi-squared statistic χ2 involves the variance σ2 of the observation, where the statistical output is weighted based on the measurement error
Here, denote the intensity measured at angle
The weight factor ω increases when the squared difference between the two expectation functions and (derived from the raw and the ICA filtered PLI data sets, respectively) is large. In case of missed components of interest, the signal strength of the ICA filtered 3D-PLI signal would be largely reduced at corresponding pixel locations. Consequently the expectation function would differ in terms of its amplitude, while the shape of the waveform may be still sinusoidal. The assumption here is that the signal power (across the rotation angles) of the signal of interest is larger compared to noise. The restriction of ω ≥ 1 is needed in order to not artificially improve the goodness-of-fit in cases where the two expectation functions and are very similar (e.g., at pixel locations with low noise).
2.2.2. Finding the optimal parameters
The optimal parameters
In Figure 4 the progress of changes in the sinusoidal profile of one exemplary basis vector is shown for all 18 iterations (blue) within
The process of updating the weight matrix and how
3. Signal enhancement in 3D-PLI data sets
3.1. Signal acquisition and preprocessing
The performance of
The sections were digitized using the polarimeter setup described in section 1.2. Each brain section was imaged at 18 equidistant rotation angles of the polarimeter covering an angle range between 0° and 170° (Figure 1b). The acquired RGB-colored images have a size of 2776 × 2080 pixels with a pixel size of 64 µm × 64 µm. The intensities were sampled with a dynamic range of 14 bits per color channel. Since the light source of the polarimeter is composed of light-emitting diodes (LED) emitting a narrow-band green wavelength spectrum (central wavelength of 525 nm), only the green channel from the RGB color triplet was used for further analysis.
Before the images were decomposed by ICA, a standardized calibration technique using flat fields was applied, in order to compensate pixel-wise for inhomogeneities across the field of view . In addition, the separation of brain tissue from the non-relevant image background was done by means of the interactive learning and segmentation toolkit (ilastik) . Thus, the following calculations and statistical analyses were solely done on basis of brain tissue measurements.
3.2. Performance on signal decomposition using
To test the performance of the signal decomposition and signal restoration
After the determination of the parameters
In general both ICA algorithms produced remarkable decomposition results, where signal enhancement (
|wrGOF < 1 (%)||3.57 %||0.35 %|
|wrGOF ≥ 10 (%)||38.8 %||80.0 %|
|wrGOF ≥ 100 (%)||15.1 %||55.6 %|
In Figure 6a pixel locations of a representative section is shown, where the test statistic (
With the introduction of the weighting factor (or penalty factor) as expressed by Equation (12), the test statistic
In recent years 3D-polarized light imaging (3D-PLI) has been shown to provide new insights into the organization of the human brain including mapping of the fiber anatomy [15,45,46]. Through advances in the experimental setup of the employed polarimeter, as well as in signal processing, this modality provides unique data sets to explore the 3D fiber architecture in the human brain at a submillimeter resolution [16,47,48]. Though the technique as described here is applicable solely to postmortem brain tissue, the comprehensive description of complex fiber orientations in distinct brain regions (e.g., prevalent fiber crossings) can be used to guide and evaluate fiber tractography algorithms based on diffusion MRI. By this means the fiber orientation maps provided by 3D-PLI might help to optimize the reliability of in-vivo diffusion MRI results. Precise information about the local individual fiber architecture of a patient is of particular interest in case of planning and performing a neurosurgical intervention, for instance.
However, in 3D-PLI the reconstruction of nerve fiber pathways strongly depends on the quality of the measured intensities represented by the so-called light intensity profiles or 3D-PLI signals, respectively. Hence, advanced signal processing tools are required to enable the precise determination of locally prevailing fiber orientations in form of unit vectors defined by the in-section direction angle
Independent component analysis (ICA) turned out to improve 3D-PLI signals significantly [22,23]. It was shown that ICA is capable of restoring the original birefringent signal by effectively removing noise and artifact components in the measured data. In addition, measures for the qualitative and quantitative evaluation of the 3D-PLI signals before and after the ICA filtering were introduced. In particular, the signal enhancement after ICA based denoising is large at the white to gray matter boundary, where the 3D-PLI signal is weak due to decreasing fiber density when approaching gray matter domains .
With the introduction of
The dedicated signal processing tool