Error analysis for Mueller matrix measurement.

The skin is the biggest human organ. It is also easily accessible and visible from the outside. Diagnosis of skin diseases such as melanoma skin cancer is often based on optical parameters, e.g. the widely known ABCDE rule of dermatoscopy. Not only cancer, but also inflammatory skin diseases are rated according to criteria which are at least partially based on optical criteria (see PASI- or EASI-scores) and often include subjective criteria. The standard instrument for an optical examination of the skin is the so-called dermatoscope. It is an epiluminescence microscope which nowadays often provides digital images of the suspected lesion. Due to the design as a contact-based device, most dermatoscopes face problems like suppression of perfusion, among others. That is why we developed a compact non-contact dermatoscope. To obtain further information about, for example, the tissue composition, an additional Mueller matrix (MM) measurement system was developed which can acquire the 4×4 MM spatially resolved for a 2D-image in a comparably short time. The MM (M <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sub> ) contains all information about the polarization changing properties of a sample. It allows to calculate the Stokes vector of the outcoming light (S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> o) for every state of incoming light (S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> i) by applying S <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> o = Mm ·s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">→</sup> i.. The concept of measuring the MM is well known in the literature [1,2]. In our approach, we investigate the potential of a MM measuring system as an extension to a non-contact dermatoscope. The ultimate goal is to realize a diagnostic camera, which can take overview and zoomed images under comparable conditions for each examination and provides additional diagnostic information by measuring the MM of the area of interest. This information is particularly interesting for inflammatory skin diseases like Morphea where contribution of the chiral molecule collagen is important. To validate the potential of the MM measurement system, samples with known chirality like sucrose where studied. For this, cuvettes of poly(methyl methacrylate) (PMMA) and polystyrene (PS) filled with different concentrations of sucrose solved in pure water were utilized. The MM reliably differentiates between different concentrations of sucrose in the water. Also, as Fig. 1 shows, the system was able to quantify the slight changes in the polarization probably induced by stress birefringence which is material dependent.

A decomposition for Mueller matrices into three physically descriptive components was recently developed by Shih-Yau Lu. The effect of experimental error on this decomposition was studied. Both analytical and numerical methods were employed. Symbolic expression of the component matrices in terms of the original Mueller matrix elements shows how errors in the original matrix propagate through the decomposition. Complete symbolic decomposition was given for non-depolarizing Mueller matrices and their associated physical parameters; however, the depolarizing case produced unmanageably large expressions, so approximations were used. For the numerical results, Mathcad was used to randomly generate Mueller matrices, incorporate matrices is proportional to the original error within the measured Mueller matrix, and that the proportional constant increases with each subsequent step in the decomposition. In addition, Cloude's method for eliminating 'noise' in a Mueller matrix was employed, and its effect on error distribution was analyzed.

For the measurement of Mueller matrix in an optical system with birefringence and small polarization-dependent loss or gain (PDL/G), we theoretically derive the statistical relationship between the Mueller matrix measurement error and three input states of polarization (SOP). Based on this theoretical relation and simulation results, it can be concluded that the three input SOPs, that are coplanar with an angle of 120 degrees between any two of them in Stokes space, can be considered as a substitute for the best input SOPs which can statistically lead to the minimum measurement error. This conclusion is valid when the PDL/G of the optical system under test is less than 0.35 dB.

Digital staining based on Mueller matrix measurements and their derivatives was investigated. Mueller matrix imaging was performed at the microscopic level on gastric tissue sections. Full Mueller matrices (4 × 4) were reconstructed using recorded images, followed by the extraction of polarization parameters. The most effective parameters and their combinations were extracted from Mueller matrix elements, principal component scores and polarization parameters respectively to classify samples into three categories - i.e. cancer, dysplasia and intestinal metaplasia/normal glands for various regions of interest sizes. It was observed that two-step classification yielded higher classification accuracy than the traditional one-step classification and that pixel classification based on Mueller matrix elements yielded higher accuracy than that based on polarization parameters and derived principal components. Moreover, Mueller matrix images with a lower spatial resolution generated higher classification accuracy but those with a higher spatial resolution revealed more morphological details.ns. The original stained image (top) and the digital staining image (bottom).

High-speed optical polarization characterization is highly desirable for a wide range of applications, including remote sensing, telecommunication, and medical diagnosis. The utilization of the Mueller matrix provides a superior systematic and comprehensive approach to represent polarization attributes when matter interacts with optical beams. However, the current measurement speed of Mueller matrix is limited to only seconds or milliseconds. In this study, we present an ultrafast Mueller matrix polarimetry (MMP) technique based on optical time-stretch and spectral encoding that enables us to achieve an impressive temporal resolution of 4.83 nanoseconds for accurate Mueller matrix measurements. The unique feature of optical time-stretch technology enables continuous, ultrafast single-shot spectroscopy, resulting in a remarkable speed of up to 207 MHz for spectral encoding Mueller matrix measurement. We have employed an effective Mueller linear reconstruction algorithm based on the measured modulation matrix, accounting for all potential non-ideal effects of polarization components like retardance error and azimuth error. To ensure high precision, prior to the actual measurement, high-order dispersion induced by time-stretch requires adjustment through proper modulation matrix design. Upon such correction, both the results of static and rapid dynamic samples measurements exhibit exceptional accuracy with root-mean-square error (RMSE) approximately equal to 0.04 and 0.07 respectively. This presented ultrafast MMP provides a significant advance over preceding endeavors, enabling superior accuracy and increased speed concurrently.

Error matrices in gas-electron diffraction: II. Influence of weight matrix

The review contains a systematization of the main approaches to the practical implementation of Mueller matrix polarimetry and the prospects for its application in biology and agriculture. The most typical optical layouts for measuring the Mueller matrix of various objects, such as disperse systems, tissues and surface structures, are discussed. Mueller matrix measurements, being integrated into standard schemes of conventional optical methods, such as scatterometry, optical coherence tomography, fluorimetry, spectrophotometry and reflectometry, can significantly expand their capabilities in the characterization of biological systems and bioorganic materials. Additionally, microwave Mueller matrix polarimetry can be used for monitoring soil conditions and crop growth. The proposed systematization is aimed at outlining the conceptual directions for the development of non-invasive diagnostic tools based on measuring the Mueller matrix, primarily with a focus on biological research and agricultural practice.

In order to control nanoimprint lithography processes to achieve good fidelity, accurate characterization of structural parameters of nanoimprinted resist patterns is highly desirable. Among the possible techniques, optical scatterometry is relatively ideal due to its high throughput, low cost, and minimal sample damage. Compared with conventional optical scatterometry, which is usually based on reflectometry and ellipsometry and obtains at most two ellipsometric angles, Mueller matrix ellipsometry (MME) based scatterometry can provide up to 16 quantities of a 4 × 4 Mueller matrix in each measurement and can thereby acquire much more useful information about the sample. In addition, MME has different measurement accuracy in different measurement configurations. It is expected that much more accurate characterization of nanoimprinted resist patterns can be achieved by choosing appropriate measurement configurations and fully using the rich information hidden in the measured Mueller matrices. Accordingly, nanoimprinted resist patterns were characterized using an in-house developed Mueller matrix ellipsometer in this work. We have experimentally demonstrated that not only more accurate quantification of line width, line height, sidewall angle, and residual layer thickness of nanoimprinted resist patterns can be achieved, but also the residual layer thickness variation over the illumination spot can be directly determined, when performing MME measurements in the optimal configuration and meanwhile incorporating depolarization effects into the optical model. The comparison of MME-extracted imprinted resist profiles has also indicated excellent imprint pattern fidelity.

The analysis of critical dimensions of nanostructured areas should ideally give both their size with nanometer resolution and the size distribution over a large area, two contradicting goals for most microscopic techniques. Here, we demonstrate that Mueller-matrix metrology is a fast and nondestructive technique able to identify deviations, as small as 1 to 2 nm, from the desired shape and the statistical distribution of sizes on a potentially unlimited area. Tiny shape deviations are accessible from Mueller-matrix measurements over a complete azimuthal rotation after the subsequent differential decomposition of the matrices and elucidated from small anisotropies. The size distribution is hidden in the depolarization and is extracted by comparing the simulated and measured sum-decomposed Mueller matrices. This study especially demonstrates that depolarization concealed in measured Mueller matrices contains valuable information about the sample.

Mueller matrix-based optimization of reflective type twisted nematic liquid crystal SLM at oblique incidences

Mueller matrix polarimetry (MMP) provides valuable structural insights into tissue and holds promise for medical diagnostics. However, its clinical adoption is hindered by labor-intensive data collection and annotation. This study examines the use of MMP data collected in reflection from ex vivo human brain tissue to identify neoplastic regions. Using a custom-built single-wavelength MMP imaging system, we compare deep learning models trained on Mueller matrix measurements against Lu-Chipman feature maps. Our networks achieve segmentation accuracy comparable to multi-spectral polarimetry, highlighting the potential of real-time MMP for brain tumor differentiation. We further provide a qualitative analysis discussing challenges and opportunities for neurosurgical MMP applications.

A double-beam polarization-sensitive system based on optical coherence tomography was built to measure the Mueller matrix of scattering biological tissue with high spatial resolution. The Jones matrix of a sample can be determined with a single scan and subsequently converted into an equivalent nondepolarizing Mueller matrix. As a result, the system can be used to measure the Mueller matrix of an unstable sample, such as soft tissue. The polarization parameters of a porcine tendon, including magnitude and orientation of birefringence and diattenuation, were extracted by decomposition of the measured Mueller matrix.

Recently, with the emergence of new light sources, polarization devices, and detectors, together with a prominent increase in data processing capability, polarization techniques find more and more applications in various areas, one of which is biomedicine. For probing the characteristic features of complex biomedical specimen, Mueller matrix polarimetry has demonstrated distinctive advantages. Mueller matrix polarimetry can be achieved on other optical techniques by adding the polarization state generator and analyzer to their existing optical paths appropriately. Common biomedical optical equipment, such as microscopes and endoscopes, can be upgraded to fulfill Mueller matrix imaging and measurement abilities. Compared with traditional non-polarization optical methods, Mueller matrix polarimetry can provide far more information to characterize the samples, including the anisotropic optical properties, such as birefringence and diattenuation, as well as the distinctive features of various scattering particles and microstructures. Also, Mueller matrix polarimetry is more sensitive to scattering by sub-wavelength microstructures. As a label-free and non-invasive tool, Mueller matrix polarimetry has broad application prospects in biomedical studies and clinical diagnosis. In this review, we provide an introduction to the Mueller matrix methodology, including the Stokes-Mueller formalism, and also the decomposition and transformation methods to derive new parameters. We also summarize the status of the Mueller matrix polarimetric field, including recent improvements, both in instrumentation and data analysis. The current and future applications of Mueller matrix polarimetry in biomedicine are provided and discussed.

We show that the antisymmetric Mueller generator provides a universal algebraic kernel for geometric phase in classical polarization optics and in quantum two-level systems. For any ideal retarder, the antisymmetric 3×3 block of its Mueller matrix (the antisymmetric generator of the adjoint SU(2) action on the Stokes vector) encodes the angular-velocity vector that drives the tangential motion on the Poincaré sphere and fully determines the Pancharatnam-Berry phase, while the symmetric block is geometrically neutral. The same antisymmetric generator governs the evolution of pure qubit states on the Bloch sphere. This unified viewpoint yields operational criteria to identify and control geometric-phase contributions from measured Mueller matrices and from qubit process tomography.

In the era of big data, data may come from multiple sources, known as multi-view data. Multi-view clustering aims at generating better clusters by exploiting complementary and consistent information from multiple views rather than relying on the individual view. Due to inevitable system errors caused by data-captured sensors or others, the data in each view may be erroneous. Various types of errors behave differently and inconsistently in each view. More precisely, error could exhibit as noise and corruptions in reality. Unfortunately, none of the existing multi-view clustering approaches handle all of these error types. Consequently, their clustering performance is dramatically degraded. In this paper, we propose a novel Markov chain method for Error-Robust Multi-View Clustering (EMVC). By decomposing each view into a shared transition probability matrix and error matrix and imposing structured sparsity-inducing norms on error matrices, we characterize and handle typical types of errors explicitly. To solve the challenging optimization problem, we propose a new efficient algorithm based on Augmented Lagrangian Multipliers and prove its convergence rigorously. Experimental results on various synthetic and real-world datasets show the superiority of the proposed EMVC method over the baseline methods and its robustness against different types of errors.