Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of R d . For restrictions to the Euclidean ball in odd dimensions, to the rotation group SO ( 3 ) , and to the Grassmannian manifold G 2 , 4 , we compute the kernels' Fourier coefficients and determine their asymptotics. The L 2 -discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For SO ( 3 ) , the nonequispaced fast Fourier transform is publicly available, and, for G 2 , 4 , the transform is derived here. We also provide numerical experiments for SO ( 3 ) and G 2 , 4 .

Supervised learning and dimension reduction techniques for quantification of retinal fluid in optical coherence tomography images

PurposeThe purpose of the present study is to develop fast automated quantification of retinal fluid in optical coherence tomography (OCT) image sets.MethodsWe developed an image analysis pipeline tailored towards OCT images that consists of five steps for binary retinal fluid segmentation. The method is based on feature extraction, pre-segmention, dimension reduction procedures, and supervised learning tools.ResultsFluid identification using our pipeline was tested on two separate patient groups: one associated to neovascular age-related macular degeneration, the other showing diabetic macular edema. For training and evaluation purposes, retinal fluid was annotated manually in each cross-section by human expert graders of the Vienna Reading Center. Compared with the manual annotations, our pipeline yields good quantification, visually and in numbers.ConclusionsBy demonstrating good automated retinal fluid quantification, our pipeline appears useful to expert graders within their current grading processes. Owing to dimension reduction, the actual learning part is fast and requires only few training samples. Hence, it is well-suited for integration into actual manufacturer's devices, further improving segmentation by its use in daily clinical life.

Modeling Photo-Bleaching Kinetics to Create High Resolution Maps of Rod Rhodopsin in the Human Retina

We introduce and describe a novel non-invasive in-vivo method for mapping local rod rhodopsin distribution in the human retina over a 30-degree field. Our approach is based on analyzing the brightening of detected lipofuscin autofluorescence within small pixel clusters in registered imaging sequences taken with a commercial 488nm confocal scanning laser ophthalmoscope (cSLO) over a 1 minute period. We modeled the kinetics of rhodopsin bleaching by applying variational optimization techniques from applied mathematics. The physical model and the numerical analysis with its implementation are outlined in detail. This new technique enables the creation of spatial maps of the retinal rhodopsin and retinal pigment epithelium (RPE) bisretinoid distribution with an ≈ 50μm resolution.

Modifications of iterative schemes used for curvature correction in noninvasive biomedical imaging

Iterative polynomial fitting along image rows and columns has recently been used to remove curvature bias in multispectral image sets of the human forearm and phantoms. However, this method is only applicable if foreground and background features satisfy strong separation conditions. In this method, we verify that the iterative polynomial approach converges toward bivariate polynomial fitting, and, hence, the resulting fit corresponds to low-pass filtering the image. In contrast to the iterative fitting, the bivariate polynomial fit can be performed on images with missing or excluded parts. Indeed, our observation enables us to modify the scheme and significantly weaken the required assumptions on foreground/background separation allowing a wider range of application.

Schroedinger Eigenmaps for the analysis of biomedical data

We introduce Schroedinger Eigenmaps (SE), a new semi-supervised manifold learning and recovery technique. This method is based on an implementation of graph Schroedinger operators with appropriately constructed barrier potentials as carriers of labeled information. We use our approach for the analysis of standard biomedical datasets and new multispectral retinal images.

Locally learning biomedical data using diffusion frames

Diffusion geometry techniques are useful to classify patterns and visualize high-dimensional datasets. Building upon ideas from diffusion geometry, we outline our mathematical foundations for learning a function on high-dimension biomedical data in a local fashion from training data. Our approach is based on a localized summation kernel, and we verify the computational performance by means of exact approximation rates. After these theoretical results, we apply our scheme to learn early disease stages in standard and new biomedical datasets.

Quantitative principal component model for skin chromophore mapping using multi-spectral images and spatial priors

We describe a novel reconstruction algorithm based on Principal Component Analysis (PCA) applied to multi-spectral imaging data. Using numerical phantoms, based on a two layered skin model developed previously, we found analytical expressions, which convert qualitative PCA results into quantitative blood volume and oxygenation values, assuming the epidermal thickness to be known. We also evaluate the limits of accuracy of this method when the value of the epidermal thickness is not known. We show that blood volume can reliably be extracted (less than 6% error) even if the assumed thickness deviates 0.04mm from the actual value, whereas the error in blood oxygenation can be as large as 25% for the same deviation in thickness. This PCA based reconstruction was found to extract blood volume and blood oxygenation with less than 8% error, if the underlying structure is known. We then apply the method to in vivo multi-spectral images from a healthy volunteer's lower forearm, complemented by images of the same area using Optical Coherence Tomography (OCT) for measuring the epidermal thickness. Reconstruction of the imaging results using a two layered analytical skin model was compared to PCA based reconstruction results. A point wise correlation was found, showing the proof of principle of using PCA based reconstruction for blood volume and oxygenation extraction.

RedundancyMiner: De-replication of redundant GO categories in microarray and proteomics analysis

BACKGROUND

The Gene Ontology (GO) Consortium organizes genes into hierarchical categories based on biological process, molecular function and subcellular localization. Tools such as GoMiner can leverage GO to perform ontological analysis of microarray and proteomics studies, typically generating a list of significant functional categories. Two or more of the categories are often redundant, in the sense that identical or nearly-identical sets of genes map to the categories. The redundancy might typically inflate the report of significant categories by a factor of three-fold, create an illusion of an overly long list of significant categories, and obscure the relevant biological interpretation.

RESULTS

We now introduce a new resource, RedundancyMiner, that de-replicates the redundant and nearly-redundant GO categories that had been determined by first running GoMiner. The main algorithm of RedundancyMiner, MultiClust, performs a novel form of cluster analysis in which a GO category might belong to several category clusters. Each category cluster follows a "complete linkage" paradigm. The metric is a similarity measure that captures the overlap in gene mapping between pairs of categories.

CONCLUSIONS

RedundancyMiner effectively eliminated redundancies from a set of GO categories. For illustration, we have applied it to the clarification of the results arising from two current studies: (1) assessment of the gene expression profiles obtained by laser capture microdissection (LCM) of serial cryosections of the retina at the site of final optic fissure closure in the mouse embryos at specific embryonic stages, and (2) analysis of a conceptual data set obtained by examining a list of genes deemed to be "kinetochore" genes.

Direct curvature correction for noncontact imaging modalities applied to multispectral imaging

Noncontact optical imaging of curved objects can result in strong artifacts due to the object's shape, leading to curvature biased intensity distributions. This artifact can mask variations due to the object's optical properties, and makes reconstruction of optical/physiological properties difficult. In this work we demonstrate a curvature correction method that removes this artifact and recovers the underlying data, without the necessity of measuring the object's shape. This method is applicable to many optical imaging modalities that suffer from shape-based intensity biases. By separating the spatially varying data (e.g., physiological changes) from the background signal (dc component), we show that the curvature can be extracted by either averaging or fitting the rows and columns of the images. Numerical simulations show that our method is equivalent to directly removing the curvature, when the object's shape is known, and accurately recovers the underlying data. Experiments on phantoms validate the numerical results and show that for a given image with 16.5% error due to curvature, the method reduces that error to 1.2%. Finally, diffuse multispectral images are acquired on forearms in vivo. We demonstrate the enhancement in image quality on intensity images, and consequently on reconstruction results of blood volume and oxygenation distributions.

Principal component model of multispectral data for near real-time skin chromophore mapping

Multispectral images of skin contain information on the spatial distribution of biological chromophores, such as blood and melanin. From this, parameters such as blood volume and blood oxygenation can be retrieved using reconstruction algorithms. Most such approaches use some form of pixelwise or volumetric reconstruction code. We explore the use of principal component analysis (PCA) of multispectral images to access blood volume and blood oxygenation in near real time. We present data from healthy volunteers under arterial occlusion of the forearm, experiencing ischemia and reactive hyperemia. Using a two-layered analytical skin model, we show reconstruction results of blood volume and oxygenation and compare it to the results obtained from our new spectral analysis based on PCA. We demonstrate that PCA applied to multispectral images gives near equivalent results for skin chromophore mapping and quantification with the advantage of being three orders of magnitude faster than the reconstruction algorithm.