1,369 research outputs found
Left-Invariant Diffusion on the Motion Group in terms of the Irreducible Representations of SO(3)
In this work we study the formulation of convection/diffusion equations on
the 3D motion group SE(3) in terms of the irreducible representations of SO(3).
Therefore, the left-invariant vector-fields on SE(3) are expressed as linear
operators, that are differential forms in the translation coordinate and
algebraic in the rotation. In the context of 3D image processing this approach
avoids the explicit discretization of SO(3) or , respectively. This is
particular important for SO(3), where a direct discretization is infeasible due
to the enormous memory consumption. We show two applications of the framework:
one in the context of diffusion-weighted magnetic resonance imaging and one in
the context of object detection
Unsupervised Anomaly Detection in Medical Images Using Masked Diffusion Model
It can be challenging to identify brain MRI anomalies using supervised
deep-learning techniques due to anatomical heterogeneity and the requirement
for pixel-level labeling. Unsupervised anomaly detection approaches provide an
alternative solution by relying only on sample-level labels of healthy brains
to generate a desired representation to identify abnormalities at the pixel
level. Although, generative models are crucial for generating such anatomically
consistent representations of healthy brains, accurately generating the
intricate anatomy of the human brain remains a challenge. In this study, we
present a method called masked-DDPM (mDPPM), which introduces masking-based
regularization to reframe the generation task of diffusion models.
Specifically, we introduce Masked Image Modeling (MIM) and Masked Frequency
Modeling (MFM) in our self-supervised approach that enables models to learn
visual representations from unlabeled data. To the best of our knowledge, this
is the first attempt to apply MFM in DPPM models for medical applications. We
evaluate our approach on datasets containing tumors and numerous sclerosis
lesions and exhibit the superior performance of our unsupervised method as
compared to the existing fully/weakly supervised baselines. Code is available
at https://github.com/hasan1292/mDDPM.Comment: Accepted in MICCAI 2023 Workshop
Locally Adaptive Frames in the Roto-Translation Group and their Applications in Medical Imaging
Locally adaptive differential frames (gauge frames) are a well-known
effective tool in image analysis, used in differential invariants and
PDE-flows. However, at complex structures such as crossings or junctions, these
frames are not well-defined. Therefore, we generalize the notion of gauge
frames on images to gauge frames on data representations defined on the extended space of positions and
orientations, which we relate to data on the roto-translation group ,
. This allows to define multiple frames per position, one per
orientation. We compute these frames via exponential curve fits in the extended
data representations in . These curve fits minimize first or second
order variational problems which are solved by spectral decomposition of,
respectively, a structure tensor or Hessian of data on . We include
these gauge frames in differential invariants and crossing preserving PDE-flows
acting on extended data representation and we show their advantage compared
to the standard left-invariant frame on . Applications include
crossing-preserving filtering and improved segmentations of the vascular tree
in retinal images, and new 3D extensions of coherence-enhancing diffusion via
invertible orientation scores
Bayesian Inference with Combined Dynamic and Sparsity Models: Application in 3D Electrophysiological Imaging
Data-driven inference is widely encountered in various scientific domains to convert the observed measurements into information that cannot be directly observed about a system. Despite the quickly-developing sensor and imaging technologies, in many domains, data collection remains an expensive endeavor due to financial and physical constraints. To overcome the limits in data and to reduce the demand on expensive data collection, it is important to incorporate prior information in order to place the data-driven inference in a domain-relevant context and to improve its accuracy.
Two sources of assumptions have been used successfully in many inverse problem applications. One is the temporal dynamics of the system (dynamic structure). The other is the low-dimensional structure of a system (sparsity structure). In existing work, these two structures have often been explored separately, while in most high-dimensional dynamic system they are commonly co-existing and contain complementary information.
In this work, our main focus is to build a robustness inference framework to combine dynamic and sparsity constraints. The driving application in this work is a biomedical inverse problem of electrophysiological (EP) imaging, which noninvasively and quantitatively reconstruct transmural action potentials from body-surface voltage data with the goal to improve cardiac disease prevention, diagnosis, and treatment. The general framework can be extended to a variety of applications that deal with the inference of high-dimensional dynamic systems
Integral curves of noisy vector fields and statistical problems in diffusion tensor imaging: nonparametric kernel estimation and hypotheses testing
Let be a vector field in a bounded open set .
Suppose that is observed with a random noise at random points that are independent and uniformly distributed in The problem
is to estimate the integral curve of the differential equation
starting at a given
point and to develop statistical tests for the hypothesis that
the integral curve reaches a specified set We develop an
estimation procedure based on a Nadaraya--Watson type kernel regression
estimator, show the asymptotic normality of the estimated integral curve and
derive differential and integral equations for the mean and covariance function
of the limit Gaussian process. This provides a method of tracking not only the
integral curve, but also the covariance matrix of its estimate. We also study
the asymptotic distribution of the squared minimal distance from the integral
curve to a smooth enough surface . Building upon this, we
develop testing procedures for the hypothesis that the integral curve reaches
. The problems of this nature are of interest in diffusion tensor
imaging, a brain imaging technique based on measuring the diffusion tensor at
discrete locations in the cerebral white matter, where the diffusion of water
molecules is typically anisotropic. The diffusion tensor data is used to
estimate the dominant orientations of the diffusion and to track white matter
fibers from the initial location following these orientations. Our approach
brings more rigorous statistical tools to the analysis of this problem
providing, in particular, hypothesis testing procedures that might be useful in
the study of axonal connectivity of the white matter.Comment: Published in at http://dx.doi.org/10.1214/009053607000000073 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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