Geodesic Regression on the Grassmannian

Hong, Yi; Kwitt, Roland; Singh, Nikhil; Davis, Bradley Moore; Vasconcelos, Nuno; Niethammer, Marc

doi:10.1007/978-3-319-10605-2_41

Cited by 25 publications

(18 citation statements)

References 26 publications

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“…A complete listing of ADNI investigators can be found at: http://adni.loni.usc.edu/wp-content/uploads/how_ to_apply/ADNI_Acknowledgement_List.pdf Image registration is a key task in medical image analysis to study deformations between images. Building on image registration approaches, image regression models [4,5,6,7,8,9,10,11,12,13] have been developed to analyze deformation trends in longitudinal imaging studies. One such approach is geodesic regression (GR) [4,7,8] which (for images) build on the large displacement diffeomorphic metric mapping model (LDDMM) [14].…”

Section: Introductionmentioning

confidence: 99%

Fast Predictive Simple Geodesic Regression

Ding¹,

Fleishman²,

Yang³

et al. 2017

Lecture Notes in Computer Science

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Deformable image registration and regression are important tasks in medical image analysis. However, they are computationally expensive, especially when analyzing large-scale datasets that contain thousands of images. Hence, cluster computing is typically used, making the approaches dependent on such computational infrastructure. Even larger computational resources are required as study sizes increase. This limits the use of deformable image registration and regression for clinical applications and as component algorithms for other image analysis approaches. We therefore propose using a fast predictive approach to perform image registrations. In particular, we employ these fast registration predictions to approximate a simplified geodesic regression model to capture longitudinal brain changes. The resulting method is orders of magnitude faster than the standard optimization-based regression model and hence facilitates large-scale analysis on a single graphics processing unit (GPU). We evaluate our results on 3D brain magnetic resonance images (MRI) from the ADNI datasets.

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Section: Introductionmentioning

confidence: 99%

Fast Predictive Simple Geodesic Regression

Ding¹,

Fleishman²,

Yang³

et al. 2017

Lecture Notes in Computer Science

Self Cite

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“…In a more traditional learning setting, there has been work using geodesic regression, a generalization of linear regression, on Riemannian manifolds [15,14,38,22,27], where a geodesic curve is computed such that the average distance (on the manifold) from the data points to the curve is minimized. This involves computing gradients on the manifold.…”

Section: Related Workmentioning

confidence: 99%

Learning Invariant Riemannian Geometric Representations Using Deep Nets

Lohit

Turaga²

2017

2017 IEEE International Conference on Computer Vision Workshops (ICCVW)

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Non-Euclidean constraints are inherent in many kinds of data in computer vision and machine learning, typically as a result of specific invariance requirements that need to be respected during high-level inference. Often, these geometric constraints can be expressed in the language of Riemannian geometry, where conventional vector space machine learning does not apply directly. The central question this paper deals with is: How does one train deep neural nets whose final outputs are elements on a Riemannian manifold? To answer this, we propose a general framework for manifold-aware training of deep neural networks -we utilize tangent spaces and exponential maps in order to convert the proposed problem into a form that allows us to bring current advances in deep learning to bear upon this problem. We describe two specific applications to demonstrate this approach: prediction of probability distributions for multi-class image classification, and prediction of illumination-invariant subspaces from a single faceimage via regression on the Grassmannian. These applications show the generality of the proposed framework, and result in improved performance over baselines that ignore the geometry of the output space. In addition to solving this specific problem, we believe this paper opens new lines of enquiry centered on the implications of Riemannian geometry on deep architectures.

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“…There exists several studies addressing regression in the setting that the independent variable is a point in Euclidean space and the dependent variable is a point in non-flat manifold space such as Riemann and Grassmann manifolds. Based on their methodology, we can group these studies into three categories: (i) Parametric approaches like [5,7,9,13,15,25] usually try to find a formulation for the geodesic and then provide a numerical solution for its estimation. (ii) Semi-parametric approach, e.g., [27] uses a link function to map from Euclidean to Riemannian space.…”

Section: Manifold-valued Data Regressionmentioning

confidence: 99%

Zero-Shot Domain Adaptation via Kernel Regression on the Grassmannian

Yang¹,

Hospedales²

2015

Procedings of the Proceedings of the 1st International Workshop on DIFFerential Geometry in Computer Vision for Analysis of Sha

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Most visual recognition methods implicitly assume the data distribution remains unchanged from training to testing. However, in practice domain shift often exists, where real-world factors such as lighting and sensor type change between train and test, and classifiers do not generalise from source to target domains. It is impractical to train separate models for all possible situations because collecting and labelling the data is expensive. Domain adaptation algorithms aim to ameliorate domain shift, allowing a model trained on a source to perform well on a different target domain. However, even for the setting of unsupervised domain adaptation, where the target domain is unlabelled, collecting data for every possible target domain is still costly. In this paper, we propose a new domain adaptation method that has no need to access either data or labels of the target domain when it can be described by a parametrised vector and there exits several related source domains within the same parametric space. It greatly reduces the burden of data collection and annotation, and our experiments show some promising results.

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Geodesic Regression on the Grassmannian

Cited by 25 publications

References 26 publications

Fast Predictive Simple Geodesic Regression

Fast Predictive Simple Geodesic Regression

Learning Invariant Riemannian Geometric Representations Using Deep Nets

Zero-Shot Domain Adaptation via Kernel Regression on the Grassmannian

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