Manifold Valued Statistics, Exact Principal Geodesic Analysis and the Effect of Linear Approximations

Sommer, Stefan; Lauze, François; Hauberg, Søren; Nielsen, Mads

doi:10.1007/978-3-642-15567-3_4

Cited by 66 publications

(72 citation statements)

References 20 publications

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“…The last step of training involves performing of the Principal Geodesic Analysis (Fletcher et al, 2004) (see also Sommer et al, 2010), PGA in short, in the feature space. This is necessary as the extracted features belong to a torus S dim(n) where…”

Section: Trainingmentioning

confidence: 99%

Principal Geodesic Analysis Boundary Delineation With Superpixel-Based Constraints

Baran

Tabor

2017

Image Anal Stereol

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In this paper an algorithm for accurate delineation of object boundaries is proposed. The method employs a superpixel algorithm to obtain an oversegmentation of the input image, used as a constraint in the task. A shape model is built by applying Principal Geodesic Analysis on angular representation of automatically placed uniformly distant landmark points. The shape model is used to detect the boundaries of an object on a given image by iterative elongation of a partial boundary along borders of superpixels. Contrary to many stateof-the-art methods, the proposed approach does not need an initial boundary. The algorithm was tested on two natural and two synthetic sets of images. Mean Dice coefficients between 0.91 and 0.97 were obtained. In almost all cases the object was found. In areas of relatively high gradient magnitude the borders are delineated very accurately, though further research is needed to improve the accuracy in areas of low gradient magnitude and automatically select the parameters of the proposed error function.

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Section: Trainingmentioning

confidence: 99%

Principal Geodesic Analysis Boundary Delineation With Superpixel-Based Constraints

Baran

Tabor

2017

Image Anal Stereol

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“…[3,42,11,29,13,20], we deal with the nonlinearity of the space of interest by utilizing the linear tangent space at the identity. Though not explored here, other types of statistics on manifolds [32,12,34,38,36,21,19,41,28,44,10,23,14] may also be applicable.…”

Section: Related Workmentioning

confidence: 99%

Highly-Expressive Spaces of Well-Behaved Transformations: Keeping it Simple

Freifeld

Hauberg

Batmanghelich

et al. 2015

2015 IEEE International Conference on Computer Vision (ICCV)

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“…Thereafter many attitudes to the shape analysis were raised and the application of this new statistical field has motivated many researchers to employ it in other sciences. Some instances are statistical shape analysis of brain (Free et al 2001), discovering variability of DNA molecules (Dryden 2002), medial image processing (Fletcher et al 2003), facial gender classification (Wu et al 2007) and study of vertebral fractures (Sommer et al 2010). Other applications of this subject can be found in Dryden and Mardia (1998).…”

Section: A Brief Review Of Shape Analysismentioning

confidence: 99%

“…Although these approaches might provide some reasonable answers to the questions on hand, in some circumstances the induced errors are high (Huckemann and Hotz 2009) and it sounds the better choice is to perform statistical analysis directly on the manifold. The advantages of this will be consistency in representation, dimensionality reduction, and accuracy in measurements (Sommer et al 2010). Along with some useful definitions, a comprehensive treatment of how to perform statistical analysis directly on manifolds is given in Pennec (2006).…”

Section: A Brief Review Of Shape Analysismentioning

confidence: 99%

Highly resistant gradient descent algorithm for computing intrinsic mean shape on similarity shape spaces

Fotouhi

Golalizadeh

2014

Stat Papers

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Among many algorithms, gradient descent algorithm (GDA) is a simple tool to derive an optimal quantity in dealing with an optimization problem in the linear space. Apart from the initial value, the step size has a great impact on the convergence rate of this algorithm. Its affect on the geometric structure of the consecutive configurations is more crucial if one works with an optimization problem in the statistical shape analysis. In other words, if the step size of the GDA is not properly tuned, the geometry might not be preserved while the algorithm is moving forward to reach an optimal mean shape. In order to improve the performance of the GDA, we introduce a dynamic step size and a new criterion both to check the geometry in each step of the algorithm and to accelerate the convergence rate. These lead to a new robust algorithm on deriving the intrinsic mean on the shape space. We compare the performance of our proposed procedure to the usual GDA using a real shape data accompanied with simulation studies.

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Manifold Valued Statistics, Exact Principal Geodesic Analysis and the Effect of Linear Approximations

Cited by 66 publications

References 20 publications

Principal Geodesic Analysis Boundary Delineation With Superpixel-Based Constraints

Principal Geodesic Analysis Boundary Delineation With Superpixel-Based Constraints

Highly-Expressive Spaces of Well-Behaved Transformations: Keeping it Simple

Highly resistant gradient descent algorithm for computing intrinsic mean shape on similarity shape spaces

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