Manifolds, Active Models, and Deformable Templates

2006 ◽  
Author(s):  
Ulf Grenander ◽  
Michael I. Miller
Keyword(s):  
Image Analysis ◽  
Random Fields ◽  
Transport Theory ◽  
Structural Models ◽  
Gaussian Random Fields ◽  
Deformable Templates ◽  
Infinite Products ◽  
Matrix Groups ◽  
The Matrix ◽  
Deformable Shapes

To study shape we introduce manifolds and submanifolds examined in the continuum as the generators. Transformations are constructed which are built from the matrix groups and infinite products. This gives rise to many of the widely used structural models in image analysis often termed active models, essentially the deformable templates. These deformations are studied as both diffeomorphisms as well as immersions. A calculus is introduced based on transport theory for activating these deformable shapes by taking variations with respect to the matrix groups parameterizing them. Segmentation based on activating these manifolds is examined based on Gaussian random fields and variations with respect to the parameterizations.

scholarly journals MODELS OF COVARIANCE FUNCTIONS OF GAUSSIAN RANDOM FIELDS ESCAPING FROM ISOTROPY, STATIONARITY AND NON NEGATIVITY

2014 ◽  
Vol 33 (1) ◽  
pp. 75
Author(s):  
Pablo Gregori ◽  
Emilio Porcu ◽  
Jorge Mateu
Keyword(s):  
Image Analysis ◽  
Random Fields ◽  
Fourier Transforms ◽  
Gaussian Random Fields ◽  
Space Time ◽  
Linear Combinations ◽  
Covariance Functions ◽  
Recent Advances ◽  
Special Cases ◽  
Covariance Models

This paper represents a survey of recent advances in modeling of space or space-time Gaussian Random Fields (GRF), tools of Geostatistics at hand for the understanding of special cases of noise in image analysis. They can be used when stationarity or isotropy are unrealistic assumptions, or even when negative covariance between some couples of locations are evident. We show some strategies in order to escape from these restrictions, on the basis of rich classes of well known stationary or isotropic non negative covariance models, and through suitable operations, like linear combinations, generalized means, or with particular Fourier transforms.

Comparison of Nonlinear Spatial Correlation Models by the Influence of the Data Augmentation to the Classification Risk

2002 ◽  
Vol 7 (1) ◽  
pp. 31-42
Author(s):  
J. Šaltytė ◽  
K. Dučinskas
Keyword(s):  
Spatial Correlation ◽  
Random Fields ◽  
Data Augmentation ◽  
Gaussian Random Fields ◽  
Classification Rule ◽  
Numerical Comparison ◽  
First Order ◽  
Bayesian Risk ◽  
Correlation Models

The Bayesian classification rule used for the classification of the observations of the (second-order) stationary Gaussian random fields with different means and common factorised covariance matrices is investigated. The influence of the observed data augmentation to the Bayesian risk is examined for three different nonlinear widely applicable spatial correlation models. The explicit expression of the Bayesian risk for the classification of augmented data is derived. Numerical comparison of these models by the variability of Bayesian risk in case of the first-order neighbourhood scheme is performed.

scholarly journals Homogenisation Efficiency Assessed with Microstructure Analysis and Hardness Measurements in the EN AW 2011 Aluminium Alloy

Metals ◽  
2021 ◽  
Vol 11 (8) ◽  
pp. 1211
Author(s):  
Maja Vončina ◽  
Aleš Nagode ◽  
Jožef Medved ◽  
Irena Paulin ◽  
Borut Žužek ◽  
...  
Keyword(s):  
Experimental Data ◽  
Image Analysis ◽  
Phase Equilibrium ◽  
Aluminium Alloy ◽  
Phase Formation ◽  
Aluminium Alloys ◽  
Eutectic Phase ◽  
The Matrix ◽  
Equilibrium Solidification ◽  
Homogenisation Process

When extruding the casted rods from EN AW 2011 aluminium alloys, not only their homogenized structure, but also their extrudable properties were significantly influenced by the hardness of the alloy. In this study, the object of investigations was the EN AW 2011 aluminium alloy, and the effect of homogenisation time on hardness was investigated. First, homogenisation was carried out at 520 °C for different times, imitating industrial conditions. After homogenisation, the samples were analysed by hardness measurements and further characterised by microscopy and image analysis to verify the influence of homogenisation on the resulting microstructural constituents. In addition, non-equilibrium solidification was simulated using the program Thermo-Calc and phase formation during solidification was investigated. The homogenisation process enabled more rounded shape of the Al2Cu eutectic phase, equilibrium formation of the phases, and the precipitation in the matrix, leading to a significant increase in the hardness of the EN AW 2011 aluminium alloy. The experimental data revealed a suitable homogenisation time of 4–6 h at a temperature of 520 °C, enabling optimal extrusion properties.

scholarly journals Random Marked Sets

2012 ◽  
Vol 44 (3) ◽  
pp. 603-616 ◽  
Author(s):  
F. Ballani ◽  
Z. Kabluchko ◽  
M. Schlather
Keyword(s):  
Random Fields ◽  
Covariance Function ◽  
Point Processes ◽  
Marked Point ◽  
Positive Definiteness ◽  
Gaussian Random Fields ◽  
Marked Point Processes ◽  
Random Set ◽  
Geostatistical Methods ◽  
New Class

We aim to link random fields and marked point processes, and, therefore, introduce a new class of stochastic processes which are defined on a random set in . Unlike for random fields, the mark covariance function of a random marked set is in general not positive definite. This implies that in many situations the use of simple geostatistical methods appears to be questionable. Surprisingly, for a special class of processes based on Gaussian random fields, we do have positive definiteness for the corresponding mark covariance function and mark correlation function.

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