The paper deals with uninorms and nullnorms as basic semi-group operations which are commutative... more The paper deals with uninorms and nullnorms as basic semi-group operations which are commutative and monotone (increasing). These operations were first introduced on the unit interval and later generalized to bounded lattices. In [Kalina 2019] they were introduced on bounded posets. This contribution is a generalization and extension of the results in [Kalina 2019]. Some necessary and some sufficient conditions for the existence of idempotent uninorms and idempotent nullnorms on bounded posets are studied. Finally, some application examples are provided.
2010 3rd International Congress on Image and Signal Processing, 2010
In this paper, we describe methods designed to segment objects from 3D cell nuclei and cell membr... more In this paper, we describe methods designed to segment objects from 3D cell nuclei and cell membrane images and we point out several possible applications in analysis of the zebrafish cell image data. We show how we can segment the cell nuclei, the inner boundaries of the cells, the outer boundary of an organism and the intercellular borders. We also present a method for detection of approximate nuclei centers that is a necessary preliminary step for the segmentation. All methods that we mention are based on numerical solution of partial differential equations. We describe all mathematical models and we sketch the principles of the time and space discretization. The explanation of all methods is accompanied by illustrations and suggestions of applications in biology.
The paper deals with the nonlinear tensor diffusion which enhances image structure coherence. Fir... more The paper deals with the nonlinear tensor diffusion which enhances image structure coherence. First we briefly describe the diffusion model and provide its basic properties. Further we build a semi-implicit finite volume scheme for the above mentioned model with the help of a co-volume mesh. This method is well-known as diamond-cell method owing to the choice of co-volume as a diamond-shaped polygon. The scheme design is shown for $2$D as well as $3$D case, cf. [3,4]. Then the convergence and error estimate analysis for $2$D scheme is presented, cf. [3,2]. The convergence proof is based on Kolmogorov's compactness theorem and a bounding of a gradient in tangential direction by means a gradient in normal direction. We proved that the error of the numerical solution in $L^2$-norm is of order $h$, where $h$ is a spatial step under the natural relation $k \approx h^2$ with a time step $k$. Last part is devoted to results of computational experiments. They confirm the usefulness this...
We present a set of techniques that enable us to segment objects from 3D cell membrane images. Pa... more We present a set of techniques that enable us to segment objects from 3D cell membrane images. Particularly, we propose methods for detection of approximate cell nuclei centers, extraction of the inner cell boundaries, the surface of the organism and the intercellular borders-the so called intercellular skeleton. All methods are based on numerical solution of partial differential equations. The center detection problem is represented by a level set equation for advective motion in normal direction with curvature term. In case of the inner cell boundaries and the global surface, we use the generalized subjective surface model. The intercellular borders are segmented by the advective level set equation where the velocity field is given by the gradient of the signed distance function to the segmented inner cell boundaries. The distance function is computed by solving the time relaxed eikonal equation. We describe the mathematical models, explain their numerical approximation and finally we present various possible practical applications on the images of zebrafish embryogenesis-computation of important quantitative characteristics, evaluation of the cell shape, detection of cell divisions and others.
The paper deals with the nonlinear tensor diffusion which yields a coherence improvement. It is v... more The paper deals with the nonlinear tensor diffusion which yields a coherence improvement. It is very appropriate for images with flow-like structures. Two convolutions are used in the construction of diffusion tensor for such a model, see [Drblíková, O.—Mikula, K.: Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM J. Numer. Anal. 46 (2007), 37–60], [Weickert, J.: Coherence-enhancing diffusion filtering, Int. J. Comput. Vis. 31 (1999), 111–127]. In this paper we introduce the third supplemental convolution in order to enhance the diffusion strategy. First, we briefly present the classical coherence enhancing model and explain our modification. Then the discrete scheme is provided. The core of the paper consists in numerical experiments. Benefits of the additional convolution are discussed and illustrated in the figures.
We present nonlinear diffusion equations, numerical schemes to solve them and their application f... more We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which are artificially connected due to acquisition error intrinsically linked to physics of LSM. In all studied aspects it turned out that the nonlinear diffusion filter which is called geodesic mean curvature flow (GMCF) has the best performance.
The quantitative and systematic analysis of embryonic cell dynamics from in vivo 3Dþtime image da... more The quantitative and systematic analysis of embryonic cell dynamics from in vivo 3Dþtime image data sets is a major challenge at the forefront of developmental biology. Despite recent breakthroughs in the microscopy imaging of living systems, producing an accurate cell lineage tree for any developing organism remains a difficult task. We present here the BioEmergences workflow integrating all reconstruction steps from image acquisition and processing to the interactive visualization of reconstructed data. Original mathematical methods and algorithms underlie image filtering, nucleus centre detection, nucleus and membrane segmentation, and cell tracking. They are demonstrated on zebrafish, ascidian and sea urchin embryos with stained nuclei and membranes. Subsequent validation and annotations are carried out using Mov-IT, a custom-made graphical interface. Compared with eight other software tools, our workflow achieved the best lineage score. Delivered in standalone or web service mode, BioEmergences and Mov-IT offer a unique set of tools for in silico experimental embryology.
We propose the use of gradient schemes for the approximation of possibly non-local regularization... more We propose the use of gradient schemes for the approximation of possibly non-local regularizations of the Perona-Malik equations. We then prove the convergence of this method thanks to compactness arguments, and we apply it to a particular gradient scheme on rectangular meshes, whose advantage is to lead to nine-point stencil matrices. Numerical examples provide indications about the accuracy of the method.
We present a gradient scheme (which happens to be similar to the MPFA finite volume O-scheme) for... more We present a gradient scheme (which happens to be similar to the MPFA finite volume O-scheme) for the approximation to the solution of the Perona-Malik model regularized by a time delay and to the solution of the nonlinear tensor anisotropic diffusion equation. Numerical examples showing properties of the method and applications in image filtering are discussed.
We present nonlinear diffusion equations, numerical schemes to solve them and their application f... more We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which are artificially connected due to acquisition error intrinsically linked to physics of LSM. In all studied aspects it turned out that the nonlinear diffusion filter which is called geodesic mean curvature flow (GMCF) has the best performance.
The quantitative and systematic analysis of embryonic cell dynamics from in vivo 3D+time image da... more The quantitative and systematic analysis of embryonic cell dynamics from in vivo 3D+time image data sets is a major challenge at the forefront of developmental biology. Despite recent breakthroughs in the microscopy imaging of living systems, producing an accurate cell lineage tree for any developing organism remains a difficult task. We present here the BioEmergences workflow integrating all reconstruction steps from image acquisition and processing to the interactive visualization of reconstructed data. Original mathematical methods and algorithms underlie image filtering, nucleus centre detection, nucleus and membrane segmentation, and cell tracking. They are demonstrated on zebrafish, ascidian and sea urchin embryos with stained nuclei and membranes. Subsequent validation and annotations are carried out using Mov-IT, a custom-made graphical interface. Compared with eight other software tools, our workflow achieved the best lineage score. Delivered in standalone or web service mode, BioEmergences and Mov-IT offer a unique set of tools for in silico experimental embryology.
The paper deals with uninorms and nullnorms as basic semi-group operations which are commutative... more The paper deals with uninorms and nullnorms as basic semi-group operations which are commutative and monotone (increasing). These operations were first introduced on the unit interval and later generalized to bounded lattices. In [Kalina 2019] they were introduced on bounded posets. This contribution is a generalization and extension of the results in [Kalina 2019]. Some necessary and some sufficient conditions for the existence of idempotent uninorms and idempotent nullnorms on bounded posets are studied. Finally, some application examples are provided.
2010 3rd International Congress on Image and Signal Processing, 2010
In this paper, we describe methods designed to segment objects from 3D cell nuclei and cell membr... more In this paper, we describe methods designed to segment objects from 3D cell nuclei and cell membrane images and we point out several possible applications in analysis of the zebrafish cell image data. We show how we can segment the cell nuclei, the inner boundaries of the cells, the outer boundary of an organism and the intercellular borders. We also present a method for detection of approximate nuclei centers that is a necessary preliminary step for the segmentation. All methods that we mention are based on numerical solution of partial differential equations. We describe all mathematical models and we sketch the principles of the time and space discretization. The explanation of all methods is accompanied by illustrations and suggestions of applications in biology.
The paper deals with the nonlinear tensor diffusion which enhances image structure coherence. Fir... more The paper deals with the nonlinear tensor diffusion which enhances image structure coherence. First we briefly describe the diffusion model and provide its basic properties. Further we build a semi-implicit finite volume scheme for the above mentioned model with the help of a co-volume mesh. This method is well-known as diamond-cell method owing to the choice of co-volume as a diamond-shaped polygon. The scheme design is shown for $2$D as well as $3$D case, cf. [3,4]. Then the convergence and error estimate analysis for $2$D scheme is presented, cf. [3,2]. The convergence proof is based on Kolmogorov's compactness theorem and a bounding of a gradient in tangential direction by means a gradient in normal direction. We proved that the error of the numerical solution in $L^2$-norm is of order $h$, where $h$ is a spatial step under the natural relation $k \approx h^2$ with a time step $k$. Last part is devoted to results of computational experiments. They confirm the usefulness this...
We present a set of techniques that enable us to segment objects from 3D cell membrane images. Pa... more We present a set of techniques that enable us to segment objects from 3D cell membrane images. Particularly, we propose methods for detection of approximate cell nuclei centers, extraction of the inner cell boundaries, the surface of the organism and the intercellular borders-the so called intercellular skeleton. All methods are based on numerical solution of partial differential equations. The center detection problem is represented by a level set equation for advective motion in normal direction with curvature term. In case of the inner cell boundaries and the global surface, we use the generalized subjective surface model. The intercellular borders are segmented by the advective level set equation where the velocity field is given by the gradient of the signed distance function to the segmented inner cell boundaries. The distance function is computed by solving the time relaxed eikonal equation. We describe the mathematical models, explain their numerical approximation and finally we present various possible practical applications on the images of zebrafish embryogenesis-computation of important quantitative characteristics, evaluation of the cell shape, detection of cell divisions and others.
The paper deals with the nonlinear tensor diffusion which yields a coherence improvement. It is v... more The paper deals with the nonlinear tensor diffusion which yields a coherence improvement. It is very appropriate for images with flow-like structures. Two convolutions are used in the construction of diffusion tensor for such a model, see [Drblíková, O.—Mikula, K.: Convergence analysis of finite volume scheme for nonlinear tensor anisotropic diffusion in image processing, SIAM J. Numer. Anal. 46 (2007), 37–60], [Weickert, J.: Coherence-enhancing diffusion filtering, Int. J. Comput. Vis. 31 (1999), 111–127]. In this paper we introduce the third supplemental convolution in order to enhance the diffusion strategy. First, we briefly present the classical coherence enhancing model and explain our modification. Then the discrete scheme is provided. The core of the paper consists in numerical experiments. Benefits of the additional convolution are discussed and illustrated in the figures.
We present nonlinear diffusion equations, numerical schemes to solve them and their application f... more We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which are artificially connected due to acquisition error intrinsically linked to physics of LSM. In all studied aspects it turned out that the nonlinear diffusion filter which is called geodesic mean curvature flow (GMCF) has the best performance.
The quantitative and systematic analysis of embryonic cell dynamics from in vivo 3Dþtime image da... more The quantitative and systematic analysis of embryonic cell dynamics from in vivo 3Dþtime image data sets is a major challenge at the forefront of developmental biology. Despite recent breakthroughs in the microscopy imaging of living systems, producing an accurate cell lineage tree for any developing organism remains a difficult task. We present here the BioEmergences workflow integrating all reconstruction steps from image acquisition and processing to the interactive visualization of reconstructed data. Original mathematical methods and algorithms underlie image filtering, nucleus centre detection, nucleus and membrane segmentation, and cell tracking. They are demonstrated on zebrafish, ascidian and sea urchin embryos with stained nuclei and membranes. Subsequent validation and annotations are carried out using Mov-IT, a custom-made graphical interface. Compared with eight other software tools, our workflow achieved the best lineage score. Delivered in standalone or web service mode, BioEmergences and Mov-IT offer a unique set of tools for in silico experimental embryology.
We propose the use of gradient schemes for the approximation of possibly non-local regularization... more We propose the use of gradient schemes for the approximation of possibly non-local regularizations of the Perona-Malik equations. We then prove the convergence of this method thanks to compactness arguments, and we apply it to a particular gradient scheme on rectangular meshes, whose advantage is to lead to nine-point stencil matrices. Numerical examples provide indications about the accuracy of the method.
We present a gradient scheme (which happens to be similar to the MPFA finite volume O-scheme) for... more We present a gradient scheme (which happens to be similar to the MPFA finite volume O-scheme) for the approximation to the solution of the Perona-Malik model regularized by a time delay and to the solution of the nonlinear tensor anisotropic diffusion equation. Numerical examples showing properties of the method and applications in image filtering are discussed.
We present nonlinear diffusion equations, numerical schemes to solve them and their application f... more We present nonlinear diffusion equations, numerical schemes to solve them and their application for filtering 3D images obtained from laser scanning microscopy (LSM) of living zebrafish embryos, with a goal to identify the optimal filtering method and its parameters. In the large scale applications dealing with analysis of 3D+time embryogenesis images, an important objective is a correct detection of the number and position of cell nuclei yielding the spatio-temporal cell lineage tree of embryogenesis. The filtering is the first and necessary step of the image analysis chain and must lead to correct results, removing the noise, sharpening the nuclei edges and correcting the acquisition errors related to spuriously connected subregions. In this paper we study such properties for the regularized Perona-Malik model and for the generalized mean curvature flow equations in the level-set formulation. A comparison with other nonlinear diffusion filters, like tensor anisotropic diffusion and Beltrami flow, is also included. All numerical schemes are based on the same discretization principles, i.e. finite volume method in space and semi-implicit scheme in time, for solving nonlinear partial differential equations. These numerical schemes are unconditionally stable, fast and naturally parallelizable. The filtering results are evaluated and compared first using the Mean Hausdorff distance between a gold standard and different isosurfaces of original and filtered data. Then, the number of isosurface connected components in a region of interest (ROI) detected in original and after the filtering is compared with the corresponding correct number of nuclei in the gold standard. Such analysis proves the robustness and reliability of the edge preserving nonlinear diffusion filtering for this type of data and lead to finding the optimal filtering parameters for the studied models and numerical schemes. Further comparisons consist in ability of splitting the very close objects which are artificially connected due to acquisition error intrinsically linked to physics of LSM. In all studied aspects it turned out that the nonlinear diffusion filter which is called geodesic mean curvature flow (GMCF) has the best performance.
The quantitative and systematic analysis of embryonic cell dynamics from in vivo 3D+time image da... more The quantitative and systematic analysis of embryonic cell dynamics from in vivo 3D+time image data sets is a major challenge at the forefront of developmental biology. Despite recent breakthroughs in the microscopy imaging of living systems, producing an accurate cell lineage tree for any developing organism remains a difficult task. We present here the BioEmergences workflow integrating all reconstruction steps from image acquisition and processing to the interactive visualization of reconstructed data. Original mathematical methods and algorithms underlie image filtering, nucleus centre detection, nucleus and membrane segmentation, and cell tracking. They are demonstrated on zebrafish, ascidian and sea urchin embryos with stained nuclei and membranes. Subsequent validation and annotations are carried out using Mov-IT, a custom-made graphical interface. Compared with eight other software tools, our workflow achieved the best lineage score. Delivered in standalone or web service mode, BioEmergences and Mov-IT offer a unique set of tools for in silico experimental embryology.
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Papers by Olga Stašová