We present a new approach to approximate continuous-domain mathematical morphology operators. The approach is applicable to irregularly sampled signals. We define a dilation under this new approach, where samples are duplicated and shifted according to the flat, continuous structuring element. We define the erosion by adjunction, and the opening and closing by composition. These new operators will significantly increase precision in image measurements. Experiments show that these operators indeed approximate continuous-domain operators better than the standard operators on sampled one-dimensional signals, and that they may be applied to signals using structuring elements smaller than the distance between samples. We also show that we can apply the operators to scan lines of a two-dimensional image to filter horizontal and vertical linear structures.

The path opening is a filter that preserves bright regions in the image in which a path of a certain length L fits. A path is a (not necessarily straight) line defined by a specific adjacency relation. The most efficient implementation known scales as O(min(L, d, Q) N) with the length of the path, L , the maximum possible path length, d , the number of graylevels, Q , and the image size, N . An approximation exists (parsimonious path opening) that has an execution time independent of path length. This is achieved by preselecting paths, and applying 1D openings along these paths. However, the preselected paths can miss important structures, as described by its authors. Here, we propose a different approximation, in which we preselect paths using a grayvalue skeleton. The skeleton follows all ridges in the image, meaning that no important line structures will be missed. An H-minima transform simplifies the image to reduce the number of branches in the skeleton. A graph-based version of the traditional path opening operates only on the pixels in the skeleton, yielding speedups up to one order of magnitude, depending on image size and filter parameters. The edges of the graph are weighted in order to minimize bias. Experiments show that the proposed algorithm scales linearly with image size, and that it is often slightly faster for longer paths than for shorter paths. The algorithm also yields the most accurate results-as compared with a number of path opening variants-when measuring length distributions.

The recent development of methods for extracting precise measurements of spatial gene expression patterns from three-dimensional (3D) image data opens the way for new analyses of the complex gene regulatory networks controlling animal development. We present an integrated visualization and analysis framework that supports user-guided data clustering to aid exploration of these new complex data sets. The interplay of data visualization and clustering-based data classification leads to improved visualization and enables a more detailed analysis than previously possible. We discuss 1) the integration of data clustering and visualization into one framework, 2) the application of data clustering to 3D gene expression data, 3) the evaluation of the number of clusters k in the context of 3D gene expression clustering, and 4) the improvement of overall analysis quality via dedicated postprocessing of clustering results based on visualization. We discuss the use of this framework to objectively define spatial pattern boundaries and temporal profiles of genes and to analyze how mRNA patterns are controlled by their regulatory transcription factors.

Path openings and closings are morphological operations with flexible line segments as structuring elements. These line segments have the ability to adapt to local image structures, and can be used to detect lines that are not perfectly straight. They also are a convenient and efficient alternative to straight line segments as structuring elements when the exact orientation of lines in the image is not known. These path operations are defined by an adjacency relation, which typically allows for lines that are approximately horizontal, vertical or diagonal. However, because this definition allows zig-zag lines, diagonal paths can be much shorter than the corresponding horizontal or vertical paths. This undoubtedly causes problems when attempting to use path operations for length measurements. This paper 1) introduces a dimensionality-independent implementation of the path opening and closing algorithm by Appleton and Talbot, 2) proposes a constraint on the path operations to improve their ability to perform length measurements, and 3) shows how to use path openings and closings in a granulometry to obtain the length distribution of elongated structures directly from a gray-value image, without a need for binarizing the image and identifying individual objects.

During animal development, complex patterns of gene expression provide positional information within the embryo. To better understand the underlying gene regulatory networks, the Berkeley Drosophila Transcription Network Project (BDTNP) has developed methods that support quantitative computational analysis of three-dimensional (3D) gene expression in early Drosophila embryos at cellular resolution. We introduce PointCloudXplore (PCX), an interactive visualization tool that supports visual exploration of relationships between different genes' expression using a combination of established visualization techniques. Two aspects of gene expression are of particular interest: 1) gene expression patterns defined by the spatial locations of cells expressing a gene and 2) relationships between the expression levels of multiple genes. PCX provides users with two corresponding classes of data views: 1) Physical Views based on the spatial relationships of cells in the embryo and 2) Abstract Views that discard spatial information and plot expression levels of multiple genes with respect to each other. Cell Selectors highlight data associated with subsets of embryo cells within a View. Using linking, these selected cells can be viewed in multiple representations. We describe PCX as a 3D gene expression visualization tool and provide examples of how it has been used by BDTNP biologists to generate new hypotheses.

To allow a more rigorous understanding of animal gene regulatory networks, the Berkeley Drosophila Transcription Network Project (BDTNP) has developed a suite of methods that support quantitative, computational analysis of three-dimensional (3D) gene expression patterns with cellular resolution in early Drosophila embryos. Here we report the first components of a visualization tool, PointCloudXplore, that allows the relationships between different gene’s expression to be analyzed using the BDTNP’s datasets. PointCloudXplore uses the established visualization techniques of multiple views, brushing, and linking to support the analysis of high-dimensional datasets that describe many genes’ expression. Each of the views in PointCloud- Xplore shows a different gene expression data property. Brushing is used to select and emphasize data associated with defined subsets of embryo cells within a view. Linking is used to show in additional views the expression data for a group of cells that have first been highlighted as a brush in a single view, allowing further data subset properties to be determined. In PointCloudXplore, physical views of the data are linked to parallel coordinates. Physical views show the spatial relationships between different genes’ expression patterns within the embryo. Parallel coordinates, on the other hand, show only some features of each gene’s expression, but allow simultaneous analysis of data for many more genes than would be possible in a physical view. We have developed several extensions to standard parallel coordinates to facilitate brushing the visualization of 3D gene expression data.

The Berkeley DrosophilaTranscription Network Project (BDTNP) is developing a suite of methods that will allow a quantitative description and analysis of three dimensional (3D) gene expression patterns in an animal with cel- lular resolution. An important component of this approach are algorithms that segment 3D images of an organism into individual nuclei and cells and measure relative levels of gene expression. As part of the BDTNP, we are devel- oping tools for interactive visualization, control, and verification of these algorithms. Here we present a volume visualization prototype system that, combined with user interaction tools, supports validation and quantitative determination of the accuracy of nuclear segmentation. Visualizations of nuclei are combined with information obtained from a nuclear segmentation mask, supporting the comparison of raw data and its segmentation. It is possible to select individual nuclei interactively in a volume rendered image and identify incorrectly segmented objects. Integration with segmentation algorithms, implemented in MATLAB, makes it possible to modify a segmentation based on visual examination and obtain additional information about incorrectly segmented objects. This work has already led to significant improvements in segmentation accuracy and opens the way to enhanced analysis of images of complex animal morphologies.

When performing measurements in digitized images, the pixel pitch does not necessarily limit the attainable accuracy. Proper sampling of a band-limited continuous-domain image preserves all information present in the image prior to digitization. It is therefore (theoretically) possible to obtain measurements from the digitized image that are identical to measurements made in the continuous domain. Such measurements are sampling invariant, since they are independent of the chosen sampling grid. It is impossible to attain strict sampling invariance for filters in mathematical morphology due to their nonlinearity, but it is possible to approximate sampling invariance with arbitrary accuracy at the expense of additional computational cost. In this paper, we study morphological filters with line segments as structuring elements. We present a comparison of three known and three new methods to implement these filters. The method that yields a good compromise between accuracy and computational cost employs a (subpixel) skew to the image, followed by filtering along the grid axes using a discrete line segment, followed by an inverse skew. The staircase approximations to line segments under random orientations can be modeled by skewing a horizontal or vertical line segment. Rather than skewing the binary line segment we skew the image data, which substantially reduces quantization error. We proceed to determine the optimal number of orientations to use when measuring the length of line segments with unknown orientation.