Morphological Scale Space Research Articles

Finite Volume Schemes for the Affine Morphological Scale Space (Amss) Model

Abstract Finite volume (FV) numerical schemes for the approximation of Affine Morphological Scale Space (AMSS) model are proposed. For the scheme parameter θ, 0 ≤ θ ≤ 1 the numerical schemes of Crank-Nicolson type were derived. The explicit (θ = 0), semi-implicit, fully-implicit (θ = 1) and Crank-Nicolson (θ = 0.5) schemes were studied. Stability estimates for explicit and implicit schemes were derived. On several numerical experiments the properties and comparison of the numerical schemes are presented.

Finite Volume Scheme for AMSS Model

Abstract We propose a new finite volume numerical scheme for the approximation of the Affine Morphological Scale Space (AMSS) model. We derive the basic scheme and its iterative improvement. For both schemes, several numerical experiments using examples where the exact solution is known are presented. Then the numerical errors and experimental order of convergence of the proposed schemes is studied.

Local morphological pattern: A scale space shape descriptor for texture classification

This work sheds light on real world texture classification problem where textural information present at several scales as images are captured from different distances and zooming conditions. Inspired by the concept of mathematical morphology in image analysis, we developed a novel statistical approach to texture representation, which yields more discriminative, simple, efficient, yet robust descriptor called local morphology pattern (Lmp). The proposed approach consists of two parts: first it generates a morphological scale space representation of the texture image, called local shape vector (Lsv) using two commonly used powerful morphology operations namely, “opening” and “closing” at different scale. It perfectly localizes and well preserves the contours information of the objects. Then Lmp is formed by utilizing the relation of center pixel with its neighboring pixels in the scale space. The encoding of Lmp is done via uniform pattern scheme. The normalized histogram of Lmp (Hlmp) is computed by concatenating the histograms of Lmp at different scales. It is observed that Lmp is invariant under local Lipschitz transform and its extension is adequate to precisely differentiate between the fundamental texture primitives. Lipschitz is a very conventional transform which incorporates rotation, translation, projective transformation and also changes in view-point. Extensive experiments of texture classification on the benchmark texture databases Outex_TC-00010 (Outex_TC10), Outex_TC-00012 (Outex_TC12), KTH-TIPS, and UIUC validate that the proposed Lmp descriptor has better or comparable performance with well-known and state-of-the-art methods.

Automated Reconstruction of Building LoDs from Airborne LiDAR Point Clouds Using an Improved Morphological Scale Space

Reconstructing building models at different levels of detail (LoDs) from airborne laser scanning point clouds is urgently needed for wide application as this method can balance between the user’s requirements and economic costs. The previous methods reconstruct building LoDs from the finest 3D building models rather than from point clouds, resulting in heavy costs and inflexible adaptivity. The scale space is a sound theory for multi-scale representation of an object from a coarser level to a finer level. Therefore, this paper proposes a novel method to reconstruct buildings at different LoDs from airborne Light Detection and Ranging (LiDAR) point clouds based on an improved morphological scale space. The proposed method first extracts building candidate regions following the separation of ground and non-ground points. For each building candidate region, the proposed method generates a scale space by iteratively using the improved morphological reconstruction with the increase of scale, and constructs the corresponding topological relationship graphs (TRGs) across scales. Secondly, the proposed method robustly extracts building points by using features based on the TRG. Finally, the proposed method reconstructs each building at different LoDs according to the TRG. The experiments demonstrate that the proposed method robustly extracts the buildings with details (e.g., door eaves and roof furniture) and illustrate good performance in distinguishing buildings from vegetation or other objects, while automatically reconstructing building LoDs from the finest building points.

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

It is well-known that there are striking analogies between linear shift-invariant systems and morphological systems for image analysis. So far, however, the relations between both system theories are mainly understood on a pure convolution / erosion level. A formal connection on the level of differential or pseudodifferential equations and their induced scale-spaces is still missing. The goal of our paper is to close this gap. We present a simple and fairly general dictionary that allows to translate any linear shift-invariant evolution equation into its morphological counterpart and vice versa. It is based on a scale-space representation by means of the symbol of its (pseudo)differential operator. Introducing a novel transformation, the Cramer---Fourier transform, puts us in a position to relate the symbol to the structuring function of a morphological scale-space of Hamilton---Jacobi type. As an application of our general theory, we derive the morphological counterparts of many linear shift-invariant scale-spaces, such as the Poisson scale-space, $$\alpha $$ź-scale-spaces, summed $$\alpha $$ź-scale-spaces, relativistic scale-spaces, and their anisotropic variants. Our findings are illustrated by experiments.

Generalised morphological image diffusion

Relationships between linear and morphological scale-spaces have been considered by various previous works. The aim of this paper is to study how to generalise the discrete and continuous diffusion-based approaches in order to introduce nonlinear filters whose limit effects mimic the asymmetric behaviour of morphological dilation and erosion, as well as other multiscale filters, hybrid between the standard linear and morphological filters. A methodology based on the counter-harmonic mean is adopted here. Partial differential equations are formulated and details of numerical implementation are discussed to illustrate the various studied cases: isotropic, edge-preserving and coherence-enhancing diffusion. We also found a new way to derive the classical link between Gaussian scale-space and dilation/erosion scale-spaces based on quadratic structuring functions in the discrete and continuous setting. We have included some preliminary applications of the generalised morphological diffusion to solve image processing problems such as denoising and image enhancement in the case of asymmetric bright/dark image properties.

Shape recognition using orientational and morphological scale‐spaces of curvatures

In this study, a scale‐invariant representation for closed planar curves (silhouettes) is proposed. The orientations of all points within the Gaussian scale‐space of the curve are extracted. This orientation scale‐space is used to create the silhouette orientation image in which the positions of each pixel indicate the curve's pixel positions and scales, whereas the colour represents orientation. The representation is extracted for multiple levels of the morphological scale‐space of the silhouette. The proposed representation is invariant to scale and transformable under planar rotation. Using linear and non‐linear distance learning methods, experiments on the MPEG7, ETH80 and Kimia shape datasets were conducted, with results indicating an advanced recognition capability.

Mitosis Detection for Invasive Breast Cancer Grading in Histopathological Images.

Histopathological grading of cancer not only offers an insight to the patients' prognosis but also helps in making individual treatment plans. Mitosis counts in histopathological slides play a crucial role for invasive breast cancer grading using the Nottingham grading system. Pathologists perform this grading by manual examinations of a few thousand images for each patient. Hence, finding the mitotic figures from these images is a tedious job and also prone to observer variability due to variations in the appearances of the mitotic cells. We propose a fast and accurate approach for automatic mitosis detection from histopathological images. We employ area morphological scale space for cell segmentation. The scale space is constructed in a novel manner by restricting the scales with the maximization of relative-entropy between the cells and the background. This results in precise cell segmentation. The segmented cells are classified in mitotic and non-mitotic category using the random forest classifier. Experiments show at least 12% improvement in F1 score on more than 450 histopathological images at 40× magnification.

Hybrid ensemble of classifiers for logo and trademark symbols recognition

Abstract The paper presents a hybrid ensemble of diverse classifiers for logo and trademark symbols recognition. The proposed ensemble is composed of four types of different member classifiers. The first one compares color distribution of the logo patterns and is responsible for sifting out images of different color distribution. The second of the classifiers is based on the structural tensor recognition of local phase histograms. A proposed modification in this module consists of tensor computation in the space of the morphological scale-space. Thanks to this, more discriminative histograms describing global shapes are obtained. Next in the chain, is a novel member classifier that joins the Hausdorff distance with the correspondence measure of the log-polar patches computed around the corner points. This sparse classifier allows reliable comparison of even highly deformed patterns. The last member classifier relies on the statistical affine moment invariants which describe global shapes. However, a real advantage is obtained by joining the aforementioned base classifiers into a hybrid ensemble of classifiers, as proposed in this paper. Thanks to this a more accurate response and generalizing properties are obtained at reasonable computational requirements. Experimental results show good recognition accuracy even for the highly deformed logo patterns, as well as fair generalization properties which support human search and logo assessment tasks.

Multiscale Analysis of Similarities between Images on Riemannian Manifolds

In this paper we study the problem of comparing two patches of an image defined on a Riemannian manifold, which can be defined by the image domain with a suitable metric depending on the image. The size of the patch will not be determined a priori, and we identify it with a variable scale. Our approach can be considered as a nonlocal extension (comparing two points) of the multiscale analyses defined using the axiomatic approach by Álvarez et al. [Arch. Ration. Mech. Anal., 123 (1993), pp. 199--257]. Following this axiomatic approach, we can define a set of similarity measures that appear as solutions of a degenerate partial differential equation. This equation can be further specified in the linear case, and we observe that it contains as a particular instance the case of using weighted Euclidean distances as comparison measures. Finally, we discuss the case of some morphological scale spaces that exhibit a higher complexity.

Multiscale Analysis for Images on Riemannian Manifolds

In this paper we study multiscale analyses for images defined on Riemannian manifolds and extend the axiomatic approach proposed by Álvarez, Guichard, Lions, and Morel to this general case. This covers the case of two- and three-dimensional images and video sequences. After obtaining the general classification, we consider the case of morphological scale spaces, which are given in terms of geometric equations, and the linear case given by the Laplace--Beltrami flow. We consider in some detail the case of image metrics given in terms of the structure tensor and compute some cases of such a tensor for video. Then we comment on the connections with variational formulations of image diffusion comparing the anisotropies that appear. Finally, we include numerical experiments illustrating some of the models. Namely, we compare some examples for still images using the Laplace--Beltrami flow and some variational models. We also consider several examples in video: the mean curvature motion and the extension of the morphological and Galilean invariant scale spaces to the video manifold case, and the Laplace--Beltrami flow. We point out that the number of models that appear is huge, and we have restricted ourselves to such cases for the sake of brevity and illustration.

Erratum to: Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI

The online version of the original article can be found under doi: http://dx.doi.org/10.1007/s10851-012-0387-2

Morphological and Linear Scale Spaces for Fiber Enhancement in DW-MRI

We consider morphological and linear scale spaces on the space ℝ3⋊S 2 of 3D positions and orientations naturally embedded in the group SE(3) of 3D rigid body movements. The general motivation for these (convection-)diffusions and erosions is to obtain crossing-preserving fiber enhancement on probability densities defined on the space of positions and orientations. The strength of these enhancements is that they are expressed in a moving frame of reference attached to fiber fragments, allowing us to diffuse along the fibers and to erode orthogonal to them. The linear scale spaces are described by forward Kolmogorov equations of Brownian motions on ℝ3⋊S 2 and can be solved by convolution with the corresponding Green’s functions. The morphological scale spaces are Bellman equations of cost processes on ℝ3⋊S 2 and we show that their viscosity solutions are given by a morphological convolution with the corresponding morphological Green’s function. For theoretical underpinning of our scale spaces on ℝ3⋊S 2 we introduce Lagrangians and Hamiltonians on ℝ3⋊S 2 indexed by a parameter η∈[1,∞). The Hamiltonian induces a Hamilton-Jacobi-Bellman system that coincides with our morphological scale spaces on ℝ3⋊S 2. By means of the logarithm on SE(3) we provide tangible estimates for both the linear- and the morphological Green’s functions. We also discuss numerical finite difference upwind schemes for morphological scale spaces (erosions) of Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI), which allow extensions to data-adaptive erosions of DW-MRI. We apply our theory to the enhancement of (crossing) fibres in DW-MRI for imaging water diffusion processes in brain white matter.

A Semi-Lagrangian approximation for the AMSS model of image processing

Among the various nonlinear image filtering techniques, the Affine Morphological Scale Space (AMSS) model is characterized by especially appealing theoretical properties. In this paper we propose a Semi-Lagrangian scheme to approximate the AMSS model, prove convergence of its monotone version and apply it to some typical problem of image filtering. We also carry out a comparison with the Level Lines Affine Shortening implementation of the AMSS model, as well as with other nonlinear filtering techniques, namely the Mean Curvature equation and the Perona–Malik model.

Context-based mapping of damaged buildings from high-resolution optical satellite images

In the early stages of post-disaster response, a quick and reliable damage assessment map is essential. As time is a critical factor, automated damage mapping from remotely sensed images is the expected solution to drastically reduce data acquisition and computation time. Recently, high-resolution satellite images, such as QuickBird data, have been in high demand by damage assessment analysts and disaster management practitioners. However, the existing automated mapping approaches hardly accommodate such high-resolution data. This research aims at developing a new context-based automated approach for earthquake damage mapping from high-resolution satellite images. Relevant contextual information (including structure, shape, size, edge texture, spatial relations) describing the damage situation is formulated and up-scaled on a morphological scale-space. Speed optimization is achieved by parallel processing implementation. The developed approach was tested with two QuickBird images acquired on 26 June 2005 and 3 June 2008 over YingXiu town, Sichuan, China, which suffered the devastating 12 May 2008 earthquake. In comparison to the reference, the developed mapping approach could achieve over 80% accuracy for computation of the damage ratio. Future research is planned to test the approach on various disaster cases for both optical and radar images using a grid-computing platform towards a cost-effective damage mapping solution.

Texture classification via morphological scale-space: Tex-Mex features

We consider the problem of classifying textures. First, we consider images where the orientation of the texture is known. Then, we consider the classification of textures where the orientation is unknown. Last, classification in real scenes is considered. A wide variety of techniques are tested using the Outex framework. We introduce a new grayscale multiscale texture classification method based on a class of morphological filters called sieves. The method, denoted Tex-Mex because it extracts TEXture features using Morphological EXtrema filters, is shown to be among the best performing texture feature extraction methods. Tex-Mex features can be computed rapidly and are shown to be more robust and compact than the alternatives. Furthermore, they may be applied over windows of arbitrary size and orientation, a useful attribute when classifying texture in real scenes.

An approach to extracting information of residential areas from Beijing-1 image based on Gabor texture segmentation

According to the features of high-resolution panchromatic imagery of Beijing-1 small satellite, an approach to extracting information of residential areas is proposed in this paper based on Gabor texture segmentation. The algorithm extracts the features in different directions and different scales by building the Gabor filter, uses cluster analysis of multiple features to segment the image, and performs the fusion processing based on morphological scale space. It solves the problems in image processing resulting from low contrast between remote sensing objects and background, the blurring of image edges and high noise. It has the benefits of direction selection and frequency selection with strong self-adaptive ability. Our experiments prove the effectiveness of the approach for extracting information of residential areas from Beijing-1 high-resolution imagery.

Multi-scale solution for building extraction from LiDAR and image data

This paper presents a multi-scale solution based on mathematical morphology for extracting the building features from remotely sensed elevation and spectral data. Elevation data are used as the primary data to delineate the structural information and are firstly represented on a morphological scale-space. The behaviors of elevation clusters across the scale-space are the cues for feature extraction. As a result, a complex structure can be extracted as a multi-part object in which each part is represented on a scale depending on its size. The building footprint is represented by the boundary of the largest part. Other object attributes include the area, height or number of stories. The spectral data is used as an additional source to remove vegetation and possibly classify the building roof material. Finally, the results can be stored in a multi-scale database introduced in this paper. The proposed solution is demonstrated using the data derived from a Light Detection And Ranging (LiDAR) surveying flight over Tokyo, Japan. The results show a reasonable match with reference data and prove the capability of the proposed approach in accommodation of diverse building shapes. Higher density LiDAR is expected to produce better accuracy in extraction, and more spectral sources are necessary for further classification of building roof material. It is also recommended that parallel processing should be implemented to reduce the computation time.

Adaptive multi-scale AMSS operator for quality detection of silks

In order to de-noise the silk images corrupted by various noises, a new improved filter based on affine morphological scale space (AMSS) equation was proposed. In this new operator both the adaptive idea and scale space were fused into partial differential equations. Scale parameters were changed adaptively according to the presence or absence of noise in image which was detected by directional gradient operators with four templates. A new evaluating criterion was constructed to test the performance of the improved AMSS operators. Two parameters based on mean square error and minimum absolute error could be acquired to evaluate the effect of keeping edge information and the ability of smoothing noise. Experimental results illustrate that this improved operator can efficiently keep edge information, and the de-noising performance is more effective than other methods for the image with multi-noise.

Morphological multiscale decomposition of connected regions with emphasis on cell clusters

After binarization of cells in complex cytological and histological images the segmented structures can be rather far away from a final quantification of features of single cells since cells may overlap and cluster strongly. Separating optically, partially or totally fused entities like cells is a problem which frequently cannot be solved by a watershed segmentation or a basic morphological processing of images. However, considering different morphological scales after iterative erosion gives rise to dominant markers of singular objects. Performing a reconstruction by iterative dilation yields a scale-independent decomposition of multiple disjointed cell clumps of different sizes within an image. Accordingly we developed a technique that splits cell clumps into meaningful parts. Since this method is based on the analysis of the morphological-scale space, generated by iterative erosion, it is independent on the size of cell clusters. The detection of dominant points within the eroded scales are cell-specific markers. The converse integration of markers at different scales is obtained by a successive reconstruction based on constrained dilation of the original cell shape. The advantages of this approach are the independence of cell shapes which are clumped, the consideration of holes or background intensities within objects and the robustness with regard to convergence. An important benefit is the fitting of the operation time to the size of clusters by the size of the morphological structuring element. This means, that this approach requires only one parameter. Finally, a better match of the morphological scale space approach was found and compared with the ground truth as well as the results of the watershed technique. The primary object of this paper is to highlight the algorithm and its results by using different examples from benchmark databases, self generated images that exhibit different topological features and complex configurations of cells within histological images.