# Coherence Enhancement Based on Recursive Anisotropic Scale-Space with Adaptive Kernels
## 1 Introduction  
The extraction of significant features from images plays an important role in a variety of applications ranging from low-level image processing to high-level computer vision techniques. The reconstruction of objects of interest is often required to interpret semantic structure from visual information. Image formation is, in general, a complex process involving the interplay between the physical environment and observation conditions. However, it is often necessary to discard nuisance factors such as undesirable intensity perturbations and geometrical distortions. This naturally leads to the analysis of imagery at multiple scales based on scale-space theory [1; 2; 3; 4]. Scale-space theory provides an analytic framework in which a signal is represented at multiple scales constructed by a family of scale parameters. A typical scale-space representation is the Gaussian scale-space developed based on the solution of the heat equation [5; 6; 7; 8]. This linear scale-space has numerous attractive properties, namely, scale-space axioms that include linearity, semi-group structure, existence of an infinitesimal generator, non-creation of local extrema, rotational symmetry, and scale invariance [9; 10; 11; 12]. Several scale-space algorithms that employ wavelets [13; 14; 15] and a variety of kernels such as local phases forming monogenic signals [16] and Beltrami flows [17] were proposed. The analysis of visual information via scale-space is often beneficial because image features at a variety of scales coexist in a single image. It is desirable for features at a specific scale to be characterized at the corresponding scale. However, the feature scales are generally not known in advance for correspondences to be established between the scales and features at the characterization stage. This necessitates an alternative approachin which the analysis and feature characterization are performed in an alternative manner. One of the major drawbacks of linear scale-space approaches is the dislocation of significant image features such as edges and corners. This occurs because insignificant perturbations in appearance are suppressed while the scale increases. There is a trade-off between the degree of geometrical feature distortions and robustness against insignificant perturbations in scale-space construction [18]. It is often desirable to suppress nuisance factors such as noise while preserving geometrical properties that serve as characteristic indicators for recognizing objects of interest [19]. Thus, there have been several of proposals for overcoming the limitations arising from the isotropy of linear operations during the construction of scale-space by using anisotropic operations [20; 21; 22; 23; 24; 25] in which the diffusion process is designed to consider local geometrical properties such as the direction and contrast of linear features. In addition to the linear features, curvilinear feature characterization based on structure tensors was developed; in this case, the orientations of the local features are used in an isotropic manner in the diffusion computation [26; 27; 28; 29]. Curvilinear feature characterization was applied to the detection of curvilinear structures such as vessels [30; 31; 32; 33], and for tubular structure segmentation [34] and fingerprint recognition [35; 36; 37]. Several approaches for anisotropic diffusion were developed based on structure-adaptive anisotropic filters [38; 39] and anisotropic transformations [40]. However, static kernels have still been employed in conventional anisotropic diffusion to compute structure tensors for measuring local feature orientations. This motivates our proposed anisotropic diffusion process in scale-space based on varying the kernel shapes. In this paper, we propose an anisotropic scale-space based on data-driven kernels that adjust their sizes and shapes adaptively. The diffusion process is performed recursively based on the modified kernels, which are designed to consider the orientations of local geometrical features.