White matter tractography mapping is an important tool for neuro-surgical planning and navigation. It relies on the accurate manual delineation of anatomical seeding ROIs (sROIs) by neuroanatomy experts. Stringent pre-operative time-constraints and limited availability of experts suggest that automation tools are strongly needed for the task. In this article, we propose and compare several multi-modal fully convolutional network architectures for segmentation of sROIs. Inspired by their manual segmentation practice, anatomical information from T1w maps is fused by the network with directionally encoded color (DEC) maps to compute the segmentation. Qualitative and quantitative validation was performed on image data from 75 real tumor resection candidates for the sROIs of the motor tract, the arcuate fasciculus, and optic radiation. Favorable comparison was also obtained with state-of-the-art methods for the tumor dataset as well as the ISMRM 2017 traCED challenge dataset. The proposed networks showed promising results, indicating they may significantly improve the efficiency of pre-surgical tractography mapping, without compromising its quality.

We address the problem of people detection in top-view fisheye imaging. Even within the same top-view fisheye frame, upright people appear slanted in various directions and are distorted in different ways. Due to this variability, standard people detectors are not directly applicable to top-view fisheye frames, and dedicated people detectors for the top-view fisheye domain are hard to design. We extract features in the fisheye frame, unwrap them to a perspective-like feature map, and forward them to a people detector designed and trained for common perspective images. Extracting the features before unwrapping prevents harmful smoothing of the gradient information. To facilitate feature unwrapping, we employ dense feature extraction and compute the unwrapping Jacobian. Distortion-free unwrapping is known to be impossible. We optimize the unwrapping method for the explicit goal of people detection performance. Applying a tunable fisheye camera model to project the fisheye image plane onto a unit half sphere, followed by the stereographic map projection, we obtain people detection performance similar to the standard perspective case. We complete the solution by introducing a convenient purposive fisheye-camera calibration process, optimized for subsequent people detection performance.

The never-ending quest for lower radiation exposure is a major challenge to the image quality of advanced CT scans. Post-processing algorithms have been recently proposed to improve low-dose CT denoising after image reconstruction. In this work, a novel algorithm, termed the locally-consistent non-local means (LC-NLM), is proposed for this challenging task. By using a database of high-SNR CT patches to filter noisy pixels while locally enforcing spatial consistency, the proposed algorithm achieves both powerful denoising and preservation of fine image details. The LC-NLM is compared both quantitatively and qualitatively, for synthetic and real noise, to state-of-the-art published algorithms. The highest structural similarity index (SSIM) were achieved by LC-NLM in 8 out of 10 denoised chest CT volumes. Also, the visual appearance of the denoised images was clearly better for the proposed algorithm. The favorable comparison results, together with the computational efficiency of LC-NLM makes it a promising tool for low-dose CT denoising. These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Demosaicing is an algorithm used to reconstruct a color image from the incomplete color samples of a color filter array (CFA). Most demosaicing algorithms can be broadly classified into spatial-domain and frequency-domain approaches. Despite significant progress in the past decade, current state of the art demosaicing algorithms still tend to produce artifacts at high-saturation edges. In this paper we propose a new approach to demosaicing - example based. Comparative experimental evaluation shows that example-based demosaicing (EBD) produces visually superior, artifact-free results.

We present a novel method for recovering the 3D structure and scene flow from calibrated multi-view sequences. We propose a 3D point cloud parametrization of the 3D structure and scene flow that allows us to directly estimate the desired unknowns. A unified global energy functional is proposed to incorporate the information from the available sequences and simultaneously recover both depth and scene flow. The functional enforces multi-view geometric consistency and imposes brightness constancy and piece-wise smoothness assumptions directly on the 3D unknowns. It inherently handles the challenges of discontinuities, occlusions, and large displacements. The main contribution of this work is the fusion of a 3D representation and an advanced variational framework that directly uses the available multi-view information. The minimization of the functional is successfully obtained despite the non-convex optimization problem. The proposed method was tested on real and synthetic data.

We introduce a novel variational method for the extraction of objects with either bilateral or rotational symmetry in the presence of perspective distortion. Information on the symmetry axis of the object and the distorting transformation is obtained as a by--product of the segmentation process. The key idea is the use of a flip or a rotation of the image to segment as if it were another view of the object. We call this generated image the symmetrical counterpart image. We show that the symmetrical counterpart image and the source image are related by planar projective homography. This homography is determined by the unknown planar projective transformation that distorts the object symmetry. The proposed segmentation method uses a level-set-based curve evolution technique. The extraction of the object boundaries is based on the symmetry constraint and the image data. The symmetrical counterpart of the evolving level-set function provides a dynamic shape prior. It supports the segmentation by resolving possible ambiguities due to noise, clutter, occlusions, and assimilation with the background. The homography that aligns the symmetrical counterpart to the source level-set is recovered via a registration process carried out concurrently with the segmentation. Promising segmentation results of various images of approximately symmetrical objects are shown.

Video surveillance systems produce huge amounts of data for storage and display. Long-term human monitoring of the acquired video is impractical and ineffective. Automatic abnormal motion detection system which can effectively attract operator attention and trigger recording is therefore the key to successful video surveillance in dynamic scenes, such as airport terminals. This paper presents a novel solution for real-time abnormal motion detection. The proposed method is well-suited for modern video-surveillance architectures, where limited computing power is available near the camera for compression and communication. The algorithm uses the macroblock motion vectors that are generated in any case as part of the video compression process. Motion features are derived from the motion vectors. The statistical distribution of these features during normal activity is estimated by training. At the operational stage, improbable-motion feature values indicate abnormal motion. Experimental results demonstrate reliable real-time operation.

We consider the problem of restoring a multichannel image corrupted by blur and impulsive noise (e.g., salt-and-pepper noise). Using the variational framework, we consider the L1 fidelity term and several possible regularizers. In particular, we use generalizations of the Mumford-Shah (MS) functional to color images and gamma-convergence approximations to unify deblurring and denoising. Experimental comparisons show that the MS stabilizer yields better results with respect to Beltrami and total variation regularizers. Color edge detection is a beneficial by-product of our methods.

Shape symmetry is an important cue for image understanding. In the absence of more detailed prior shape information, segmentation can be significantly facilitated by symmetry. However, when symmetry is distorted by perspectivity, the detection of symmetry becomes non-trivial, thus complicating symmetry-aided segmentation. We present an original approach for segmentation of symmetrical objects accommodating perspective distortion. The key idea is the use of the replicative form induced by the symmetry for challenging segmentation tasks. This is accomplished by dynamic extraction of the object boundaries, based on the image gradients, gray levels or colors, concurrently with registration of the image symmetrical counterpart (e.g. reflection) to itself. The symmetrical counterpart of the evolving object contour supports the segmentation by resolving possible ambiguities due to noise, clutter, distortion, shadows, occlusions and assimilation with the background. The symmetry constraint is integrated in a comprehensive level-set functional for segmentation that determines the evolution of the delineating contour. The proposed framework is exemplified on various images of skewsymmetrical objects and its superiority over state of the art variational segmentation techniques is demonstrated.

Image restoration and segmentation are both classical problems, that are known to be difficult and have attracted major research efforts. This paper shows that the two problems are tightly coupled and can be successfully solved together. Mutual support of image restoration and segmentation processes within a joint variational framework is theoretically motivated, and validated by successful experimental results. The proposed variational method integrates semi-blind image deconvolution (parametric blur-kernel), and Mumford-Shah segmentation. The functional is formulated using the T-convergence approximation and is iteratively optimized via the alternate minimization method. While the major novelty of this work is in the unified treatment of the semi-blind restoration and segmentation problems, the important special case of known blur is also considered and promising results are obtained.

The mouse brain library (MBL) is a database of brain images, each consisting of sparse coronal or horizontal sections. To facilitate morphometric research, it is necessary to index each of these sections by its location within a canonical 3D atlas (NeuroTerrain). This is done with a 2D to 3D matching technique which was developed in this study. The registration method is imaged-based and uses a genetic algorithm to find the local-affine transformation that maximizes a mutual information metric. The average distance between the registration results achieved with the proposed method as compared with manual matching by an expert was 250 microns. This compares well with repeated manual trials where inter-trial matching distance was on average 200 microns.

Object detection and segmentation can be facilitated by the availability of a reference object. However, accounting for possible transformations between the different object views, as part of the segmentation process, remains a challenge. Recent works address this problem by using comprehensive training data. Other approaches are applicable only to limited object classes or can only accommodate similarity transformations. We suggest a novel variational approach to prior-based segmentation, which accounts for planar projective transformation, using a single reference object. The prior shape is registered concurrently with the segmentation process, without point correspondence. The algorithm detects the object of interest and correctly extracts its boundaries. The homography between the two object views is accurately recovered as well. Extending the Chan-Vese level set framework, we propose a region-based segmentation functional that includes explicit representation of the projective homography between the prior shape and the shape to segment. The formulation is derived from two-view geometry. Segmentation of a variety of objects is demonstrated and the recovered transformation is verified.

Photometric stereo is a fundamental approach in computer vision. At its core lies a set of image irradiance equations each taken with a different illumination. The vast majority of studies in this field have assumed orthography as the projection model. This paper re-examines the basic set of equations of photometric stereo, under an assumption of perspective projection. We show that the resulting system is linear (as is the case under the orthographic model; Nevertheless, the unknowns are different in the perspective case). We then suggest a simple reconstruction algorithm based on the perspective formulae, and compare it to its orthographic counterpart on synthetic as well as real images. This algorithm obtained lower error rates than the orthographic one in all of the error measures. These findings strengthen the hypothesis that a more realistic set of assumptions, the perspective one, improves reconstruction significantly.

We propose a new algorithm for dense optical flow computation. Dense optical flow schemes are challenged by the presence of motion discontinuities. In state of the art optical flow methods, over-smoothing of flow discontinuities accounts for most of the error. A breakthrough in the performance of optical flow computation has recently been achieved by Brox et al. Our algorithm embeds their functional within a contour-based segmentation framework. Piecewise-smooth flow fields are accommodated and flow boundaries are crisp. Experimental results show the superiority of our algorithm with respect to alternative techniques.

A method for representation and lossy compression of textured surfaces is presented. The input surfaces are represented by surfels (surface elements), i.e., by a set of colored, oriented, and sized disks. The position and texture of each surfel are mapped onto a sphere. The mapping is optimized for preservation of geodesic distances. The components of the resulting spherical vector-valued function are decorrelated by the Karhunen-Loeve transform and represented by spherical wavelets. Successful representation and reconstruction is demonstrated. Methods for geodesic distance computation on surfaces represented by surfels are presented.

A voxel-based method for flattening a surface in 3D space into 2D while best preserving distances is presented. Triangulation or polyhedral approximation of the voxel data are not required. The problem is divided into two main parts: Voxel-based calculation of the minimal geodesic distances between points on the surface and finding a configuration of points in 2D that has Euclidean distances as close as possible to these distances. The method suggested combines an efficient voxel-based hybrid distance estimation method, that takes the continuity of the underlying surface into account, with classical multidimensional scaling (MDS) for finding the 2D point configuration. The proposed algorithm is efficient, simple, and can be applied to surfaces that are not functions. Experimental results are shown.

Presents a novel technique for texture mapping on arbitrary surfaces with minimal distortion by preserving the local and global structure of the texture. The recent introduction of the fast marching method on triangulated surfaces has made it possible to compute a geodesic distance map from a given surface point in O(n lg n) operations, where n is the number of triangles that represent the surface. We use this method to design a surface flattening approach based on multi-dimensional scaling (MDS). MDS is a family of methods that map a set of points into a finite-dimensional flat (Euclidean) domain, where the only data given is the corresponding distance between every pair of points. The MDS mapping yields minimal changes of the distances between the corresponding points. We then solve an "inverse" problem and map a flat texture patch onto a curved surface while preserving the structure of the texture.

A computer-based method for the assessment of body-image distortions in anorexia nervosa and other eating-disorder patients is presented in this paper. At the core of the method is a realistic pictorial simulation of lifelike weight changes, applied to a real source image of the patient. The patients, using a graphical user interface, adjust their body shapes until they meet their self-perceived appearance. Measuring the extent of virtual fattening or slimming of a body with respect to its real shape and size allows direct quantitative evaluation of the cognitive distortion in body image. In a preliminary experiment involving 33 anorexia-nervosa patients, 70% of the subjects chose an image with simulated visual weight gain between 8%-16% as their "real" body image, while only one of them recognized the original body image. In a second experiment involving 30 healthy participants, the quality of the weight modified images was evaluated by pairwise selection trials. Over a weight change range from -16% to +28%, in about 30% of the trials, artificially modified images were mistakenly taken as "original" images, thus demonstrating the quality of the artificial images. The method presented is currently in a clinical validation phase, toward application in the research, diagnosis, evaluation, and treatment of eating disorders.

We present a method to recover scenes deteriorated by superposition of transparent and semi-reflected contributions, as appear in reflections of windows. Separating the superimposed contributions from the images in which either contribution is in focus is based on mutual blurring and subtraction of the perturbing components. This procedure requires the defocus blur kernels to be known. The use of uncalibrated kernels had previously led to contaminated results. We propose a method for self-calibration of the blur kernels from the raw images themselves. The kernels are sought to minimize the mutual information of the recovered layers. This relaxes the need for prior knowledge on the optical transfer function. Experimental results are presented.

Qjidaa and Radouane (1999) presented a method for robust line fitting and experimentally compared it to other methods, including a method suggested by us. The results attributed by Qjidaa and Radouane to our algorithm are incorrect. We apply our algorithm to the data used by Qjidaa and Radouane and demonstrate its robustness and accuracy.

We analyze the effect of perturbations on the estimation of Depth from Defocus (DFD) implemented by changing the focus setting (e.g., axially moving the sensor). The analysis yields the optimal change of focus setting, and the spatial frequencies for which estimation is most robust. For stable estimation at all spatial frequencies, the change in focus setting should be less than twice the depth of field. For the most robust estimation in the highest spatial frequencies the axial interval should be equal to the depth of field.

When a transparent surface is present between an observer and an object, an image reflected by the surface may be superimposed on the image of the observed object. We present a new approach to recover the scenes (layers) and to classify which is the reflected/transmitted one, based on imaging through a polarizing filter at two orientations. Estimates of the separate layers are obtained by weighted pixel-wise differences of these images, inverting the image formation process. However the weights depend on the angle of incidence, hence on the inclination of the transparent (invisible) surface. This angle is estimated by seeking the angle-value which (through the weights) leads to decorrelation of the estimated layers. Experimental results, obtained using real photos of actual objects, demonstrate the success of angle estimation and consequent layer separation and labeling. The method is shown to be superior to earlier methods where only raw optical data was used.

Consider situations where the depth at each point in the scene is multi-valued due to the presence of a virtual image semi-reflected by a transparent surface. The semi-reflected image is linearly superimposed on the image of the object that is behind the transparent surface. A novel approach is proposed for the recovery of the superimposed layers. By searching for the images in which either of the objects (layers) is focused, the transparent areas are detected and an estimate of the depth map of each layer is obtained. As a result of the focusing, an initial separation of the layers is achieved. The separation is enhanced via mutual blurring of the perturbing components in the images, based on the depths estimate and the parameters of the imaging system.

Depth from focus (DFF) and depth from defocus (DFD) methods are shown to be realizations of the geometric triangulation principle. Fundamentally, the depth sensitivities of DFF and DFD are not different than those of stereo (or motion) based systems having the same physical dimensions. Contrary to common belief DFD does not inherently avoid the matching (correspondence) problem. Basically DFD and DFF do not avoid the occlusion problem any more than triangulation techniques, but they are more stable in the presence of such disruptions. The fundamental advantage of DFF and DFD methods is the two-dimensionality of the aperture, allowing more robust estimation. These results elucidate the limitations of methods based on depth of field and provide a foundation for fair performance comparison between DFF/DFD and shape from stereo (or motion) algorithms.

In range sensing with time-multiplexed structured light, there is a trade-off between accuracy, robustness and the acquisition period. In this paper a novel structured light method is described. Adaptation of the number and form of the projection patterns to the characteristics of the scene takes place as part of the acquisition process. Noise margins are matched to the actual noise level, thus reducing the number of projection patterns to the necessary minimum. Color is used for light plane labeling. The dimension of the pattern space are thus increased without raising the number of projection patterns. It is shown that the color of an impinging light plane can be identified from the image of the illuminated scene, even with colorful scenes. Identification is local and does not rely on spatial color sequences. The suggested approach has been implemented and the theoretical results are supported by experiments.

This paper presents a new algorithm for planning the time-optimal motion of a robot travelling with limited velocity from a given location to a given destination on a surface in the presence of moving obstacles. Additional constraints such as space variant terrain traversability and fuel economy can be accommodated. A multivalued distance map is defined and applied in computing optimal trajectories. The multivalued distance map incorporates constraints imposed by the moving obstacles, surface topography, and terrain traversability. It is generated by an efficient numerical curve propagation technique.

A method for efficient detection of the dominant local reflectional symmetry in grey level images is described. The general approach is to define a local measure of reflectional symmetry that transforms the symmetry detection problem to an optimization problem, and obtain the symmetric regions by an efficient global optimization algorithm. The symmetry of a 1D function can be measured in the frequency domain as the fraction of its energy that resides in symmetric Fourier basis functions. This approach is extended to two dimensions. Locality can be formally treated in terms of the Gabor decomposition and implemented via soft windowing. The resulting measure is a complicated multimodal function of the location of the center of the supporting region, its size, and the orientation of the symmetry axis. A new probabilistic generic algorithm is applied to the determination of the global maximum of the reflectional symmetry function. Less than one thousand evaluations of the local symmetry measure are typically needed in order to locate the dominant symmetry in natural, wildlife test images.

Finding the shortest path between points on a surface is a challenging global optimization problem. It is difficult to devise an algorithm that is computationally efficient, locally accurate and guarantees to converge to the globally shortest path. In this paper a two stage coarse-to-fine approach for finding the shortest paths is suggested. In the first stage the algorithm of Ref. 10 that combines a 3D length estimator with graph search is used to rapidly obtain an approximation to the globally shortest path. In the second stage the approximation is refined to become a shorter geodesic curve, i.e., a locally optimal path. This is achieved by using an algorithm that deforms an arbitrary initial curve ending at two given surface points via geodesic curvature shortening flow. The 3D curve shortening flow is transformed into an equivalent 2D one that is implemented using an efficient numerical algorithm for curve evolution with fixed end points, introduced in Ref. 9.

This paper presents a new algorithm for planning the time-optimal motion of a robot traveling with limited velocity from a given location to a given destination on a surface in the presence of moving obstacles. Additional constraints such as space variant terrain traversability and fuel economy can be accommodated. A multilayer distance map is defined and applied in computing optimal trajectories. The multilayer distance map incorporates constraints imposed by the moving obstacles, surface topography and terrain traversability. It is generated by an efficient numerical curve propagation technique.

In probabilistic Hough transforms computation is accelerated by polling instead of voting. A small part of the data set is selected at random and used as input to the algorithm. Most probabilistic Hough algorithms use a fixed poll size. It has been experimentally demonstrated that adaptive termination of voting can lead to improved performance in terms of the error rate versus average poll size tradeoff. However, the lack of a solid theoretical foundation made general performance evaluation and optimal design of adaptive stopping rules nearly impossible. In this paper we suggest two novel adaptive stopping rules in the framework of the statistical theory of sequential hypothesis testing. The performance of the suggested stopping rules is verified using real images. It is shown that the extension suggested in this paper to Wald's one sided alternative sequential test (1947) performs better than previously available adaptive (or fixed) stopping rules.

Reliable detection of objects using the Hough transform is often possible even if just a small random poll of edge points is used for voting. This can lead to significant computational savings. To reduce the risk of errors, it is customary to preset the poll size to a value that is much larger than necessary in average conditions. An adaptive setting of the poll size in the probabilistic Hough transform is suggested. It is experimentally demonstrated that by monitoring changes in the ranks of peaks in the parameter space, sensible decisions on voting termination can be made. Adaptive stopping leads to polls that are on average smaller than the fixed poll that leads to the same error rate. In many applications the number of objects to be detected is unknown. Finding the number of appearances of an object in a noisy image is difficult, especially with partial data. The authors present an adaptive stopping rule that terminates voting as soon as any number of objects seem to be reliably detected, even though the existence of others may not be ruled out yet.>

It is customary to represent an analog image in digital form by dividing its support to pixels, and within each pixel to represent the brightness by a quantized scalar i.e., to approximate the 2-D image function by a horizontal planar patch. This paper studies a representation scheme in which the image function is represented within each pixel by an inclined planar patch. If the image function is to be represented by b bits per pixel, a bit allocation trade-off arises, and the optimal allocation of bits to the representation of the average value and of the two slope coefficients within each pixel needs to be determined. Analysis shows that allocating all the bits to represent the average brightness is not always optimal, and bits should be allocated to the representation of the slope coefficients. Similar results were obtained for the 1-D case.>

Length estimation of three dimensional digitized curves is considered. The three dimensional chain code representation is assumed. It is shown that previously suggested length estimators are biased, and alternative simple estimators that possess desired optimality properties are developed. To support optimal design of three dimensional length estimators, useful statistical properties of three dimensional chain encoded digital straight lines are derived. Although this paper focuses on simple estimators, the suggested approach can be used to design more complicated three dimensional length estimators.>

The problem of fitting a straight line to a planar set of points is reconsidered. A parameter space computational approach capable of fitting one or more lines to a set of points is presented. The suggested algorithm handles errors in both coordinates of the data points, even when the error variances vary between coordinates and among points and can be readily made robust to outliers. The algorithm is quite general and allows line fitting according to several useful optimality criteria to be performed within a single computational framework. It is observed that certain extensions of the Hough transform can be turned to be equivalent to well-known M estimators, thus allowing computationally efficient approximate M estimation.>

The problem of determining the optimal straight path between a planar set of points is considered. Each point contributes to the cost of a path a value that depends on the distance between the path and the point. The cost function, quantifying this dependence, can be arbitrary and may be different for different points. An algorithm to solve this problem using an extension of the Hough transform is described. The range of applications includes straight-line fitting to a set of points in the presence of outliers, navigation, and path planning. The proposed extended Hough transform can be tuned to equivalent to well-known robust least-squares techniques, and allows efficient, approximate M-estimation.>

A variation of the Hough Transform that is aimed at detecting digital lines has been recently suggested. Other Hough algorithms are intended to detect straight lines in the analog pre-image. These approaches arc analyzed and compared in terms of the relation between the achievable resolution and the required number of accumulators, using a definition of resolution that is based on the Geometric Probability measure of straight lines. It is shown that the "analog" approach is greatly superior in high resolution applications, where a "digital " Hough Transform would generally require an infeasibly large number of accumulators. The Hough Transform [2,4] is a well known technique for recognizing predefined features in edge maps. In this paper, the Hough Transform for detecting straight lines is considered. Most Hough algorithms consist of an incrementation stage, in which each edge point "votes " for the parameter-pairs of all possible straight lines on which it can lie, and an exhaustive search for peaks. These correspond to large collinear sets of edge-points. Originally, the slope-intercept (m,b) parametrization of straight lines had been employed in the Hough Transform. It has the advantage that an edge point corresponds to a straight line in the parameter space, thus voting is simple. Its drawback is that the parameter space is unbounded, implying some theoretical and practical difficulties. With normal (p,0) parametrization of straight lines, as suggested by [2], an edge point corresponds to a sinusoid in the parameter space, thus voting is somewhat more complex. The normal parametrization has the advantage that a bounded image leads to a bounded parameter space. Other straight-line parametrizations have also been suggested, see [4,11,17]. In most implementations of the Hough algorithm the parameter space is represented by a rectangular accumulator array, such that each accumulator corresponds to a rectangular, constant size domain in the parameter space. The quantization of the parameter space greatly influences the resolution and detection capabilities of the algorithm, as well as the computational and storage requirements; see

The author is concerned with the calculation of object features directly from the Fourier-series coefficients of boundary function r( phi ) which describes the length of the radius-vector from the origin to a boundary point. The area, the coordinates of the centroid, and the second-order moments with respect to the axes passing through the origin are determined. Given these features, the orientation of the central axes, and the central moments of inertia can be easily determined. The difficulty of calculating the perimeter in terms of the Fourier coefficients of r( phi ) is known. Hence, lower and upper bounds on the perimeter are established.>

The application of gray-scale digitizers to the digitization of binary images of straight-edged silhouettes is considered. A measure of digitization-induced ambiguity is introduced. It is shown that if the gray levels are not quantized and the sampling resolution is sufficiently high, error-free reconstruction of the original binary image from the digitized image is possible. When the total bit-count for the representation of the digitized image is limited, i.e., sampling resolution and quantization accuracy are both finite, error-free reconstruction is usually impossible. The authors' suggested bit allocation policy is then to increase the quantization accuracy as much as possible, once sufficient sampling resolution has been reached.>