Other ICP variants have come about as a result of negating the various drawbacks of the original algorithm of Besl and Mckay. Non-Rigid ICP [10] extends the ICP framework to non- rigid restoration, using an adjustable stiffness parameter, while retaining the convergence properties of the original algorithm. Scaling ICP or SICP [11] integrates a scale matrix with boundaries into the original ICP algorithm for scaling registration. Generalized ICP or GICP [12] combines the ICP and **point**-to-plane ICP algorithms into a new single probabilistic framework. This allowed for far greater robustness than the standard ICP approach. The latest techniques include Learning Anisotropic ICP, Weighted Average ICP, and ICP using Bi-unique Correspondences. Learning Anisotropic ICP (LA-ICP) [13] presents an online learning approach to 3D object registration that vastly improves the performance of **Iterative** **Closest** **Point** (ICP) methods. Weighted Average ICP (WA-ICP) [14] uses a new weighting approach to establish correspondence. ICP using Bi-Unique Correspondences [15] guarantees the uniqueness of corresponding pairs by searching multiple **closest** points. The latest trend in the field of 3D object registration and modeling is the use of RGB-D cameras which capture RGB images along with per-pixel depth information. This has been facilitated mainly by the increasing cheapness and availability of such cameras to the general public. RGB-D allows for greater resolution than Laser scanning but at the cost of reduced accuracy. An in-depth look into using depth cameras for 3D modelling in conjunction with using the ICP algorithm can be found in the work of Peter Henry et al in the field of RGB-D mapping [16]. It utilizes a novel joint optimization algorithm combining both matching using visual features and shape alignment. Sparse feature detection is carried out on the RGB images using a Scale Invariant Feature Transform (SIFT) feature detector and extractor. Random Sampling Consensus or RANSAC is used to find the best rigid transformation between the obtained feature sets. Then the ICP algorithm is carried out on the dense **point** clouds obtained from the depth data. The two results obtained are properly weighted and added to give us a finely refined 3D model of the indoor environment, which would not be

Show more
The mentioned article explains usage of non-rigid **Iterative** **Closest** **Point** for tracking patients abdominal surfaces. This approach presents a solution for a hard case of sur- face registration problem. The abdominal skin changes from convex to concave during breathing phases which under geometric distance constraint, in worst case, may lead to attraction of outlying points in source cloud to **closest** points in the center of target cloud. No extreme folding of **point** surfaces should be present in this type of deformation tracking, yet the greatest change in amplitude is present for central points in cloud. The surfaces were acquired with time-of-flight camera Swiss Ranger SR4000, which has an absolute accuracy of about 1 cm. The markers used as landmark points in our algorithm were placed on the abdominal skin before the acquisition so that they were imaged with the designated object. The markers used were square-shaped and had a size of 15 mm. The frequency of the camera was set to 30 MHz, which allowed for acquisition of within a 5 m radius. The patient was at the distance of from 1 to 1.5 meters from the camera.

Show more
18 Read more

3D: three-dimensional; EM: Expectation Maximization; PSSM: Protein Structure Superposition method for addressing the cases with Missing data; ICP: **iterative** **closest** **point**; PCA: principal component analysis; RMSD: root mean squared deviation; NCBI: National Center for Biotechnology Information; PDB: Protein Data Bank; LS: least square; CPSARST: Circular Permutation Search Aided by Ramachandran Sequential Transformation; CCP4: Collaborative Computational Project Number 4; MUSTANG: MUltiple STructural AligNment AlGorithm; CAS: Chinese Academy of Sciences.

10 Read more

SCF). We compare our algorithm with original **Iterative** **Closest** **Point** Algorithm (ICP)[4] and ICP with angular invariant feature algorithm (ICP-AIF)[9]. We know the real conjugate points between **point** clouds. Therefore, we measure RMS distance error between two ground truth **point** sets. All registration result is plotted as graphic RMS registration error (ground truth) as a function of iteration number. The tests are performed on a computer with a processor "Intel" Core (TM) 2 Duo CPU, frequency 2 GHz and a memory of 2 GB.

Show more
Pre-operative and inter-operative prostate shape may suffer deformation due to extrusion of the ultrasonic probe, inflation of the anorectal coil inside the rectum dur- ing the MR scanning, and alterations in patient position [4]. To compensate for these movements, many non-rigid registration algorithms have been proposed over the past 20 years [5–8]. Mutual information (MI) is a widely used similarity criterion in multi- modal image registration and has been independently proposed by Collignon [9] and Viola [10]. Moradi [11] created a label map registration frame (LMRF) that aligned TRUS and MR images by using 3D Slicer, which first used the **iterative** **closest** **point** (ICP) method to rigidly align the outlines of the two images and then combined MI and B-splines to elastically register binary label maps from the two images obtained by manual contouring. Although LMRF aligned contours with high accuracy and could be conveniently implemented in 3D Slicer, it used only label maps to correct the local deformation and ignored pixel intensity, resulting in a high target registration error (TRE) of 3.6 ± 1.7 mm. Mitra [12] utilized directional quadrature filter pairs to convert TRUS and MR images into texture images and then used normalized mutual informa- tion (NMI) as a similarity criterion to register the texture images. The average TRE and mean Dice Similarity coefficient (DSC) were 2.64 ± 1.47 mm and 0.943 ± 0.039, but the computation time exceeded 797 s due to the use of 4 quadrature convolutions and the L-BFGS optimization method. However, several recent studies have shown that MI- based registration can be improved in certain cases. One improvement is to calculate MI over a set of overlapping image blocks to include spatial information. Loeckx [13] pro- posed the conditional mutual information (CMI), which considered the spatial location in the reference image of each joint intensity pair as the priori condition and calculated conditional entropies between the intensities given the spatial distribution. CMI-based registration resulted in a significant improvement in theoretical, phantom and clinical data compared with MI-based registration, but it was approximately 15 times slower than MI because it regarded the block label as the third channel of the joint histogram. Furthermore, MI assumes that all pixels in the overlapping area affect the calculations equally, but it is clear that different pixels contribute differently to the computation of MI [14]. Another kind of method assigns different weights to pixels using feature detec- tion operators, e.g., the saliency measure [14] and the Harris corner detector [15]. But this method is often unable to extract effective features from TRUS images due to the low signal to noise ratio (SNR).

Show more
21 Read more

In [12] they proposed a nose-based registration scheme for better handling of occluded faces. Curvature information is utilized for automatic detection of the nose area, and an average nose model is used for fine alignment via **Iterative** **Closest** **Point** (ICP) algorithm. On the registered surfaces, occlusions are detected by analyzing the difference from the average face model, and the occlusion-removed surfaces are completed by a modified version of the Gappy PCA method. The restored faces are classified using different local masks, and multiple classifiers are fused for final identity estimation.

Show more
This paper presents a novel method for pair-wise range image registration, a backbone task in world modeling, parts inspection and manufacture, object recognition, pose estimation, robotic navigation, and reverse engineering. The method finds the most suitable homogeneous transformation matrix between two constructed range images to create a more complete 3D view of a scene. The proposed solution integrates a ray casting-based fitness estimation with a global optimization method called improved self-adaptive differential evolution. This method eliminates the fine registration steps of the well-known **iterative** **closest** **point** (ICP) algorithm used in previously proposed methods, and thus, is the first direct global registration algorithm. With its parallel implementation potential, the ray

Show more
15 Read more

In 1986, M.S. Khan et al. [13] introduced the general Mann **iterative** scheme as follows: If X is a complete metric space, T is a self-map of X and A is a lower triangular matrix with non-negative entries, zero column limits, and row sum 1, then the General Mann **iterative** schemes {x n } of T is defined by

15 Read more

The paper studies two splitting forms of generalized saddle **point** matrix to derive two alternate direction **iterative** schemes for generalized saddle **point** systems. Some convergence results are established for these two alternate direction **iterative** methods. Meanwhile, a numerical example is given to show that the proposed alternate direction **iterative** methods are much more eﬀective and eﬃcient than the existing one.

11 Read more

Abstract—We introduce a general **iterative** scheme for finding a common of the set solutions of variational inequality problems for an inverse-strongly monotone mapping and the set of common fixed points of a countable family of nonexpansive mappings in a real Hilbert space. We show that the sequence converges strongly to a common element of the above two sets under some parameters controlling conditions. The results presented in this paper improve and extend the corresponding results announced by many others.

(m − k)!k! (k − 1)! (n − k)! (2k − n)! , (24) and [q] denotes the integer part of the real number q.The parameter m is a free parameter that should be optimized by trial and error. It was seen that with increasing m accuracy of result increases up to a **point** and then owing to the rounding errors it decreases [25]. Thus, for choosing optimum m, it is beneﬁcial to apply an algorithm repeatedly for diﬀerent values of m and study its eﬀect on the solution. The other way to choose optimal value of m could be, to apply the Stehfest’s algorithm for inverting the Laplace transform of some elementry functions which are known.

Show more
The purpose of this work is to introduce and study an **iterative** method to approximate solutions of a hierarchical ﬁxed **point** problem and a variational inequality problem involving a ﬁnite family of nonexpansive mappings on a real Hilbert space. Further, we prove that the sequence generated by the proposed **iterative** method converges to a solution of the hierarchical ﬁxed **point** problem for a ﬁnite family of nonexpansive mappings which is the unique solution of the variational inequality problem. The results presented in this paper are the extension and

16 Read more

The above equivalence relation (.) reminds us to use ﬁxed **point** method to solve (.). Many authors have given a continuation of the study on the CQ algorithm and its variant form. For related work, please refer to [–]. Especially, the following regularized method was presented by Xu []:

14 Read more

Recently, the classical variational inequality (1.2) and fixed **point** problem of nonexpansive mappings have received rapid development, see, for example, [1-17] and the references therein. In this paper, we consider a viscosity approximation algorithm for variational inequality problems and fixed **point** set of nonexpansive mappings. Strong convergence theorems are established in the framework of Hilbert spaces.

An **iterative** process is considered for finding a common element in the fixed **point** set of a strict pseudocontraction and in the zero set of a nonlinear mapping which is the sum of a maximal monotone operator and an inverse strongly monotone mapping. Strong convergence theorems of common elements are established in real Hilbert spaces.

15 Read more

On the other hand the approximation of fixed points for multi-valued nonexpansive maps using Hausdorff metric was initiated by Markin [10] (see also [11]). Later, an interesting and rich fixed **point** theory for such maps was developed which has applications in control theory, convex optimization, differential inclusion and economics (see [4] and references cited therein). The theory of multi-valued nonexpansive mappings are harder than the corresponding theory of single valued nonexpansive maps. Different **iterative** processes have been used to approximate the fixed points of multi-valued nonexpansive mappings (see e.g, Khan and Yildirim [7], Pa- nyank [14], Sastry and Babu [15], Shahzad and Zegeye [18], Song and Wang [19], and Song and Cho [20]).

Show more
13 Read more

Consider, therefore, a Lagrange quadratic interpolation using the three grid points closest to, and surrounding the point z :.. and let 6 be defined by.[r]

17 Read more

It works as an extension of DBSCAN algorithm. Here, clusters are first separately formed using MinPts and Eps and are then merged into a single cluster. This algorithm overcomes the problem of handling datasets of different densities. DBSCAN required only two parameters [8], but DBCLUM requires three parameters, namely Threshold, MinPts and Eps. Eps is the radius to find neighbors, MinPts is the minimum number of points required to form a cluster. Lastly, Threshold is the value which decides whether two clusters will be merged or not. The algorithm begins with a **point** P and then calculates distances of neighbours with respect to Eps. If the number of points found are more than MinPts, a cluster is then formed and labeled [8]. Otherwise, the **point** is termed as noise. The next step includes visiting another **point** and to try and form a cluster. This process is repeated until all points are assigned as a part of the cluster or as a noise.

Show more
In this work, we deal with two-**point** Riemann–Liouville fractional boundary value problems. Firstly, we establish a new comparison principle. Then, we show the existence of extremal solutions for the two-**point** Riemann–Liouville fractional boundary value problems, using the method of upper and lower solutions. The performance of the approach is tested through a numerical example.

Find x ∗ ∈ F (T) such that h(I − S)x ∗ , x − x ∗ i ≥ 0, ∀x ∈ F (T ), (1.2) where T and S are nonexpansive mappings such that F (T ) is nonempty. It is easy to see that x ∗ is a solution of the variational inequalities (1.2) if and only if it is a fixed **point** of the nonexpansive mapping P F(T ) S, where P F (T ) stands for the metric projection on the closed convex set F (T).

10 Read more