Projection Distance Research Articles

Flame geometry of downward buoyant turbulent jet fires under cross flows: Experiments, dimensional analysis and an integral model

This study investigates the flame geometry of vertically downward facing buoyant turbulent jet fires issuing from a nozzle subject to cross flows, which deflect the jet flow by interacting with downward momentum issuing from the source and the upward buoyant forces owing to the flames. The flames turn eventually upwards as the buoyant forces dominate. The objective of this work is to understand the physics and develop non-dimensional correlations and modeling for the flame geometry of this jet fire configuration never investigated before. For this purpose, experiments are conducted having 4 circular nozzles (diameter at 3, 4, 5 and 7 mm) with propane employed as fuel under various heat release rates. The cross flows are generated by a wind tunnel having a uniform cross flow air speed varying from 0.31 m/s to 2.08 m/s. The horizontal and downward distance from the nozzle to the lowest point of the jet flame, the vertical thickness of the flame at the lowest point, the flame vertical height from lowest point to the flame tip, the flame tip horizontal projection distance are measured. Three characteristic length scales developed by dimensional analysis considering initial downward momentum M0, flame buoyancy g′, inertial cross flow force ua2 together with the normalized air required for complete combustion Sm˙f/ρaua×(ua2/g′)2, are found to represent well the aforementioned properties of the flame geometry. An integral model, considering the mass and momentum conservations, is developed to predict the trajectory and geometric properties of the downward jet flame under cross flow showing good agreement with the experimental results.

Efficient locality-sensitive hashing over high-dimensional streaming data

Approximate nearest neighbor (ANN) search in high-dimensional spaces is fundamental in many applications. Locality-sensitive hashing (LSH) is a well-known methodology to solve the ANN problem. Existing LSH-based ANN solutions typically employ a large number of individual indexes optimized for searching efficiency. Updating such indexes might be impractical when processing high-dimensional streaming data. In this paper, we present a novel disk-based LSH index that offers efficient support for both searches and updates. The contributions of our work are threefold. First, we use the write-friendly LSM-trees to store the LSH projections to facilitate efficient updates. Second, we develop a novel estimation scheme to estimate the number of required LSH functions, with which the disk storage and access costs are effectively reduced. Third, we exploit both the collision number and the projection distance to improve the efficiency of candidate selection, improving the search performance with theoretical guarantees on the result quality. Experiments on four real-world datasets show that our proposal outperforms the state-of-the-art schemes.

Efecto de los parámetros de manufactura en la calidad del recubrimiento cerámico de barrera térmica después de envejecido por tratamiento térmico

Ceramic coatings of zirconium oxide (ZrO2) were prepared and stabilized with yttrium oxide (7-8 wt% Y2O3) by plasma-aided thermal projection (APS), at different injection distance and carrier gas flow conditions. Ageing thermal treatments were performed to know the corresponding effects on the quality and performance of the thermal barrier. Determination of layer thickness of the thermal barrier, evolution of the thermally grown oxide (TGO) layer and the evaluation of the mechanical properties through nanoindentation was carried out by the micro-structural study. Results shown out that the effect of the projection distance is not significant. However, a significant effect of the gas flow, namely, the lower the flow is, the higher the thickness of the ceramic layer. Apparently, the layers coatings micro-structure is not significantly altered (the form of the splats) due to the increase in the treatment time, but the distribution of the defects and the exogenous particles (pollutants) presented clear shifts. Consequently, a decrease for the values of the mechanical properties of the layers which produce the thermal barrier was observed; also, thickness of the TGO layer was increased because of the thermal treatment time. During undertaken studies, the TGO layer never exceed the critical limit.

Compulsory perspective interaction projection mapping with elements of ink paintings

With the introduction of new media and the rapid development of electronic software technology in recent times, the single and boring method of creating traditional art no longer satisfies the high aesthetic expectations of today’s viewer. Pure technology is not enough to realize its value if it does not integrate strong and connotative creative content. In such an artistic environment, the integration and innovation of multiple fields and elements has become inevitable. This article discusses the characteristics and the combination of multiple elements of ink painting, projection, interaction, and compulsory distance method in the field of new media, with examples, used to create corresponding works.

An optimized fringe generator of 3D pavement profilometry based on laser interference fringe

As pavement distresses have more negative influences on the driving safety and comfortableness,pavement distress detection and maintenance becomes more significant. 3D pavement profilometry can help analyze the type and severity of the pavement distress, which provides a basis for pavement maintenance. This paper has employed laser interference fringe on 3D pavement profilometry. This new technology has great research value and prospect on improving the measurement accuracy.However,the fringe generators used in previous 3D profilometry technologies performed close-range projection (< 1 m) with laser interference fringe on the object surfaces being measured which have small sizes (diameter < 0.1 m), so the area of the fringe region was small.To extract the 3D profile information of the pavement surface,it is imperative to increase the projection distance to enlarge the area of fringe region. However, increasing the projection distance will cause the fringe spatial frequency and the measurement accuracy to decrease.In order to solve the problem,this study optimizes the configuration of the fringe generator based on previous researches, that enables it to emit interference fringes with higher spatial frequencies. It is proved that the new fringe generator makes it possible to extract the 3D pavement profile.After validation, the results show that the measurement error reaches ±0.38 mm when the fiber core interval is 0.75 mm and the projection distance is 1200 mm.

18‐3: Impact Study of Windshield Geometry on the Subjective Customer Perception for Augmented Reality Head‐up Displays (AR HUD)

New Generations of Head‐up Displays (HUD) are expected to grow in size and projection distance. For these reasons, even smallest deviations in the windshield geometry may cause new optical image defects. These image defects need to be specified, measured, and correlated with subjective customer perceptions. Based on a customer study (n=54), this paper gives first indications that heterogeneous image quality disturb customers.

Multi-body 3D–2D registration for image-guided reduction of pelvic dislocation in orthopaedic trauma surgery

Surgical reduction of pelvic dislocation is a challenging procedure with poor long-term prognosis if reduction does not accurately restore natural morphology. The procedure often requires long fluoroscopic exposure times and trial-and-error to achieve accurate reduction. We report a method to automatically compute the target pose of dislocated bones in preoperative CT and provide 3D guidance of reduction using routine 2D fluoroscopy. A pelvic statistical shape model (SSM) and a statistical pose model (SPM) were formed from an atlas of 40 pelvic CT images. Multi-body bone segmentation was achieved by mapping the SSM to a preoperative CT via an active shape model. The target reduction pose for the dislocated bone is estimated by fitting the poses of undislocated bones to the SPM. Intraoperatively, multiple bones are registered to fluoroscopy images via 3D–2D registration to obtain 3D pose estimates from 2D images. The method was examined in three studies: (1) a simulation study of 40 CT images simulating a range of dislocation patterns; (2) a pelvic phantom study with controlled dislocation of the left innominate bone; (3) a clinical case study investigating feasibility in images acquired during pelvic reduction surgery. Experiments investigated the accuracy of registration as a function of initialization error (capture range), image quality (radiation dose and image noise), and field of view (FOV) size. The simulation study achieved target pose estimation with translational error of median 2.3 mm (1.4 mm interquartile range, IQR) and rotational error of 2.1° (1.3° IQR). 3D–2D registration yielded 0.3 mm (0.2 mm IQR) in-plane and 0.3 mm (0.2 mm IQR) out-of-plane translational error, with in-plane capture range of ±50 mm and out-of-plane capture range of ±120 mm. The phantom study demonstrated 3D–2D target registration error of 2.5 mm (1.5 mm IQR), and the method was robust over a large dose range, down to 5 Gy/frame (an order of magnitude lower than the nominal fluoroscopic dose). The clinical feasibility study demonstrated accurate registration with both preoperative and intraoperative radiographs, yielding 3.1 mm (1.0 mm IQR) projection distance error with robust performance for FOV ranging from 340 × 340 mm2 to 170 × 170 mm2 (at the image plane). The method demonstrated accurate estimation of the target reduction pose in simulation, phantom, and a clinical feasibility study for a broad range of dislocation patterns, initialization error, dose levels, and FOV size. The system provides a novel means of guidance and assessment of pelvic reduction from routinely acquired preoperative CT and intraoperative fluoroscopy. The method has the potential to reduce radiation dose by minimizing trial-and-error and to improve outcomes by guiding more accurate reduction of joint dislocations.

Pengenalan Wajah Bebas Ekspresi Menggunakan Metode Nearest Feature Line dengan Representasi Ciri dalam Ruang Eigen

Pengenalan wajah dengan ekspresi berbeda adalah salah satu bagian dari masalah pengenalan pola yang cukup kompleks jika dibandingkan dengan pengenalan pola pada profil normal.Metode pengenalan wajah bebas ekspresi menggunakan teknik Nearest Feature Line (NFL) bekerja dengan menemukan jarak proyeksi terdekat antara vektor ciri, dengan asumsi bahwa semakin dekat jarak proyeksi vektor ciri (wajah) ke vektor ciri (wajah) lainnya, maka lebih mirip sifat akan vektor ciri fisik (wajah) yang berdekatan. Perhitungan jarak NFL dilakukan pada ruang dimensi eigen dengan tujuan agar dimensi ciri vektor (wajah) memiliki dimensi yang jauh lebih kecil untuk meningkatkan tingkat akurasi pengenalan dan mempercepat waktu komputasi.Hasil pengujian yang diperoleh menunjukkan metode NFL memberikan tingkat akurasi pengenalan yang cukup baik pada nilai rata-rata akurasi 76,7% dengan keunggulan waktu komputasi yang rendah jika dibandingkan dengan metode cerdas lainnya seperti sistem jaringan saraf tiruan.

Anatomical Factors that Can Predict the Structure of Lamina Papyracea for Endoscopic Sinus Surgery.

This study investigated morphological variations of lamina papyracea, the structure that should be carefully considered when opening posterior ethmoid sinus during endoscopic sinus surgery, to avoid injury. This study employed axial, coronal, and sagittal computed tomography. Using computed tomography images of 228 face-sides, various anatomical parameters were determined: distances of the anterior and posterior ethmoid arteries from the skull base, and from the third lamella; changes in the angles of the lamina papyracea at the anterior and posterior ethmoid sinuses; and presence or absence of supraorbital ethmoid cell (SECs), Onodi cell, and Haller cell. The relationship between the distances which indicate the point of maximum projection by the lamina papyracea among third lamina and posterior ethmoid artery into the posterior ethmoid sinus and these anatomical factors were analyzed statistically. The projection distance of lamina papyracea into the posterior ethmoid sinus was -2.6 mm to 3.4 mm, and in 41.2% of cases, projection in the direction of the nasal cavity was greater than that of the lamina papyracea at the anterior ethmoid sinus. This distance increased with increasing distance of the maximum projection point from the skull base and increasing floating distances of the anterior and posterior ethmoid arteries. The number of subjects with large projection distances was increased among those with floating posterior ethmoid arteries. In addition, subjects with SECs had significantly greater projection distances. Particular care should be taken to avoid injury to the lamina papyracea when opening the posterior ethmoid sinus in subjects with floating anterior or posterior ethmoid arteries, and/or SEC. 4 Laryngoscope, 131:E19-E25, 2021.

Image Set Classification Using a Distance-Based Kernel Over Affine Grassmann Manifold.

Modeling image sets or videos as linear subspaces is quite popular for classification problems in machine learning. However, affine subspace modeling has not been explored much. In this article, we address the image sets classification problem by modeling them as affine subspaces. Affine subspaces are linear subspaces shifted from origin by an offset. The collection of the same dimensional affine subspaces of [Formula: see text] is known as affine Grassmann manifold (AGM) or affine Grassmannian that is a smooth and noncompact manifold. The non-Euclidean geometry of AGM and the nonunique representation of an affine subspace in AGM make the classification task in AGM difficult. In this article, we propose a novel affine subspace-based kernel that maps the points in AGM to a finite-dimensional Hilbert space. For this, we embed the AGM in a higher dimensional Grassmann manifold (GM) by embedding the offset vector in the Stiefel coordinates. The projection distance between two points in AGM is the measure of similarity obtained by the kernel function. The obtained kernel-gram matrix is further diagonalized to generate low-dimensional features in the Euclidean space corresponding to the points in AGM. Distance-preserving constraint along with sparsity constraint is used for minimum residual error classification by keeping the locally Euclidean structure of AGM in mind. Experimentation performed over four data sets for gait, object, hand, and body gesture recognition shows promising results compared with state-of-the-art techniques.

How far actually is the Galactic Center IRS 13E3 from Sagittarius A*?

Abstract The Galactic Center IRS 13E cluster is a very intriguing infrared object located at ${\sim } 0.13$ pc from Sagittarius A$^\ast$ (Sgr A$^\ast$) in projection distance. There are arguments both for and against the hypothesis that a dark mass like an intermediate mass black hole (IMBH) exists in the cluster. We recently detected the rotating ionized gas ring around IRS 13E3, which belongs to the cluster, in the H30$\alpha$ recombination line using ALMA. The enclosed mass is derived to be $M_{\mathrm{encl.}}\simeq 2\times 10^{4}\, M_\odot$, which agrees with an IMBH and is barely less than the astrometric upper limit mass of an IMBH around Sgr A$^\ast$. Because the limit mass depends on the true three-dimensional (3D) distance from Sgr A$^\ast$, it is very important to determine it observationally. However, the 3D distance is indefinite because it is hard to determine the line-of-sight (LOS) distance by usual methods. We attempt here to estimate the LOS distance from spectroscopic information. The CH$_3$OH molecule is easily destroyed by the cosmic rays around Sgr A$^{\ast }$. However, we detected a highly excited CH$_3$OH emission line in the ionized gas stream associated with IRS 13E3. This indicates that IRS 13E3 is located at $r\gtrsim 0.4$ pc from Sgr A$^{\ast }$.

Clinical application of autologous costal cartilage transplantation in repair of upper lip depression in secondary repair of cleft lip

To discuss the clinical application and effectiveness of autologous costal cartilage transplantation in the repair of upper lip depression in the secondary repair of cleft lip. The clinical data of 10 patients of secondary repair of upper lip depression with cleft lip by autologous costal cartilage transplantation between January 2017 and January 2019 were retrospectively analysed. There were 7 males and 3 females with an average age of 24 years (range, 18-33 years). There were 8 cases of bilateral lip fissure and 2 cases of unilateral lip fissure. All of them underwent early lip repair at the age of 2-3 years old. The change of the profile of the soft tissue profile of the upper lip of the patient was quantitatively analyzed before operation and at immediate after operation, including the soft tissue facial angle (G-Sn-Pg'), the nasolabial angle (Cm-Sn-UL), the mentolabial angle (UL-LL-Pg'), the distance between the most salient point of the upper lip to the aesthetic plane (UL-E), the highest point of the frontal part, the projection distance of the most salient point of the upper lip on the plane of the orbital ear (G-UL), and the upper lip protrusion (ULP). One case had incision infection and healed after dressing change, and in the other 9 patients, the incisions healed by first intention, and no acute infection and other complications occurred. The appearance of the lateral morphology of the upper lip at immediate after operation was significantly improved when compared with that before operation. The value of UL-LL-Pg', UL-E, G-UL, and ULP were significantly increased and G-Sn-Pg' was significantly decreased when compared with preoperative ones ( P<0.05). There was no significant difference in Cm-Sn-UL between pre- and post-operation ( t=0.821, P=0.433). All the 10 patients were followed up 6-24 months, with an average of 15 months. During the follow-up, the soft tissue morphology of the upper lip was good. No long-term complications such as cartilage absorption and cartilage displacement were found. Autologous costal cartilage transplantation is a safe and effective treatment for upper lip depression in the secondary repair of cleft lip.

Depth-of-field reduction due to blurring in a relayed plenoptic camera and mitigation via deconvolution

The projection of an externally-mounted microlens array (MLA) onto a sensor via relay lenses is oft adopted by the scientific community in lieu of traditional plenoptic cameras with embedded on-sensor MLAs. In most cases a relayed design better supports prototyping and temporary setups, while in other cases such as intensified plenoptic imaging the extended projection distance is necessary due to physical constraints. However, relaying the MLA inevitably blurs the raw image, often by 1.5 px or more. We observed an unusual behavior where fixed blurring at the raw image level produced blurs that are worsened by depth within refocused images – enough to reduce the plenoptic system’s depth-of-field by 60% in most cases. To address this problem, we hereby put forth a model to elucidate the mechanism of this depth-dependent blurring, and propose a Lucy–Richardson deconvolution procedure for mitigation. Discussions are undertaken in the contexts of regular and intensified plenoptic photography, as well as application in 3D particle image velocimetry for fluid dynamics. Results were validated on synthetic as well as experimental images.

Dimensionality Reduction of Tensors Based on Local Homeomorphism and Global Subspace Projection Distance Minimum

Tensors are powerful tools for representing and processing multidimensional data. In this paper, an algorithm of dimension reduction for multidimensional data is proposed. We consider both the global and local structures of the data to fully extract important features. Multiplying a tensor by a matrix in mode-n can change the size of a certain dimension of the tensor. Therefore, for the global structure, we use a matrix to span a subspace to minimize the distance between the tensor data and its projection on the subspace for each mode. For local structure, we construct a dimensionality reduction algorithm for tensor data based on local homeomorphism to keep the continuous dependency relationship of the original high-dimensional tensor data unchanged after dimensionality reduced in each locality. The proposed algorithm achieves dimensionality reduction by combining the global subspace projection distance minimum and local homeomorphism, which realizes global variance maximization and maintains the local non-linear geometric structure at the same time. The proposed algorithm performs good feasibility while compared with other advanced algorithms in classification and clustering experiments.

Developing a Reassembling Algorithm for Broken Objects

The research on reassembling broken objects has many important applications, such as cultural relics restoration, medical surgery and solving puzzle. Because of the complicated surfaces of the fractured object pieces, it is not easy to extract salient features from them. It becomes even more difficult and very time-consuming to reassemble broken objects when the fragments are severely corroded or some of them are lost. In order to improve the accuracy and speed of 3D fragment reassembling, an effective and efficient fragment reassembling algorithm based on point clouds is proposed this article. This method first extracts keypoints and their concavity and convexity according to the symbolic projection distance of the point cloud, and then uses the local neighborhood information of the keypoints to construct a multi-scale covariance matrix descriptor. Furthermore, by calculating the similarity of the covariance matrix descriptors, the initial pairs of match points are obtained. Finally, the geometric constraints are gradually added to optimize the sampling so as to find good hypotheses as quickly as possible. By doing so, the search space is narrowed continuously in each iteration of the process to speed up the hypothesis test. We have conducted extensive experiments. The results show that the proposed method can fuse multiple features of the fragments effectively and achieve an outstanding matching effect on the defected fragments, and that the proposed method is faster than the existing methods in literature.

Convergently evolved muscle architecture enables high-performance ballistic movement in salamanders.

Elastically powered ballistic movements, such as tongue projection, are common in nature, likely due to benefits such as increased acceleration and distance of movement, and decreased thermal sensitivity imparted by elastic mechanisms. Within Plethodontidae, both muscle-powered and elastically powered ballistic tongue projection occur. Thus, we examine how elastically powered ballistic tongue projection morphology has evolved from muscle powered projection at the level of the projector muscles (m. subarcualis rectus [SAR]). We find that two main SAR morphologies have evolved within Plethodontidae. The first SAR morphology is conducive to elastically powered ballistic projection. This ballistic SAR morphology has evolved multiple, independent times within Plethodontidae, and results from the correlated evolution of several traits including increased collagen aponeuroses, larger SAR muscles, and the loss of inner myofibers attaching directly to the tongue skeleton. While the independent evolution of ballistic SAR morphology has arrived at a similar anatomical design, other tongue structures such as tongue attachment and skeleton folding type varies among species with a ballistic SAR morphology. The second morphology is conducive to muscle-powered projection and is similar to morphology found in an outgroup, Salamandridae. The SAR of these species have inner myofibers that attach to the tongue skeleton, limiting projection distance, coupled with reduced collagen aponeuroses present in relatively small projector muscles. This SAR morphology has likely been retained from ancestors or may be related to feeding ecology. Overall, a ballistic SAR morphology has evolved repeatedly and independently due to the correlated evolution of several SAR traits, including the loss of inner myofibers, which is likely a defining feature of ballistic projection.

Effects of Single-Foot Centered and Double-Foot Centered X-ray Projection on Hallux Valgus Measurement.

ObjectiveTo investigate whether use of single‐foot centered and double‐foot centered weight‐bearing X‐rays has an impact on the relevant indicators of hallux valgus.MethodsA total of 55 female patients from the Department of Ankle Surgery of Beijing Tongren Hospital with hallux valgus (110 feet) were collected from September to December 2015. The age of these patients ranged from 18 to 43 years, with an average age of 47.9 ± 8.5 years. All selected patients fit the diagnostic criteria of hallux valgus and had weight‐bearing single foot centered and double foot centered radiographs taken. During the projection, all patients were instructed to stand on the X‐ray box, with the knee joint straightened and legs perpendicular to the floor. The projection center of the single foot was directed at the lateral part of the scaphoid bone of the foot, while the projection center was directed at the position between the scaphoid bones of both feet for the double‐foot shooting. The hallux valgus angle (HAV), the intermetatarsal angle between the first and second metatarsals (IMA), the intermetatarsal angle between the first and fifth metatarsals (IM1‐5), and the metatarsal adduction angle (MAA) were measured and examined. The difference between these two shooting conditions was compared and analyzed.ResultsThe differences in X‐ray measurement results (IMA, HAV, IM1‐5, and MMA) between different measures for the same patient were not statistically significant. The values of HAV, IMA, IM1‐5, and MAA are common indexes for evaluating hallux valgus. The average IMA was 15.9° for single‐foot centered and 14.1° for double‐foot centered X‐rays. The average HAV was 30.2° for single‐foot centered and 29.7° for double‐foot centered X‐rays. The average IM1‐5 was 31.1° for single‐foot centered and 29.7° for double‐foot centered X‐rays. The average of metatarsal adduction angle was 13.8° for single‐foot centered and 14.1° for double‐foot centered X‐rays. The differences between single‐foot centered and double‐foot centered X‐rays were statistically significant in terms of the measurement index (P < 0.05). In addition, compared with double‐foot centered weight‐bearing X‐rays, the focus of single‐foot centered X‐rays was located on the lateral part of the scaphoid bone of the foot, and the ray was closer to the vertical foot in the single‐foot centered weight‐bearing X‐ray.ConclusionWhen the weight‐bearing position and projection distance are the same, the single‐foot centered weight‐bearing X‐ray is more effective in evaluating the severity of hallux valgus compared with the double‐foot centered weight‐bearing X‐ray.

Can sectioning the posterior communicating artery be predicted with computed tomography angiography in the microsurgical clipping of basilar apex aneurysms?

Microsurgical clipping is a viable option for managing high-riding basilar apex aneurysms (BAXs) using a transsylvian approach. Cutting the posterior communicating artery (PCoA) at the perforator-free zone provides sufficient exposure of BAX, although it is not always safe. The aim was to qualitatively predict sectioning of the interfering PCoA by using preoperative computed tomographic angiography (CTA). A virtual trajectory from the lesser sphenoid wing to the neck of the aneurysm was simulated on CTA in 19 consecutive patients operated on BAX aneurysms from 2013 to 2018. The angles on CTA resembled the typical view through the extended pterional craniotomy. Next, the ipsilateral PCoA was tracked from the internal carotid to its end at the posterior cerebral artery in the trajectory. The opticocarotid and carotid-oculomotor window served as the corridors for entering the interpeduncular fossa in 16 patients (84.2%) and in 3 patients (15.8%), respectively. The PCoA was intentionally cut during nine surgeries and preserved in 10 of them. The PCoA was cut more often when positioned toward the cranial base and more medially in the corridor; both the Kernel density analysis and the data points representing the position of the PCoA supported this finding (p < 0.01). The CTA-based position of either ends of the PCoA in the trajectory did not differ between cut and not-cut groups (p = 0.19-0.96). Aneurysm projection, rupture, size, PCoA diameter, length, and other distances on CTA were not related to the sectioning of PCoA (p > 0.05). The probability of PCoA sectioning is higher when the PCoA is positioned medially and closer to the cranial base. If the anatomical course of the PCoA is recognized beforehand, it can contraindicate surgery. Preoperative CTA-based planning facilitates the surgery of BAX aneurysms. Therefore, CTA can make microsurgical clipping planning more predictable for the neurosurgeon and safer for the patient.

Discussion of temperature, TV distortion, and lateral color of a 4-megapixel DLP projector lens

This paper explores lens design through thermal analysis and correction of 4-megapixel DLP projector lenses at temperatures ranging from 10°C to 80°C, and analyzes and discusses the TV distortion and lateral color with a focal length of 24 mm, F/# of 1.71, projection distance of 7.2 m, and projection screen size of 200 inches. Appropriate lens materials are selected and changes are made to the lens barrel materials to achieve athermalization. At temperatures from 10°C to 80°C, the image quality of lens is as follows: MTF (93 lp/mm) is greater than 0.554, lateral color is less than 1.39 µm, optical distortion is less than 0.147%, TV distortion is less than 0.079%, and relative illumination less than 92.52%.

Confidence Region of Singular Subspaces for Low-Rank Matrix Regression

Low-rank matrix regression refers to the instances of recovering a low-rank matrix based on specially designed measurements and the corresponding noisy outcomes. Numerous statistical methods have been developed over the recent decade for efficiently reconstructing the unknown low-rank matrices. It is often interesting, in certain applications, to estimate the unknown singular subspaces. In this paper, we revisit the low-rank matrix regression model and introduce a two-step procedure to construct confidence regions of the singular subspaces. We investigate distributions of the joint projection distance between the empirical singular subspaces and the unknown true singular subspaces. We prove asymptotical normality of the joint projection distance with data-dependent centering and normalization when $r^{3/2}(m_{1}+m_{2})^{3/2}=o(n/\log n)$ where $m_{1}, m_{2}$ denote the matrix row and column sizes, $r$ is the rank and $n$ is the number of independent random measurements. Consequently, data-dependent confidence regions of the true singular subspaces are established which attain pre-determined confidence levels asymptotically. Additionally, non-asymptotic convergence rates are also established. Numerical results are presented to show the merits of our methods.