Small Rectangles Research Articles

METHODS FOR THE PRESERVATION AND RESTORATION OF DUNHUANG WALL PAINTINGS: FOREIGN EXPERIENCE

This article examines the genesis of planning solutions and ceiling designs of Dunhuang cave shrines. Based on the methods of defining features, graphic-and-analytical and system-and-structural analysis, the spread of types of plans and ceilings from dynasty to dynasty has been traced. Using auxiliary methods of historical, mathematical, and comparative analysis, a general picture of the development of the planning structure and constructive schemes of Dunhuang shrines from the Northern Lian period to the Yuan period has been created. By counting the types of plans and ceilings in each period, a group of the most common types of plans was determined: with a combination of larger and smaller rectangles (where the smaller is the altar part of a square or rectangular shape), a plan in the form of a square with a protruding trapezoidal altar, an altar in outline close to a rectangle or trapezoids with rounded corners, a horizontally elongated plan rectangle, based on squares with protrusions. The most common type of ceiling is fu-dou-ding with modifications.

Genesis of the Planning Structure And Ceilings of Dunhuang Sanctuaries

The article is devoted to the study of the genesis of the plans and ceilings of the Dunhuang cave sanctuaries. Based on the methods of graphic-and-analytical, defining features and system-structural analysis, the spread of types of plans and ceilings from dynasty to dynasty has been traced. With the help of a group of auxiliary methods of historical analysis, mathematical analysis and comparative analysis, a general picture of the development of the floating structure and constructive schemes of the Dunhuang sanctuaries from the Northern Liang period to the Yuan period was created. By counting the types of plans and ceilings in each period, a group of the most common types of plans was determined: with a combination of larger and smaller rectangles (where the smaller is the altar part of a square or rectangular shape), a plan in the form of a square with a protruding trapezoidal altar, an altar in outline close to a rectangle or trapezoids with rounded corners, a horizontally elongated plan rectangle, based on squares with protrusions. The most common type of ceilings is fu-dow-din with modifications.

16 Development of a Systematic Scoring System to Measure Adherence to a Temporal-Spatial Heuristic when Completing the Rey Complex Figure Task

Objective:It has been established that capturing how an individual draws the Rey Complex Figure Task (RCF) is as important as assessing what is drawn (Rey, 1941, Osterrieth, 1944). Despite the development of multiple systems that have been designed to measure these qualitative characteristics there are still no systematic means to measure adherence to the temporal-spatial heuristic that represents a typical drawing practice in healthy, neurotypical adults (Visser, 1973; Hamby et al, 1993).This study sought to develop a system for scoring temporal-spatial adherence when drawing the figure to provide objective, continuous data.Participants and Methods:Fifty-three English-speaking adults (mean age 44.61 yrs, SD 12.48; 44 female) were recruited. Exclusion criteria included vision and hearing impairment not corrected by aids; neurodivergent, neurological or psychiatric diagnosis, cancer or brain injury history. Participants completed the RCF copy phase as part of an extended neuropsychological battery. The RCF drawing process was recorded via video and a ball-point pen that digitally recorded drawing. Order data for the 18 RCF elements (Osterrieth, 1944,Taylor, 1959) was recorded by two scorers and analysed via Principal Component Analysis (PCA) with an equimax rotation to identify elements typically drawn together by a healthy, neurotypical adult. Using scoring methodology adapted from Geary et al (2011), the extent to which participants drew consecutively the member elements of each factor or 'strategy cluster’ was calculated and recorded. Strategy Cluster Scores across the population sample were examined to understand normative performance.Results:Order data was examined for interrater reliability via Pearson’s correlation coefficient and was considered good (r2 = 0.78, p < 0.001). PCA identified four factors or 'strategy clusters’ that were statistically robust and accounted for 67.34% of total variation. The strategy clusters were Core Structure (rectangle, diagonal, horizontal, vertical); Triangular Structure (triangle, horizontal in triangle, vertical in triangle, diamond); Internal Left-Hand Side (four horizontal lines, smaller rectangle, horizontal in top-left quad); and Internal Right-Hand Side (five lines, circle, vertical top-right quad, small triangle). The mean RCF Strategy Cluster Score was 6.23 (SD 1.94; possible range: 2.75 to 10). Population data spread indicated that healthy neurotypical adults only partially observed a temporal-spatial heuristic, rather than strict, absolute adherence.Conclusions:Four strategy clusters were identified where cluster members were typically drawn consecutively. RCF Cluster Strategy scoring was shown to measure the temporal-spatial heuristic objectively, providing continuous data that lends itself to clinical standardisation. Further, the study demonstrated that whilst healthy, neurotypical adults copy the RCF using a temporal-spatial heuristic, it is only partially adhered to. Traditionally deviation from strict adherence to the four strategy clusters during drawing was deemed to be indicative of cognitive dysregulation, however our findings demonstrate a normal distribution of typical population performance. These findings have important implications for interpreting how RCF drawing strategy informs clinical assessment and diagnosis as both very strict and very weak adherence to a temporal-spatial heuristic can be indicative of atypical function. The study supports this novel scoring system as a fast and reliable means to systematically measure RCF Cluster Strategy that with further validation could be adopted within clinical practice.

“Into the Widest Imagination”

“Into the Widest Imagination”

Early warning of Diaporthe infection in kiwifruit soft rot by plasmonic dimer-enhanced Raman spectroscopy.

Diaporthe genus is the dominant pathogens of kiwifruit soft rot with long incubation period and rapid onset and very hard to detect in advance. It is of great significance to develop point-of-care tests for disease prevention and field management. Here we screen a pair of genus-level primers for constructing plasmonic dimer Rayleigh/Raman spectroscopy, which enables rapid, specific, and sensitive detection of Diaporthe genus in real kiwifruits. Dark-filed Rayleigh scattering clearly visualizes the target-induced dimer assembly of nanoprobes. Plasmonic dimer-enhanced Raman spectroscopy has high specificity for 12 common isolates of Diaporthe genus in one batch samples with highly accurate results and much higher sensitivity than polymerase chain reaction (PCR). It realizes the recognition of Diaporthe infection at an early stage of 24h when the kiwifruits do not have any noticeable symptoms. It demonstrates a bright prospect of point-of-care tests for early warning of kiwifruit soft rot infection and quality control.

The Floating-Cuts model: a general and flexible mixed-integer programming model for non-guillotine and guillotine rectangular cutting problems

Cutting and packing problems are challenging combinatorial optimization problems that have many relevant industrial applications and arise whenever a raw material has to be cut into smaller parts while minimizing waste, or products have to be packed, minimizing the empty space. Thus, the optimal solution to these problems has a positive economic and environmental impact.In many practical applications, both the raw material and the cut parts have a rectangular shape, and cutting plans are generated for one raw material rectangle (also known as plate) at a time. This is known in the literature as the (two-dimensional) rectangular cutting problem. Many variants of this problem may arise, led by cutting technology constraints, raw-material characteristics, and different planning goals, the most relevant of which are the guillotine cuts. The absence of the guillotine cuts imposition makes the problem harder to solve to optimality.Based on the Floating-Cuts paradigm, a general and flexible mixed-integer programming model for the general rectangular cutting problem is proposed. To the best of our knowledge, it is the first mixed integer linear programming model in the literature for both non-guillotine and guillotine problems. The basic idea of this model is a tree search where branching occurs by successive first-order non-guillotine-type cuts. The exact position of the cuts is not fixed, but instead remains floating until a concrete small rectangle (also known as item) is assigned to a child node. This model does not include decision variables either for the position coordinates of the items or for the coordinates of the cuts. Under this framework, it was possible to address various different variants of the problem.Extensive computational experiments were run to evaluate the model’s performance considering 16 different problem variants, and to compare it with the state-of-the-art formulations of each variant. The results confirm the power of this flexible model, as, for some variants, it outperforms the state-of-the-art approaches and, for the other variants, it presents results fairly close to the best approaches. But, even more importantly, this is a new way of looking at these problems which may trigger even better approaches, with the consequent economic and environmental benefits.

Parallel border tracking in binary images using GPUs

Border tracking in binary images is an important kernel for many applications. There are very efficient sequential algorithms, most notably, the algorithm proposed by Suzuki et al., which has been implemented for CPUs in well-known libraries. However, under some circumstances, it would be advantageous to perform the border tracking in GPUs as efficiently as possible. In this paper, we propose a parallel version of the Suzuki algorithm that is designed to be executed in GPUs and implemented in CUDA. The proposed algorithm is based on splitting the image into small rectangles. Then, a thread is launched for each rectangle, which tracks the borders in its associated rectangle. The final step is to perform the connection of the borders belonging to several rectangles. The parallel algorithm has been compared with a state-of-the-art sequential CPU version, using two different CPUs and two different GPUs for the evaluation. The computing times obtained show that in these experiments with the GPUs and CPUs that we had available, the proposed parallel algorithm running in the fastest GPU is more than 10 times faster than the sequential CPU routine running in the fastest CPU.

High gain of 28 GHz transparent antenna for 5G NR Networks

A new compact transparent patch antenna for 5G new radio (NR) is presented in this paper. Small rectangles are cut out from the patch to produce the desired frequency 28 GHz. The AgHT-8 material is used to from the radiator and the ground of the antenna while the glass is used as a substrate with a relative permittivity of εr = 4.6, tanδ = 0.0036 and thickness of 1.1mm. The suggested antenna occupies a small footprint of only 104,72 mm2 (11mm x 9.52 mm). The frequency bandwidth operation of the antenna is 3.624GHz (26.369 - 29.993 GHz) centered at 28GHz, the efficiency is 76% and the gain is about 5.1 dBi. The excellent performance allows the transparent antenna suitable for 5G applications.

A Bounding Box-Based Radiomics Model for Detecting Occult Peritoneal Metastasis in Advanced Gastric Cancer: A Multicenter Study

PurposeTo develop a bounding box (BBOX)-based radiomics model for the preoperative diagnosis of occult peritoneal metastasis (OPM) in advanced gastric cancer (AGC) patients.Materials and Methods599 AGC patients from 3 centers were retrospectively enrolled and were divided into training, validation, and testing cohorts. The minimum circumscribed rectangle of the ROIs for the largest tumor area (R_BBOX), the nonoverlapping area between the tumor and R_BBOX (peritumoral area; PERI) and the smallest rectangle that could completely contain the tumor determined by a radiologist (M_BBOX) were used as inputs to extract radiomic features. Multivariate logistic regression was used to construct a radiomics model to estimate the preoperative probability of OPM in AGC patients.ResultsThe M_BBOX model was not significantly different from R_BBOX in the validation cohort [AUC: M_BBOX model 0.871 (95% CI, 0.814–0.940) vs. R_BBOX model 0.873 (95% CI, 0.820–0.940); p = 0.937]. M_BBOX was selected as the final radiomics model because of its extremely low annotation cost and superior OPM discrimination performance (sensitivity of 85.7% and specificity of 82.8%) over the clinical model, and this radiomics model showed comparable diagnostic efficacy in the testing cohort.ConclusionsThe BBOX-based radiomics could serve as a simpler reliable and powerful tool for the preoperative diagnosis of OPM in AGC patients. And M_BBOX-based radiomics is simpler and less time consuming.

A robust framework combined saliency detection and image recognition for garbage classification

Using deep learning to solve garbage classification has become a hot topic in computer version. The most widely used garbage dataset Trashnet only has garbage images with a white board as background. Previous studies based on Trashnet focus on using different networks to achieve a higher classification accuracy without considering the complex backgrounds which might encounter in practical applications. To solve this problem, we propose a framework that combines saliency detection and image classification to improve the generalization performance and robustness. A saliency network Salinet is adopted to obtain the garbage target area. Then, a smallest rectangle containing this area is created and used to segment the garbage. A classification network Inception V3 is used to identify the segmented garbage image. Images of the original Trashnet are fused with complex backgrounds of the other saliency detection datasets. The fused and original Trashnet are used together for training to improve the robustness to noises and complex backgrounds. Compared with the image classification networks and classic target detection algorithms, the proposed framework improves the accuracy of 0.50% − 15.79% on the testing sets fused with complex backgrounds. In addition, the proposed framework achieves the best performance with a gain of 4.80% in accuracy on the collected actual dataset. The comparisons prove that our framework is more robust to garbage classification in complex backgrounds. This method can be applied to smart trash cans to achieve automatic garbage classification.

"Craniotomy Within Craniotomy" Technique for Safe Resection of an Intracranial Arteriovenous Malformation With Frontal Bony Erosion: 2-Dimensional Operative Video.

Surgical resection is one option in the treatment of large high-grade brain arteriovenous malformations (AVMs). Resection of AVMs with skull-eroding components can be challenging due to the risk of excessive bleeding from these components during craniotomy and bone flap removal. We present a case of a 25-yr-old woman who presented with an acute onset right-sided frontal headache. She was found to have a large, frontal Spetzler-Martin grade IV AVM with an associated dural AVM. The AVM had caused focal erosions of the right frontal bone by a venous varix traversing the region of the calvarial defect. An elective staged endovascular embolization followed by surgical resection was recommended considering the patient's young age and the large size of the AVM located in a noneloquent area. Given the high risk of intraoperative hemorrhage during the craniotomy portion of the procedure, a "craniotomy within craniotomy" approach was planned. During this approach, a small rectangle of bone, including the portion eroded by the venous varix, was left in place, while the larger bone flap surrounding it was removed for an initial approach to the AVM. The small bony piece was safely removed at later stages of resection once the arterial feeders had been reasonably obliterated. Immediate postoperative catheter angiogram demonstrated good filling of the intracranial vascular territories with no residual AVM. The patient developed mild left facial and left hand weakness postoperatively, which resolved after 2 wk of follow-up. The patient remained neurologically intact on further follow-up.

Broadband Dual-Polarized Multidipole Antenna for Base Station Applications

A wideband dual-polarized multidipole antenna for base station applications is proposed. It consists of a pair of large square-shaped loop dipoles and a pair of small rectangle loop dipoles as radiation elements. A pair of small rectangle loop dipoles is fed by T-shaped feed structure which is in the large square-shaped loop dipoles radiating arm so that the antenna generates an additional resonance and obtains a wider bandwidth. The proposed antenna was fabricated and measured, and the results show that the antenna achieves a wide impedance bandwidth of 63.7 % with VSWR<1.5 covering the frequency range from 1.55 to 3 GHz. A high isolation is better than 29 dB within the operating frequency bandwidth. Moreover, an average gain 8 dBi and a stable radiation pattern with 3 dB beamwidth of 69° ± 4° at H-plane are obtained.

A Bounding Box-Based Radiomic Model for Detecting Occult Peritoneal Metastasis in Advanced Gastric Cancer: A Multicenter Study

Background: To develop a radiomics model for bounding box-based region of interest (ROI) annotation for the preoperative diagnosis of occult peritoneal metastasis (OPM) in advanced gastric cancer (AGC) patients. Methods: 599 AGC patients from 3 centers were retrospectively enrolled and were divided into training, validation, and testing cohorts. The ROIs for the largest tumor area, the minimum circumscribed rectangle of the ROIs (BBOX), the nonoverlapping area between the ROIs and BBOX (peritumoral area; PERI) and the smallest rectangle that could completely contain the lesion determined by a radiologist (M_BBOX) were used as inputs to extract radiomic features. Multivariate logistic regression was used to construct a radiomics model to estimate the preoperative probability of OPM in GC patients. Findings: The BBOX radiomics model had the highest area under the receiver operating characteristic curve (AUC) of 0.873 [0.820-0.940], outperforming the other radiomics models. The M_BBOX model was not significantly different from BBOX in the validation cohort (AUC: M_BBOX model 0.871 [0.814-0.940] vs. BBOX model 0.873 [0.820-0.940]; p = 0.937). M_BBOX was selected as the final radiomics model because of its extremely low annotation cost and superior OPM discrimination performance (sensitivity of 85.7% and specificity of 82.8%) over the clinical model, and this radiomics model showed comparable diagnostic efficacy in the testing cohort. Interpretation: The radiomics model based on bounding box annotation achieved higher OPM detection accuracy in AGC patients than the clinical model and had potential generalizability. Funding Information: This work was supported by the 1*3*5 Project for Disciplines of Excellence, West China Hospital, Sichuan University (No. ZY2017304). Declaration of Interests: The authors have no relevant conflicts of interest to disclose. Ethics Approval Statement: This multicenter retrospective study was approved by the institutional review board of each center, and the informed consent requirement was waived.

Effects of tie detailing configurations on reinforcement buckling and seismic performance of high-strength RC columns

The premature buckling and subsequently low-cycle fatigue fracture of longitudinal bars is one of the main failure modes of reinforced concrete (RC) columns under earthquake loading. The use of high-strength steel bars (HSSB) in RC columns reduces rebar diameter or increases tie spacing, thus increases the vulnerability of bar buckling. In view of this, this paper studies the effects of transverse reinforcement detailing (referred hereafter also as ‘tie detailing’) configurations on reinforcement buckling and then on the seismic performance of high-strength RC columns. Five rectangular RC columns with different tie detailings and axial load ratios were tested under constant axial load and reversed horizontal load. All three tie detailings studied have the same large rectangle closed-hoop, while small hoops/ties are different: including two cross-ties (Type Ⅰ), a small rhombus closed-hoop (Type Ⅱ), or two small rectangle closed-hoops (Type Ⅲ). The axial load ratio includes 0.06 and 0.18. The anti-buckling constraint mechanisms of different tie detailings are proposed, and then the global buckling length of longitudinal bars in RC columns with different tie detailings are calculated. On this basis, the effects of tie detailing on bar buckling and on the seismic performance of RC columns are studied. The observed results of bar buckling verify the reliability of the proposed anti-buckling constraint mechanisms and reinforcement buckling length calculation model. Type Ⅲ tie detailing provide better anti-buckling constraint, and thus reduces buckling length of longitudinal bars of RC columns than that with Type Ⅰ & Type Ⅱ tie detailings. Meanwhile, the use of Type Ⅲ tie detailing enhances the seismic performance of RC columns (ultimate drift ratio and hysteretic dissipated energy) than that with Type Ⅰ & Type Ⅱ tie detailings. The effects of tie detailing on seismic performance of RC columns are more significant under high axial load ratio. As the longitudinal bars are more susceptible to buckle under high axial compression, thus Type Ⅲ tie detailing can more prominently play its advantages of delaying bar buckling and improving the seismic performance of RC columns.

Characterising the influence of milk fat towards an application for extrusion-based 3D-printing of casein−whey protein suspensions via the pH−temperature-route

This study presents the design and characterisation of casein−whey protein suspensions (8.0/10.0% (w/w) casein and 2.0/2.5% (w/w) whey protein) mixed with dairy fat (1.0, 2.5 and 5.0% (w/w) total fat) processed via the pH−temperature-route in preparation for 3D-printing. Mechanical treatment was applied to significantly decrease the particle size of the milk fat globules and increase surface area, creating small fat globules (<1 μm) covered with proteins, which could act as pseudo protein particles during gelation. Different proteins covered the fat globule surface after mechanical treatment, as a result of differences in the pH adjusted just prior to heating (6.55, 6.9 or 7.1). The protein-fat suspensions appeared similar by transmission electron cryogenic microscopy and the zeta-potential of all particles was unchanged by the heating pH, with a similar charge to the solution (~−20 mV) occurring after acidification (pH 4.8/5.0) at low temperatures (2 °C). A low heating pH (6.55) resulted in increased sol−gel transition temperatures (G՛ = 1 Pa) and a decreased rate of aggregation for protein−fat suspensions. A higher heating pH (6.9 and 7.1) caused an increased rate of aggregation (aggregation rate ≥ 250 Pa/10 K), resulting in materials more promising for application in extrusion-based printing. 3D-printing of formulations into small rectangles, inclusive of a sol−gel transition in a heated nozzle, was conducted to relate the aggregation rate towards printability.

Knowledge Granularity for Continuous Parameters

In the community of Granular Computing, knowledge is interpreted as one classification ability of realistic or abstract objects. Generally, the concept of granularity is used for characterizing such an ability, which has been widely explored in literatures. To calculate the parameterized granularity, a naive approach is to find the granularity in terms of the parameter one by one. Nevertheless, such approach can only generate the single parameter based knowledge granularity, and the difference of knowledge granularities among different parameters may be slight. It follows that the knowledge granularity derived from single parameter may be lack of representativeness. In this paper, the continuous parameters based knowledge granularity is proposed, and the corresponding calculation approach is presented. Inspired by the thinking of definite integral in mathematical problems, the calculation approach is mainly implemented by following steps: firstly, the graph formed by granularity and parameter interval is divided into several small rectangles whose length of interval tends to be 0; secondly, the sum of area values of all the small rectangles is calculated; finally, the obtained area value divided by the whole length of parameter interval can be considered as the continuous parameters based knowledge granularity. This study suggests a new trend of handling problems related to knowledge from the viewpoint of continuity.

Исследование орнаментированных изделий из кости стоянки Ольская с применением трехмерного моделирования

This paper discusses the practical application of three-dimensional modelling technologies in studying bone artefacts with engraved images of the paleometal era. To achieve the study goal, we carried out a comprehensive analysis of the antler crest obtained from excavations at the Olskaya site, one of the key archaeological sites of the Tokarev culture in the Northern Priokhotye. The artefact under discussion has a complex geometric pattern, which made it possible to use it to test various tools of 3D modelling technology. As a result of our research, we reconstructed the operational sequence of the item production, which involved splitting a reindeer antler to obtain a blank, planing, double-sided cutting of teeth, surface preparation by grinding, ornamentation, and perforation. The linear ornamentation on the crest makes up a composition of horizontal motifs as stripes of small rectangles and zigzags made in double lines. 3D scanning technologies made it possible to obtain accurate data on the morphometric parameters of decorative elements, the sequence of their application and the tools used. As a result of our measurements, we found out that the metric characteristics ofwhole rectangles in the ornamented composition differ significantly. The study results showed a minimum difference in the cut lines ’depth, both in rectangles of all rows and diagonal stripes. The revealed features made it possible to assume that one master who used the same tool cut the ornament. The nature of the processing traces shows that the design of the teeth and decoration were made with different tools. The obtained data indicate the presence of specialization in the toolkit of bone carving production and a generally high level of bone processing technologies among the Tokarev culture people.

A Study of Crofton Type Formulas for Surface Area in Three-Dimension

Every three-dimensional object can be computed by a two-dimensional plane with the help of integration. Inspired by Crofton formulas and the Cavalieri's Principle, this work derives a general method for the surface area of a polygonal in three-dimensional space. In fact, the surface area of a three-dimensional object can be subdivided into a finite number of small rectangle. This research represents the area of a rectangle by the number of the intersection point between the rectangle and the line passing through the rectangle in all directions. Next, this research computes the proportionality constant D of integration. Eventually, this research extends the result to a boarder discussion on the application of the obtained result to a smooth surface in R^3. Within the process of integration , the volume of a four-dimensional object in TS2 is calculated. This research jumps out from the conventional representation of the surface area using the one-dimension integral geometry. The reader will realize another technique of representing the surface area with the integration of the number of intersection points in a sub-divided parallelograms. Moreover, not only does the research extent the concept of Cavalieri's principle to a three-dimensional application, but also the solution incites a possible way using the intersection point to explore the volume or the surface area of an object in a higher dimension world.

New water mites of Torrenticolidae (Acari, Hydrachnidia) from Jiangxi Province, P. R. China

Three torrenticolid species new to science, i.e. Monatractides trilaminatus sp. nov., Torrenticola lushanensis sp. nov., and T. planusirostrum sp. nov. are described from Jiangxi Province, P. R. China. M. trilaminatus sp. nov. can be distinguished by the shoulder and frontal platelets fused together and formed into a pair of platelets; infracapitular bay U-shaped, extremely deep and narrow; the tip of coxae I with a small rectangle extension. T. lushanensis sp. nov. can be distinguished by epimeroglandularia 4 at the same level as the 4th pair of acetabula; anal pore on the same line with ventroglandularia 2, and posterior to ventroglandularia 1. T. planusirostrum sp. nov. can be distinguished by infracapitular dorsal apodeme almost absent, ventral apodeme blunt and long, rostrum flat and parallel to the ventral apodeme. Detailed descriptions and illustrations of these species are given in this paper.

Positions and covering: A two-stage methodology to obtain optimal solutions for the 2d-bin packing problem.

We present a two-stage methodology called Positions and Covering (P&C) to solve the two-dimensional bin packing problem (2D-BPP). The objective of this classical combinatorial NP-hard problem is to pack a set of items (small rectangles) in the minimum number of bins (larger rectangles). The first stage is the key-point of the Positions and Covering, where for each item, it is generated in a pseudo-polynomial way a set of valid positions that indicate the possible ways of packing the item into the bin. In the second stage, a new set-covering formulation, strengthen with three sets of valid inequalities, is used to select the optimal non-overlapping configuration of items for each bin. Experimental results for the P&C method are presented and compared with some of the best algorithms in the literature for small and medium size instances. Furthermore, we are considering both cases of the 2D-BPP, with and without rotations of the items by 90°. To the best of our knowledge, this is one of the first exact approaches to obtain optimal solutions for the rotation case.