Shape Model Research Articles

Global optimization of underwater vehicle hull shape on flow resistance

For underwater vehicle with axisymmetric revolution body, the hull shape in the rotation axis section is the key issue on flow resistance. In this paper, we propose to use the elliptic Fourier descriptors for arbitrary hull shape modeling. By doing so, we can achieve global optimization of underwater vehicle hull shape on flow resistance. Experimental results using the computational fluid dynamics show superior improvement compared to the extensive used Myring shape with only first order elliptic Fourier descriptors under the same displacement constraint. In addition, the proposed method can be easily extend to the global optimization on flow noise.

Development of a handheld integrating sphere snow grain sizer (HISSGraS)

Abstract We developed a Handheld Integrating Sphere Snow Grain Sizer (HISSGraS) for field use to measure the specific surface area (SSA) of snow. In addition to snow samples, HISSGraS can directly measure snow surfaces and snow pit walls. The basic measurement principle is the same as that of the IceCube SSA instrument. The retrieval algorithm for SSA from reflectance employs two conversion equations formulated using spherical and nonspherical grain shape models. We observed SSAs using HISSGraS, IceCube and the gas adsorption method in a snowfield in Hokkaido, Japan. Intercomparison of the results confirmed that with HISSGraS direct measurement, SSA profile observations can be completed in just ~1/10 the time required for measurement of snow samples. Our results also suggest that HISSGraS and IceCube have similar accuracy when the same snow samples are measured using the same grain shape model. However, SSAs of near-surface snow layers measured using the three techniques exhibited some biases, possibly due to rapid snow metamorphism or melting during measurement and some technical issues with optical techniques. When excluding SSA data for the surface layer, which metamorphosed remarkably during measurement, IceCube- and HISSGraS-derived SSAs correlated strongly with those obtained by gas adsorption and HISSGraS accuracy is 21–34%.

Automated surgical planning in spring-assisted sagittal craniosynostosis correction using finite element analysis and machine learning.

Sagittal synostosis is a condition caused by the fused sagittal suture and results in a narrowed skull in infants. Spring-assisted cranioplasty is a correction technique used to expand skulls with sagittal craniosynostosis by placing compressed springs on the skull before six months of age. Proposed methods for surgical planning in spring-assisted sagittal craniosynostosis correction provide information only about the skull anatomy or require iterative finite element simulations. Therefore, the selection of surgical parameters such as spring dimensions and osteotomy sizes may remain unclear and spring-assisted cranioplasty may yield sub-optimal surgical results. The aim of this study is to develop the architectural structure of an automated tool to predict post-operative surgical outcomes in sagittal craniosynostosis correction with spring-assisted cranioplasty using machine learning and finite element analyses. Six different machine learning algorithms were tested using a finite element model which simulated a combination of various mechanical and geometric properties of the calvarium, osteotomy sizes, spring characteristics, and spring implantation positions. Also, a statistical shape model representing an average sagittal craniosynostosis calvarium in 5-month-old patients was used to assess the machine learning algorithms. XGBoost algorithm predicted post-operative cephalic index in spring-assisted sagittal craniosynostosis correction with high accuracy. Finite element simulations confirmed the prediction of the XGBoost algorithm. The presented architectural structure can be used to develop a tool to predict the post-operative cephalic index in spring-assisted cranioplasty in patients with sagittal craniosynostosis can be used to automate surgical planning and improve post-operative surgical outcomes in spring-assisted cranioplasty.

Automated tooth crown design with optimized shape and biomechanics properties

Despite the large demand for dental restoration each year, the design of crown restorations is mainly performed via manual software operation, which is tedious and subjective. Moreover, the current design process lacks biomechanics optimization, leading to localized stress concentration and reduced working life. To tackle these challenges, we develop a fully automated algorithm for crown restoration based on deformable model fitting and biomechanical optimization. From a library of dental oral scans, a conditional shape model (CSM) is constructed to represent the inter-teeth shape correlation. By matching the CSM to the patient’s oral scan, the optimal crown shape is estimated to coincide with the surrounding teeth. Next, the crown is seamlessly integrated into the finish line of preparation via a surface warping step. Finally, porous internal supporting structures of the crown are generated to avoid excessive localized stresses. This algorithm is validated on clinical oral scan data and achieved less than 2 mm mean surface distance as compared to the manual designs of experienced human operators. The mechanical simulation was conducted to prove that the internal supporting structures lead to uniform stress distribution all over the model.

In vivo marker of brainstem myelin is associated to quantitative sleep parameters in healthy young men

The regional integrity of brain subcortical structures has been implicated in sleep–wake regulation, however, their associations with sleep parameters remain largely unexplored. Here, we assessed association between quantitative Magnetic Resonance Imaging (qMRI)-derived marker of the myelin content of the brainstem and the variability in the sleep electrophysiology in a large sample of 18-to-31 years healthy young men (N = 321; ~ 22 years). Separate Generalized Additive Model for Location, Scale and Shape (GAMLSS) revealed that sleep onset latency and slow wave energy were significantly associated with MTsat estimates in the brainstem (pcorrected ≤ 0.03), with overall higher MTsat value associated with values reflecting better sleep quality. The association changed with age, however (MTsat-by-age interaction—pcorrected ≤ 0.03), with higher MTsat value linked to better values in the two sleep metrics in the younger individuals of our sample aged ~ 18 to 20 years. Similar associations were detected across different parts of the brainstem (pcorrected ≤ 0.03), suggesting that the overall maturation and integrity of the brainstem was associated with both sleep metrics. Our results suggest that myelination of the brainstem nuclei essential to regulation of sleep is associated with inter-individual differences in sleep characteristics during early adulthood. They may have implications for sleep disorders or neurological diseases related to myelin.

Implicit Shape Model Trees: Recognition of 3-D Indoor Scenes and Prediction of Object Poses for Mobile Robots

Implicit Shape Model Trees: Recognition of 3-D Indoor Scenes and Prediction of Object Poses for Mobile Robots

Transplanted seedling age and watering effects on the field performance of Senegalia macrostachya (Rchb. ex DC.) Kyal. & Boatwr., a high-valued indigenous fruit tree species in Burkina Faso

Senegalia macrostachya, ex-Acacia macrostachya is a multiple-use species providing many benefits to populations in arid and semi-arid regions. Despite its importance, limited research has been undertaken on its silviculture, growth, and productivity in plantations or in the wild. In this paper, we widely deployed the Generalized Additive Models for Location, Scale and Shape to identify factors controlling the temporal variability in S macrostachya annual pod and seed production. Data (survival rate, basal diameter, total height of each individual tree, and weight of pods and seeds produced) were collected from a factorial experiment testing different nursery production periods and the field performance of S. macrostachya under well-watered and water-stressed conditions. Long-term climate data were analyzed and the Standardized Precipitation–Evapotranspiration Index (SPEI) was calculated. Results showed that seedling survival and both height and radial growth were significantly influenced by seedling age at outplanting, watering, and climate aridity. The watered seedlings had the highest survival rate and grew taller than the non-watered seedlings. The watered seedlings also had the largest basal diameters, along with the highest pod and seed weights. There was temporal variation in S. macrostachya pod and seed production over time, indicating a cyclic production, depending mainly on climate factors. Seedling age at outplanting, early watering of outplanted seedlings, and climate aridity significantly influence not only seedling survival and growth parameters but also determine the levels of pod and seed production of S. macrostachya. Our results provide field-based evidence of the performance of this important species in the production of pods and seeds. The results also highlight the potential for domestication of S. macrostachya including use in plantations to help restore degraded rural landscapes in arid and semi-arid regions.

DeepSSM: A blueprint for image-to-shape deep learning models

Statistical shape modeling (SSM) characterizes anatomical variations in a population of shapes generated from medical images. Statistical analysis of shapes requires consistent shape representation across samples in shape cohort. Establishing this representation entails a processing pipeline that includes anatomy segmentation, image re-sampling, shape-based registration, and non-linear, iterative optimization. These shape representations are then used to extract low-dimensional shape descriptors that are anatomically relevant to facilitate subsequent statistical analyses in different applications. However, the current process of obtaining these shape descriptors from imaging data relies on human and computational resources, requiring domain expertise for segmenting anatomies of interest. Moreover, this same taxing pipeline needs to be repeated to infer shape descriptors for new image data using a pre-trained/existing shape model. Here, we propose DeepSSM, a deep learning-based framework for learning the functional mapping from images to low-dimensional shape descriptors and their associated shape representations, thereby inferring statistical representation of anatomy directly from 3D images. Once trained using an existing shape model, DeepSSM circumvents the heavy and manual pre-processing and segmentation required by classical models and significantly improves the computational time, making it a viable solution for fully end-to-end shape modeling applications. In addition, we introduce a model-based data-augmentation strategy to address data scarcity, a typical scenario in shape modeling applications. Finally, this paper presents and analyzes two different architectural variants of DeepSSM with different loss functions using three medical datasets and their downstream clinical application. Experiments showcase that DeepSSM performs comparably or better to the state-of-the-art SSM both quantitatively and on application-driven downstream tasks. Therefore, DeepSSM aims to provide a comprehensive blueprint for deep learning-based image-to-shape models.

A Competitive Segmentation of the Human Brain using Artificial Neural Network Approach Towards MRI

The diagnosis of brain abnormalities, prognosis monitoring, and treatment evaluation all rely heavily on the magnetic resonance scan of brain tissue segmentation. Although numerous automated or semi-automatic methods have been proposed in the literature to reduce the need for human intervention, the degree of accuracy is frequently still significantly lower than that of manual segmentation. We give a clever technique for fragmenting the cerebrum utilizing a managed counterfeit brain organization system called artificial neural network (ANN) and volumetric shape models. In the beginning, in addition to the usual spatial-based and intensity-based image features, a level-set oriented brain boundary fitting technique is used to accomplish this. This is controlled by the picture intensity. The ANN is then informed of the number of important structures. Additionally, rather than directly applying standard guidelines to local appearances, this ANN learns local adaptive feature classification conditions. The outcomes demonstrate that the proposed strategy achieves competitive results in a relatively shorter time spent training.

Revisiting Dimorphos formation: A pyramidal regime perspective and application to Dinkinesh’s satellite

Dimorphos’ oblate shape challenges formation models. Landslides on Didymos, induced by YORP effect, probably created a debris ring from which Dimorphos would have formed. Nonetheless, ring-derived satellites typically form with a prolate lemon shape. In light of the newest Dimorphos shape model, we revisit our previous work, Madeira et al. (2023a), and conducted new 1D simulations with material deposition in an extended region from 1.0 to 1.5 Didymos radii. An instantaneous landslide leads to a fast formation of a prolate Dimorphos directly from the ring. Now, if Didymos progressively deposits mass, Dimorphos grows through low-velocity impacts with large impactor-to-target mass ratio (pyramidal regime growth). Even during rapid, day-scale depositions, Dimorphos experiences one of these impacts, while for slower depositions, the satellite formation is primarily via pyramidal impacts. This process might reshape the satellite into an oblate shape (Leleu et al., 2018) or even in a contact-binary shape, a scenario worthy of investigation that should be studied in the future with more suitable tools. Our conclusions can be applied to Dinkinesh’s satellite, recently discovered by NASA’s Lucy mission.

ERS technical standard: Global Lung Function Initiative reference values for exhaled nitric oxide fraction (F ENO50 ).

Elevated exhaled nitric oxide fraction at a flow rate of 50 mL·s-1 (F ENO50 ) is an important indicator of T-helper 2-driven airway inflammation and may aid clinicians in the diagnosis and monitoring of asthma. This study aimed to derive Global Lung Function Initiative reference equations and the upper limit of normal for F ENO50 . Available individual F ENO50 data were collated and harmonised using consensus-derived variables and definitions. Data collected from individuals who met the harmonised definition of "healthy" were analysed using the generalised additive models of location, scale and shape (GAMLSS) technique. Data were retrospectively collated from 34 782 individuals from 34 sites in 15 countries, of whom 8022 met the definition of healthy (19 sites, 11 countries). Overall, height, age and sex only explained 12% of the between-subject variability of F ENO50 (R2=0.12). F ENO device was neccessary as a predictor of F ENO50 , such that the healthy range of values and the upper limit of normal varied depending on which device was used. The range of F ENO50 values observed in healthy individuals was also very wide, and the heterogeneity was partially explained by the device used. When analysing a subset of data in which F ENO50 was measured using the same device and a stricter definition of health (n=1027), between-site heterogeneity remained. Available F ENO50 data collected from different sites using different protocols and devices were too variable to develop a single all-age reference equation. Further standardisation of F ENO devices and measurement are required before population reference values might be derived.

Reliably and accurately estimate energy in super-capacitor via a model of cyclic voltammetry

As effective energy storage device super-capacitors have been widely applied in energy storage field. Cyclic voltammetry (CV) test is utilized to characterize the electrochemical performance of super-capacitors. Even if there are basic formulas to estimate specific capacitance by integral of CV, the integrable model of CV was not given in these literatures. Meanwhile, storage energy during one CV has not been known up to now. Researchers are keen to be able to calculate energy from CV. However, most current studies have not already provided integrable model of CV which is in any shape. The establishment of the accurate CV model is an important basis to realize the reliable estimation of energy density in super-capacitors via CV measurement, and also a necessary work to develop the super-capacitors management system. To fill this gap of knowledge, correct basic formula for calculation energy (density) is provided in this work for the first time in this work. To reliably and accurately calculate energy (density), a novel equivalent circuit model as dynamic model in explicit functional form of quasi ellipse shape of CV is established in this work. This quasi ellipse shape model of CV was validated on a set of thirty-five experimental CV curves, the results calculated from quasi ellipse model agree well with the experimental data because the essential novelty of the proposed model is that it is from Kirchhoff's current law and this law tells us how to calculate the total current density of a parallel circuit. Because this quasi ellipse shape model of CV is confirmed through experimental results and the integral of the model is also in explicit functional form, reliable and accurate energy (density) of each CV curve was also obtained. As another essential innovation of the proposed methodology is that quantitative relationship between the scan rate, potential window, intrinsic performance parameters of super-capacitor, measurement temperature, bending angle and energy (density) is quantitatively searched, this work makes a significant practical contribution and it makes impact on the research work on the research community. Hence, more scientific and practical significance of CV can be gained from the passage of this article.

Generation of synthetic aortic valve stenosis geometries for in silico trials.

In silico trials are a promising way to increase the efficiency of the development, and the time to market of cardiovascular implantable devices. The development of transcatheter aortic valve implantation (TAVI) devices, could benefit from in silico trials to overcome frequently occurring complications such as paravalvular leakage and conduction problems. To be able to perform in silico TAVI trials virtual cohorts of TAVI patients are required. In a virtual cohort, individual patients are represented by computer models that usually require patient-specific aortic valve geometries. This study aimed to develop a virtual cohort generator that generates anatomically plausible, synthetic aortic valve stenosis geometries for in silico TAVI trials and allows for the selection of specific anatomical features that influence the occurrence of complications. To build the generator, a combination of non-parametrical statistical shape modeling and sampling from a copula distribution was used. The developed virtual cohort generator successfully generated synthetic aortic valve stenosis geometries that are comparable with a real cohort, and therefore, are considered as being anatomically plausible. Furthermore, we were able to select specific anatomical features with a sensitivity of around 90%. The virtual cohort generator has the potential to be used by TAVI manufacturers to test their devices. Future work will involve including calcifications to the synthetic geometries, and applying high-fidelity fluid-structure-interaction models to perform in silico trials.

Improved models for the near-Earth asteroids (2100) Ra-Shalom, (3103) Eger, (12711) Tukmit, and (161989) Cacus

ABSTRACT We present 24 new dense light curves of the near-Earth asteroids (3103) Eger, (161989) Cacus, (2100) Ra-Shalom, and (12711) Tukmit, obtained with the Instituto Astrofísico Canarias 80 and Telescopio Abierto Remoto 2 telescopes at the Teide Observatory (Tenerife, Spain) during 2021 and 2022, in the framework of projects visible NEAs observations survey and NEO Rapid Observation, Characterization and Key Simulations. The shape models and rotation state parameters (P, λ, β) were computed by applying the light curve inversion method to the new data altogether with the archival data. For (3013) Eger and (161989) Cacus, our shape models and rotation state parameters agree with previous works, though they have smaller uncertainties. For (2100) Ra-Shalom, our results also agree with previous studies. Still, we find that a Yarkovsky–O’Keefe–Radzievskii–Paddack acceleration of υ = (0.223 ± 0.237) × 10−8 rad d−2 slightly improves the fit of the light curves, suggesting that (2100) Ra-Shalom could be affected by this acceleration. We also present for the first time a shape model for (12711) Tukmit, along with its rotation state parameters (P = 3.484900 ± 0.000031 h, λ = 27° ± 8°, β = 9° ± 15°).

BladeDISC: Optimizing Dynamic Shape Machine Learning Workloads via Compiler Approach

Compiler optimization plays an increasingly important role to boost the performance of machine learning models for data processing and management. With increasingly complex data, the dynamic tensor shape phenomenon emerges for ML models. However, existing ML compilers either can only handle static shape models or expose a series of performance problems for both operator fusion optimization and code generation in dynamic shape scenes. This paper tackles the main challenges of dynamic shape optimization: the fusion optimization without shape value, and code generation supporting arbitrary shapes. To tackle the fundamental challenge of the absence of shape values, it systematically abstracts and excavates the shape information and designs a cross-level symbolic shape representation. With the insight that what fusion optimization relies upon is tensor shape relationships between adjacent operators rather than exact shape values, it proposes the dynamic shape fusion approach based on shape information propagation. To generate code that adapts to arbitrary shapes efficiently, it proposes a compile-time and runtime combined code generation approach. Finally, it presents a complete optimization pipeline for dynamic shape models and implements an industrial-grade ML compiler, named BladeDISC. The extensive evaluation demonstrates that BladeDISC outperforms PyTorch, TorchScript, TVM, ONNX Runtime, XLA, Torch Inductor (dynamic shape), and TensorRT by up to 6.95×, 6.25×, 4.08×, 2.04×, 2.06×, 7.92×, and 4.16× (3.54×, 3.12×, 1.95×, 1.47×, 1.24×, 2.93×, and 1.46× on average) in terms of end-to-end inference speedup on the A10 and T4 GPU, respectively. BladeDISC's source code is publicly available at https://github.com/alibaba/BladeDISC.

Corrective Osteotomy in a Patient With Congenital Absence of Pronation Based on Three-Dimensional Statistical Shape Modeling.

We present a new indication of a three-dimensional statistical shape model (SSM): a patient with bilateral impaired forearm rotation due to a congenital variance in bone shape. A corrective osteotomy was planned and performed to best match the SSM created by computed tomography (CT) scans of 18 peers. Postoperatively, pronation increased by 70°, and the patient was pain-free. A CT scan showed accurate correction of the deformity and union of all osteotomies. This technique offers opportunities for patients with bilateral nontraumatic osseous forearm pathology.

What do brain endocasts tell us? A comparative analysis of the accuracy of sulcal identification by experts and perspectives in palaeoanthropology.

Palaeoneurology is a complex field as the object of study, the brain, does not fossilize. Studies rely therefore on the (brain) endocranial cast (often named endocast), the only available and reliable proxy for brain shape, size and details of surface. However, researchers debate whether or not specific marks found on endocasts correspond reliably to particular sulci and/or gyri of the brain that were imprinted in the braincase. The aim of this study is to measure the accuracy of sulcal identification through an experiment that reproduces the conditions that palaeoneurologists face when working with hominin endocasts. We asked 14 experts to manually identify well-known foldings in a proxy endocast that was obtained from an MRI of an actual invivo Homo sapiens head. We observe clear differences in the results when comparing the non-corrected labels (the original labels proposed by each expert) with the corrected labels. This result illustrates that trying to reconstruct a sulcus following the very general known shape/position in the literature or from a mean specimen may induce a bias when looking at an endocast and trying to follow the marks observed there. We also observe that the identification of sulci appears to be better in the lower part of the endocast compared to the upper part. The results concerning specific anatomical traits have implications for highly debated topics in palaeoanthropology. Endocranial description of fossil specimens should in the future consider the variation in position and shape of sulci in addition to using models of mean brain shape. Moreover, it is clear from this study that researchers can perceive sulcal imprints with reasonably high accuracy, but their correct identification and labelling remains a challenge, particularly when dealing with extinct species for which we lack direct knowledge of the brain.

Using 3D MRI Bone Shape to Predict Pre-Osteoarthritis of the Knee 2 Years After Anterior Cruciate Ligament Reconstruction

Background: Anterior cruciate ligament (ACL) injury increases risks for osteoarthritis (OA), a poorly modifiable and disabling condition. Joint changes of potentially reversible pre-OA have been described just 2 years after ACL reconstruction (ACLR) when early bone shape changes have also been reported. Purpose: This study evaluates relationships between interlimb differences in tibiofemoral bone shape derived from statistical shape modeling (SSM) of magnetic resonance imaging (MRI) and participant factors on patient-reported outcomes 2 years after unilateral ACLR. Study Design: Cross-sectional study; Level of evidence, 3. Methods: SSM-derived tibiofemoral bone shape and subchondral bone area were assessed from bilateral knee MRI scans of 72 participants with unilateral ACLR (mean age, 34 ± 11 years; 32 women) and compared with a reference cohort of 398 older individuals without OA (mean age, 50 ± 3 years; 213 women). Multivariable logistic regression models examined relationships between participant and surgical factors with interlimb differences in bone shapes or subchondral bone areas. Relationships between patient-reported outcomes and the interlimb differences in bone shape and subchondral area were examined using similar models. Results: Bone shape scores and subchondral bone areas were greater (more OA-like) in ACLR knees than uninjured contralateral knees in every bone metric tested (P≤ .001). Interlimb differences in femur shape scores of participants with ACLR were 65% greater (P < .001) than those of the significantly older reference cohort. Taller height, medial meniscal tears, and decreasing age were associated with larger interlimb differences in shape scores and subchondral areas (P < .05). Bone–patellar tendon–bone (BPTB) autograft recipients demonstrated greater interlimb subchondral area differences compared with allograft recipients (P < .05). Interlimb differences for hamstring autograft recipients did not differ from those with BPTB or allograft. Greater interlimb differences in medial femur subchondral areas were associated with worse patient-reported Knee injury and Osteoarthritis Outcome Score Symptoms (R = 0.27; P = .040). Conclusion: Even in the absence of radiographic OA, just 2 years after unilateral ACLR patients showed greater bone shape scores and subchondral areas consistent with pre-OA in their ACLR knees. Furthermore, greater medial femur bone areas were weakly associated with worse symptoms. Patients who are younger, are taller, have meniscal tears, or have BPTB grafts may be at increased risk for bony asymmetries 2 years after ACLR.

An experiment-informed discrete element modelling study of knife milling for flexural biomass feedstocks

A discrete element method (DEM) based approach is used to study the relationships between material attributes (MAs), processing parameters (PPs), and quality attributes (QAs) for the knife milling of maize stalks. An approximate DEM shape model was conceptualized based on real maize stalks and calibrated based on experimental bending test data for flexural properties (elastic bending stiffness, elastic bending angle limit, elastoplastic ratio, etc.). DEM simulations of maize stalk comminution in a Jordan Reduction Solutions (“JRS”) knife mill were performed to investigate the relationships between the MAs (maize stalk size and breakage stress limit), PPs (impeller rotational speed), and QAs (mass throughput and output particle size distribution (PSD)). The DEM results suggest that stalk length has little influence on mass throughput and PSD, whilst stalks with larger cross sections tend to generate larger sizes of milled particles given the same breakage stress limit. Both the DEM and experimental results show that faster impeller rotation (or higher power) does not necessarily generate higher throughput or smaller output PSD, especially for maize stalks of higher breakage stress limit. The correlations between these MAs, PPs and QAs are found highly stochastic, though breakage stress limit dictates mass throughput, regardless of stalk size. The DEM-predicted output particle size tended to match the experimental data with coarse PSDs based on sieve size but showed weakened fidelity with finer material, indicating the potential for further model improvement.

Automated Process Planning System for Machining Injection Molding Dies Using CAD Models of Product Shapes in STL Format

In this study, we developed a method for automatically generating computer-aided design (CAD) models of injection molding dies. The method only required 3D CAD models of products in the Standard Triangulated Language (STL) format as the input information. We also developed a system for automatically generating numerical control (NC) programs by automating the system process planning necessary for machining the injection molding dies. The method generated CAD models of the injection molding dies by dividing the STL files of the products into triangular meshes on a specified split plane. For injection molding dies with several free curved surfaces, we acquired the tool positions of a ball end mill (as approximated by a spherical shape) and flat drill (as approximated by a cylindrical shape) from the geometrical relationships of the triangles constituting the CAD model. We generated a CAD model of an injection molding die using the proposed method with respect to the CAD model of a product shape to verify the validity of the developed system. Then, we machined the product based on the NC programs and tool position. In addition, we injection molded a product with a machined die to mold it into its original product shape.