a poisson equation-based method for 3d reconstruction of

Research ArticleA Poisson Equation-Based Method for 3D Reconstruction ofAnimated Images

Ziang Lei

School of Digital Art, Xi’an University of Posts and Telecommunications, Xian 710121, China

Correspondence should be addressed to Ziang Lei; [email protected]

Received 8 September 2021; Revised 21 October 2021; Accepted 22 October 2021; Published 27 November 2021

Academic Editor: Miaochao Chen

Copyright © 2021 Ziang Lei. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

3D reconstruction techniques for animated images and animation techniques for faces are important research in computergraphics-related fields. Traditional 3D reconstruction techniques for animated images mainly rely on expensive 3D scanningequipment and a lot of time-consuming postprocessing manually and require the scanned animated subject to remain in afixed pose for a considerable period. In recent years, the development of large-scale computing power of computer-relatedhardware, especially distributed computing, has made it possible to come up with a real-time and efficient solution. In thispaper, we propose a 3D reconstruction method for multivisual animated images based on Poisson’s equation theory. Thecalibration theory is used to calibrate the multivisual animated images, obtain the internal and external parameters of thecamera calibration module, extract the feature points from the animated images of each viewpoint by using the corner pointdetection operator, then match and correct the extracted feature points by using the least square median method, andcomplete the 3D reconstruction of the multivisual animated images. The experimental results show that the proposed methodcan obtain the 3D reconstruction results of multivisual animation images quickly and accurately and has certain real-time andreliability.

1. Introduction

With the development of computer image technology, therequirements for computer hardware are getting higherand higher. It is common to use 3D models to analyze indus-trial problems; three-dimensional acquisition and displaytechnology advances also make the object with rich geomet-ric. The need for model reconstruction techniques for 3Dsolids with rich geometric properties is growing [1]. ThePoisson equation-based 3D surface reconstruction techniqueis used to study the algorithm for reconstructing a triangularsurface model with solid geometric surface information ofan object based on the existing 3D point cloud model withnormal vector information. The steps of the algorithminclude preprocessing the input point cloud informationwith normal vector information, discretizing the globalproblem, solving the discretized subdata, extracting theequivalent surface after solving the Poisson problem, andpostoptimization processing. For the algorithm of 3D sur-face reconstruction, the Poisson surface reconstruction algo-

rithm combines the advantages of global and local methodsand uses implicit fitting to obtain the implicit equation of thesurface information described by the point cloud model itsrepresentative by solving the Poisson equation and by per-forming the equivalence surface extraction on this equation,to obtain the final desired surface model with rich 3D geo-metric solid information [2].

The model reconstructed by this method has a water-tight closure feature with good geometric surface propertiesand detail properties. The paper carries out related researchwork to address some defects inherent in the original Pois-son reconstruction algorithm. Firstly, for the Poisson recon-struction algorithm of the equivalence surface extractionalgorithm, the model hole problem is greatly improved byadding [3] the duality processing and index list expansionmethod. Animated image 3D reconstruction, as well as facialanimation, is two important types of research in the field ofcomputer graphics. Since 3D models have complex spatialstructures and rich subtle features, 3D reconstructions ofanimated images and face animation techniques have

HindawiAdvances in Mathematical PhysicsVolume 2021, Article ID 1547630, 11 pageshttps://doi.org/10.1155/2021/1547630

https://orcid.org/0000-0002-9082-1165

https://creativecommons.org/licenses/by/4.0/

https://doi.org/10.1155/2021/1547630

become a major challenge in the field of computer graphics.Nowadays, the research in the field of animation picture rec-ognition can achieve good results to a certain extent, but inpractical application, it often encounters various problemssuch as the angle, pose, makeup, and lighting of the anima-tion picture that have obvious influence on the recognitioneffect, and the recognition effect is worrying [4]. To avoidthe unnecessary influence of these factors on animationimage recognition and to improve recognition accuracy,3D animation image reconstruction also plays an increas-ingly important role in the research of animation imagerecognition.

To reconstruct 3D models with detailed character fea-tures and achieve more realistic face animation techniques,a lot of research methods have been proposed by academiaand industry and applied in cartoon animation production,film and television special effect production, medical visual-ization, virtual reality (VR), and other fields, while meetingthe growing demand for 3D animation image models, mak-ing 3D face reconstruction and related animation techniquesgradually mature: High-quality face-driven animation tech-niques for film and television special effects require captur-ing the facial expressions and semantic information ofactors and reproducing this information on virtual charactermodels such as cartoons. Using advanced 3D scanning andfacial expression motion capture equipment, realistic facialanimation effects can be created in film and game produc-tions [5].

2. Related Work

For the high accuracy solution of the three-dimensionalPoisson equation, the literature proposes a multi-grid-based solution, where they combine a compact high-orderdifference approximation to the multi-grid V-cycle algo-rithm to solve the two-dimensional Poisson equationDirichlet boundary conditions. This method, along with sev-eral different orders of grid space and projection operations,has shown significant improvements in computational accu-racy when compared with the five-point formulation testedon different machines. Solid model reconstruction ofdirected point sets is a solution to surface reconstructionproposed in the literature. In this paper, it is proposed howto reconstruct a directed point set into a solid, watertightmodel, and the literature computes the eigenfunction ofthe solid model by using the Stokes Theorem method. Thiseigenfunction has a value of 1 inside the model body and 0outside the model body [6]. An efficient method is also pro-posed to compute the Fourier coefficients of the eigenfunc-tion, which uses only the sampled surface information andthe normal vector to obtain the eigenfunction by computingthe inverse Fourier transform and then extracting a solidmodel using an equivariant surface extraction technique.This approach does not require the establishment of com-plex proximity relations for the sampled points or the itera-tive solution of linear equations for large systems and alsoproposes methods to patch the postreconstruction holes.The Poisson surface reconstruction is proposed in the litera-ture. This reconstruction method does not use heuristic spa-

tial partitioning or blending but takes all points into accountat once, thus providing better resilience to data noise. Unlikethe radial basis function (RBF, radial basis function)method, the Poisson method allows for a hierarchy that sup-ports local basis functions, so this solution is for the case ofsparse linear systems with relatively good support. A multi-scale spatially adaptive algorithm is described on this basis,with time and space complexity are proportional to the sizeof the reconstructed model.

The Poisson surface reconstruction algorithm is reiter-ated in the literature and used in practical applications.The literature proposes a parallelized Poisson surface recon-struction. This approach partitions multiple grid domains,and in their implementation, they compare models withmultiple processor data sharing and find that parallel execu-tion of distributed storage provides better scalability. Usingtheir approach, a hundred million data point sets can beprocessed in parallel on 12 processors on three machinesfor model reconstruction [7], with the data display providingmore than nine times the speed up without sacrificing theaccuracy of the reconstruction. New scanning and dataacquisition techniques allow for a dramatic increase in thesize of the dataset for surface reconstruction. The proposedmasked Poisson surface reconstruction algorithm in the lit-erature interpolates constraints on the input point clouddata based on the original Poisson surface reconstructionalgorithm, and this extension can be interpreted as a gener-alized representation of the masked Poisson equation in amathematical sense, where the masked sequence is chosenover a sparse set of points rather than the entire domain incomparison to other image and geometry processing calcu-lations. The paper finds that these sparse constraints canbe handled efficiently. Because the modified linear systemstill retains the same discretization of the finite elements,the sparse structure is unchanged, and the system can stillbe solved using the multiple mesh approach. Also, they pro-pose algorithms that are effective in reducing the time com-plexity of solving for the number of linear points and thusnow faster [8], higher quality surface reconstruction. The lit-erature presents a framework for surface reconstructionbased on scattered point clouds, an algorithm for implicitfunction surface reconstruction, and the proposed methodfirst builds a series of unoriented surfaces into a hierarchyof surface approximations, represented as a weighted combi-nation of radial basis functions. By treating local hiddenblocks as nodes, globally consistent orientations are treatedas graph optimization problems. This approach makes itpossible to iterate over more points to improve the fittingaccuracy and the efficiency of updating the RBF coeffi-cients [9].

In the field of reconstructing 3D face models based onsingle 2D images, the 3D deformation model 3DMM hasbeen proposed in the literature. As a classical statisticalmodel of 3D faces, 3DMM learns explicitly the underlyingprior knowledge of 3D face models in a statistical analysisapproach that represents a 3D face model as a linear combi-nation of a set of underlying 3D face models, where this setof underlying 3D face models is obtained by performing theprincipal component analysis (PCA) on a set of densely

2 Advances in Mathematical Physics

aligned 3D face models. 3DMM treats the 3D reconstructionproblem as a kind of 3D model to a 2D image fittingproblem. 3DMM-based methods tend to iteratively seeksolutions in a parameter space defined by statistical 3Dfacial and image models. The class of methods starts withan initial set of parameter values and is based on specificchanges in the facial feature points of the face iterativelyupdate the parameters, and the final reconstructed 3D facemodel can be obtained from the optimal parameter values.In contrast to the method that seeks an optimal solutionin specific parameter space, another method based on cor-relation analysis first performs a transformation of the fea-ture space, further exploits the connection between the 3Dface model and the 3D reconstruction of the animatedimage and learns a mapping relationship from the 3Dreconstructed facial features of the animated image to thefacial features of the 3D face model in the transformedfeature space.

3. 3D Reconstruction Method for AnimatedImages Based on Poisson’s Equation

This thesis focuses on introducing a 3D model reconstruc-tion technique based on Poisson’s equation for animatedimages, using a regression-based method that is widely usedin 3D model reconstruction to calculate the adjustmentamount of 3D face mesh model feature point location infor-mation from the change of key point location information offace pictures in 2D space, and further projecting 3D facemodel feature points onto face images in 2D space. Theself-defined feature point dragging tool continuously reducesthe error between the 3D reconstructed feature point posi-tion information of the animated image and the positioninformation of the 3D face model feature points mappedon the 2D image, to iteratively optimize the reconstructed3D face model. Among them, the complex mapping rela-tionship between the location information of the key pointsof the face in the 3D reconstruction of the animated imageand the location information of the key points of the corre-

sponding 3D face model is mainly studied, to calculate the3D spatial location information of the feature points ofthis 3D face model, i.e., to reconstruct the 3D face modelof this user [10]. Poisson’s equation is a partial differentialequation commonly found in electrostatics, mechanicalengineering, and theoretical physics in mathematics. Pois-son first obtains the Poisson equation without a gravita-tional source, ΔΦ = 0 (that is, Laplace equation); whenconsidering the gravitational field, there is ΔΦ = f (f isthe mass distribution of the gravitational field). Later, itwas extended to an electric field, magnetic field, and ther-mal field distribution. The equation is usually solved byGreen’s function method, and it can also be solved bythe method of separation of variables and the method ofcharacteristic lines.

3.1. Poisson’s Equation. The Poisson equation was developedfrom the Laplace equation and is therefore closely related tothe Laplace equation; they are both a type of partial differen-tial equation. Laplace’s equation, also known as the summa-tion equation, or the potential equation, is a type of partialdifferential equation, named after the French mathematicianLaplace, who was the first to study the equation and arrive atthe conclusion that it is resolvable in the region where theequation holds, and if any two functions each satisfyLaplace’s equation [11], do the summation operation onthe two functions, which can be any linear combination,and the result of the summation also satisfies the equationexpressed earlier. The basic principle of this equation is rep-resented in Figure 1.

The problem of its solution is a mathematical problemthat is frequently encountered in related scientific fields suchas electromagnetic, astronomical, and hydrodynamic doc-trines. In general, for the three-dimensional case, the prob-lem of Laplace’s equation can be reduced to solving asecond-order differentiable real function for real indepen-dent variables, and this problem is then described in the fol-lowing form [12]. As the first formula of the article, it canbetter describe the Poisson algorithm mentioned in the

6×6

6×6

ALLE

ALLE

H(x) = sinx⁎ex

–20 20

20A(x) = sinx⁎ex

7×7

7×7

Figure 1: The underlying principle of Poisson’s equation.

3Advances in Mathematical Physics

article and can better connect the formula mentioned in thearticle.

∂2φ∂x2

+ ∂2φ∂y2

+ ∂2φ∂z2

= 0: ð1Þ

And as the above equation can often be expressed in asimplified way as

div grad φ = 0: ð2Þ

Based on Poisson’s equation, this further extends theconcept of 3D reconstruction based on Poisson’s algorithmfor animated images. Figure 2 shows the framework diagramof Poisson’s arithmetic.

3.2. Solution of Poisson’s Equation. There are many methodsfor solving Poisson’s equation, which in general can bedivided into the following according to the specific imple-mentation: direct solution method, iterative solutionmethod, and multiple grid solution methods [13].

(1) Direct solution method

There are many ways to solve Poisson’s problem directly;typical solution methods are decomposition methods. Thereason for matrix decomposition is that when the amountof data is very large, decomposing a matrix into the productof several matrices can make a large saving in storage spacebut also can significantly reduce the amount of computationwhen processing the real problem because for an algorithmto deal with fewer elements, its completion time is corre-spondingly faster. This is because the fewer elements analgorithm has to deal with, the faster it will take to completethe operation [14]. The principle of the formula is

A = 1n〠n

i=1XiYi +

x − μ

σ: ð3Þ

(2) Multiple grid solution methods

Many scholars have extensively and profoundlyresearched the multigrid algorithm, explained the meaning

Data pre-processing

TrainingTAB

Trainingsample ASD cell

Reverseadjustment

Datafeature

�e topreturn

Reverseadjustment

VCE cell Trainingoutput

Trained ASDparameter

Sensor data

Test data ASD cell VCE Predict theoutput

Uncertaintyoutput

Phase of training

Test phase

Finaldatabase

Poisson algorithm

Trained VCEparameter

Figure 2: Algorithm framework diagram.


and basic principles of multigrid, applied the multigrid algo-rithm to many fields, and finally obtained the optimal effi-ciency of numerical operations. The convergence problemhas also been well solved, and it is active in many numericalcomputation fields today. Multiple grids are widely used insolving Poisson’s equation because of their relatively fastsolution speed, especially in solving Poisson’s equation aris-ing from graphical image problems, which is unsurpassed byother methods. For the two-dimensional Poisson equation:

A = 〠n

i=1Xi:

x − μ

σ

� �: ð4Þ

Iteratively solving the set of equations consisting of par-tial differential equations after discretization because for acertain grid, it is relatively easy to remove the error compo-nents corresponding to the wavelength and the step lengthof the grid, so the multiple grids solving Poisson’s equationproblem are a fast computational method, which mainlyuses grid cells with different scale sizes, for different sparsityof the grid cells to remove the error components of differentwavelengths, respectively. It solves the problem of rough gridcorrection, mainly the error correction subproblem on therough grid and the fine grid difference error correctionmethod. A large amount of time is guaranteed to do themain operational work of the rough grid. One of the simplemultiple-grid V-cycle iterations also uses other techniquesfor error smoothings, such as the Gauss-Seidel method orthe Jacobi method. To obtain an approximation of thesmoothed error equation for the rough grid, this interpola-tion needs to be added to the error correction for the roughgrid by first interpolating the error correction for the finegrid [15].

(3) Poisson’s equation for shielding

The shielded Poisson equation remains a type of partialdifferential equation, which takes the form of something like.

B =a11 a12 a13

a21 a22 a23

a31 a32 a33

0BB@

1CCA: ð5Þ

In order not to lose the generality of the equation, thealgorithm takes the value of A to be nonnegative, and theshielded Poisson equation degenerates to the Poisson equa-tion when the value of A is zero; that is, when A = 0

B = arccos θe− x−μð Þ/σ: ð6Þ

Thus, in the three-dimensional case, when the value ofA issmall, the shielded Poisson equation represents the weightedsuperposition of functions homologous to the function.

B = 1n〠n

i=1XiYi − 〠

n

i=1Xi: ð7Þ

3.3. Equivalence Surface Extraction for Animated Images. Theequivalent surface visualization method is a surface recon-struction method to extract the equivalent curvature fromscattered points in the process of drawing human body struc-ture image in 3D spatial data field. It is also widely used in themethod of surface reconstruction for extracting a value equiv-alent surface from scattered points and the method of extract-ing intermediate frames from keyframe animation. There aremany methods and techniques of contour surface extraction,and the followingmethods are briefly summarized for contoursurface extraction in 3D spatial data fields [16].

(1) Contour-based equivalence surface extractionmethod

Contour tracing is an early contour surface extractionalgorithm that extracts sequences of exclusive contour linesof interest to the user based on specified requirements andthen reconstructs the contour surface based on thesesequences, by using triangular surface slices to trace contourlines belonging to the same object mapped to its neighboringslices, by extracting each [17] two-dimensional informationslice of contour lines and thus obtain the equivalence surfacerepresented by the region of interest. The principle of theformulation is

B = 〠n

i=1X2i +

x − μ

σ: ð8Þ

The problem faced by this method is that for adjacentslices that may have complex structural situations perhapscausing an unavoidable large number of splicing errors, thisis due to the polysemy of correspondence of contour lineson adjacent slices, and also for the same slice, there may bemultiple different closed contour lines, so the method of con-necting contour lines on adjacent slices faces many difficulties.The procedure of the slice contour-based surface reconstruc-tion algorithm is shown in Figure 3.

(2) Voxel-based isosurface extraction method

The earliest voxel-based method for equivalence surfaceextraction is the cubic block method (Cuberille), whichbinarizes the volume data by equivalence values, connectingall externally facing boundary faces for each voxel (i.e., cubicblock) that is on the boundary, using the six faces of theboundary cube to fit the equivalence surface, removing onlythe faces of the boundary cube that duplicate each other[18], and by connecting faces that do not overlap with eachother. Even if the speed of rendering is not a concern, areconstructed rendering can be achieved directly by display-ing the boundary voxels nontransparently, without remov-ing the inner faces of the cube. The principle of theformula is

N = 1n〠n

i=1X2i +

1n〠n

i=1XiYi: ð9Þ


The algorithm is characterized by the fact that there isno interpolation between voxels, so it is very simple toimplement and facilitates parallel processing due to thehigh degree of independence between the data. But theproblem faced by the method is that the final obtainedequivalence surface is a block-shaped surface composedof small planes of adjacent voxels perpendicular to eachother, which will have a large walk for the surface infor-mation representation [19], not smooth and not reflectthe detailed information of the object well. Table 1 showsthe comparison of voxel-based equivalence surface extrac-tion algorithms.

3.4. 3D Animation Image Reconstruction Technology Basis.The reconstruction process involves a variety of spatial coor-dinate systems, including object coordinate system, worldcoordinate system, camera coordinate system, and screencoordinate system, where the object coordinate system isthe local coordinate system of the object, which is the initialposition before applying any coordinate transformation, andthe object coordinates can be mapped to the camera coordi-nate system by translation, rotation, and scaling, and to thescreen coordinate system by perspective projection ororthogonal projection. The specific complex spatial transfor-mation relationship is a necessary step for 3D model coordi-nate transformation, and a clear transformation idea oftenmakes the development much more efficient. As opposedto a fixed parametric 3D face model, this paper uses a 3Dregression algorithm [20], which learns an input-to-outputmapping relationship using the 3D reconstruction of an ani-

mated image as input and the spatial location information ofthe 3D face model feature points as output. This method issignificantly better than the above methods in reconstructingthe detailed features of model expressions. Considering thedegree of effective understanding of face feature points to3D face model features, therefore using face feature pointsto reconstruct the 3D model can effectively reconstruct thedetail features of the facial model; the method can design aregression function to achieve the mapping between 2D facefeature points and the adjustment amount of the 3D facemodel. The principle of its formula is

N = x − μ

σ

� �:dydx

: ð10Þ

In a conventional 1D linear sensor, data from one col-umn (row) of sensor cells is used to reconstruct a 2D image,and different scans can be achieved by selecting differentapertures and turning different sensor cells on or off. Simi-larly, two-dimensional planar sensors are capable of volu-metric scanning.

For two-dimensional ultrasound transducers, the trans-ducer material often has a large impact on the overall sam-pling frequency and reconstruction frame rate. Although2D ultrasound transducers are capable of dynamicallyreconstructing 3D images in real-time, the impedance ofeach transducer cell is much larger than that of a 1D linearray, which makes impedance matching for 2D surfacearray transducers particularly difficult. In addition, to avoidmutual interference between sensor cells, each sensor cellneeds to be at least half a wavelength away from each other[21], which in turn makes the size of the sensor cells limited.The principle of the equation is

M = ∂2Ω∂v2

:dydx

: ð11Þ

Contourextraction

Extractcontour

feature points

Outline of thecorresponding

OutlinestitchingCurve fitting

Y processing

Find thefeaturepoints

Poissonalgorithm

Digitalsplicing

Poissonalgorithm

Figure 3: Slice contour-based reconstruction.

Table 1: Comparison of voxel-based equivalence surface extractionalgorithms.

Different methodsTraditionalmethod

Poisson equationalgorithm

Calculation amount Big Small

Methodological Can Can

Geometric difficultyoperation

Difficult Simple


The flexibility and portability of handheld sensors allowclinicians to scan areas of ROI (region of interest) in spe-cific orientations and positions, allowing the clinician toselect the best view and scan plane. The primary issues tobe addressed in 3D imaging are positioning and orientation.There are four main common localization methods, involv-ing acoustic localization, optical localization, articulatedarm localization, and magnetic localization. In addition tothese, there are reconstruction algorithms that are basedsolely on images and are independent of the sensor posi-tion. After being able to acquire 3D data accurately andquickly, the speed and accuracy of 3D body reconstructiondirectly affect the speed and accuracy of the overall 3Dreconstruction. Many reconstruction methods have beenable to scan and image in real-time, and most of thesemethods are based on traditional 3D reconstruction algo-rithms and parallel computing. Specifically, real-time 3Dvoxel-based reconstruction algorithms can be divided intothree categories: voxel-based method (VBM), pixel-basedmethod (PBM), and function-based method (FBM). Allthese methods are based on the interpolation of 2D imagesreconstructed from 1D line array sensors combined withsensor localization into 3D regions; thus, the algorithmsdiscussed here are the process of interpolating 2D imagesinto 3D. Figure 4 represents the basis of the 3D animatedimage reconstruction technique.

Structure from motion (SFM) reconstruction is amethod to recover the external parameters of the cameraand the external scene structure under the condition thatthe internal parameters of the camera are known. The incre-mental reconstruction method is a widely used reconstruc-tion method, which restores the scene structure by addingnew images step by step. The incremental reconstructionshows scene drift in large reconstructed scenes due to theaccumulation of positional errors. The principle of its for-mula is

T = ∂2Ω∂v2

:∂2Ω∂u∂v

: ð12Þ

Also, the method increases the number of parametersto be estimated as the number of images increases, result-ing in slower reconstruction. To speed up the reconstruc-tion, the reconstruction process is accelerated by usingparallel computing hardware such as GPU and FPGA.The advantage of the global reconstruction method is thatit does not drift, but the method requires high featurematching accuracy to improve the quality of feature points,which leads to a decrease in reconstruction integrity. Com-pared to incremental reconstruction, the number of globalreconstruction iterations is reduced, and there is no cumu-lative error, but it is sensitive to external points, and thereconstruction accuracy is not as high as incrementalreconstruction. The hybrid reconstruction method estimatesthe rotation matrix by taking the global reconstruction, andthe incremental method estimates the camera displacement.The reconstruction is accelerated while avoiding scene struc-ture drift.

4. Experimental Results and Analysis

For 3D reconstruction of ordered images, the initial pointcloud is first calculated according to the order of imageacquisition, and then, the newly obtained images are addedto the point cloud, and the order of addition is now followedby the time of image capture. However, in large scenes, thereconstructed image set is likely to have images withoutcommon feature points; when encountering the situationthat there is no feature point matching with the currentnewly added image in the reconstructed point cloud, the3D reconstruction according to the reconstruction processof ordered images will fail to be reconstructed. The principleof its formula is

T = ΔyΔx

:dydx

: ð13Þ

If the reconstructed scene is the image taken by the carduring driving, i.e., the image sequence is guaranteed to cor-respond to the scene sequence, then the scene can be

Figure 4: Fundamentals of 3D animated image reconstruction techniques.


reconstructed by using 3D reconstruction of ordered images.However, such a reconstruction is unstable, and to design amore robust reconstruction procedure, the reconstructionof unordered 3D images needs to be considered, and theexperiments conducted in this paper are based on this, withthe experimental platform based in Figure 5 shown in.

The Poisson equation-based 3D surface reconstructiontechnology is based on the existing sparse or dense 3D pointcloud model obtained through image sequences or othermethods of 3D point cloud reconstruction. Here, we adoptthe method of three-dimensional spatial partitioning of theoctree, placing each part on the subnodes of different depthsof the octree, conducting the process of Poisson’s equationfor the whole surface represented by the point cloud, andthen carrying out the surface reconstruction process ofequivalent surface extraction to finally obtain a 3D spatialsurface solid model with 3D surface information and a richdegree of detail presentation. The principle of the equation is

h = ∂2Ω∂u2

:∂2Ω∂u∂v

:dydx

: ð14Þ

To sufficiently verify the feasibility of the algorithmdescribed in the thesis, to prevent the interference of unnec-essary factors, to avoid the existence of errors caused by toomany discrete point data due to errors in the process of col-lecting point cloud models, and to avoid the problemscaused by wrong point cloud information or too little infor-mation in the point cloud data, the experimental data arecollected here mainly through the existing mature andready-made 3D point cloud models. The main objective isto obtain the experimental data by using the established3D point cloud model, supplemented by a simple pointcloud model with a small number of points. The principle

of the formula is

R = δyδx

:dydx

:ΔyΔx

: ð15Þ

4.1. Experimental Results. In the specific experiments, differ-ent parameters are set for several models, and the originalPoisson’s equation-based surface reconstruction algorithmis compared with the optimized shielded Poisson’sequation-based surface reconstruction algorithm. After theoptimization of the algorithm, the overall model detailreduction is more perfect, and the model hole problem gen-erated in the reconstruction process is significantly reduced.The experimental results of the data are shown in Figure 6.

It can be concluded that the overall algorithm has beenoptimized to obtain a greater improvement than the originalalgorithm. Similarly, after practical experiments, it is provedthat the improved Poisson reconstruction algorithm basedon the shielded Poisson equation, by using interpolationconstraints on the input information for preprocessing,postprocessing after the optimization of the equivalence sur-face extraction algorithm, differs from the original Poissonreconstruction algorithm based on the Poisson equation.The postprocessing, after optimization of the equivalencesurface extraction algorithm, is significantly different fromthe original Poisson equation-based Poisson reconstructionalgorithm.

4.2. Analysis of Experimental Results. After several functionaltests, although this system was able to achieve the nominalfunctions, some existing problems were found at the sametime during the use. The graph of its final experimentalresults is shown in Figure 7.

In the viewer interface, if the user performs a batch addoperation, occasionally the program does not respond and

4⁎4

Ours Culvigridoctree

MessindedCubs

MessindedRecoh

Loyloader

ASPlatformMirect4F

ASL

Graphicmevice

PlatformAlgorithm

Technology

Application

User

Algorithm user

Moyut-sethist

Figure 5: Experimental platform structure.


the system loads slowly. In the face feature detection mod-ule, it takes some time to load because some libraries inPython are called, but the user may think that the functionbutton is not clicked and thus repeat the operation. To pre-vent similar misuse, a progress bar is added to check the cur-rent running status of the system. Second, personalizationneeds to be improved. This system uses 2D images of facesunder unconstrained conditions as input and outputs 3Dmodels of faces. And the reconstruction technology isextended to practical applications so that more people havemore practical experience in areas such as stereo visionand human-computer interaction. Although the existingface 3D reconstruction system has functions such as brows-ing and feature point viewing of loaded images, it does notprovide further explanation and demonstration in the actual

3D reconstruction process, and the detailed introduction ofthis system enables users to have a deeper knowledge andunderstanding of the face 3D reconstruction process.

The current version of the system must continue toenhance the personalization design because there are notmany software systems that perform 3D reconstruction offaces through unconstrained images and visualize them,and by further exploration and enhancement, the face 3Dreconstruction technology can be brought into the lives ofmore ordinary people. With rich categories of 3D recon-struction and more processes, each step deserves more in-depth exploration to better extend and improve related func-tions and enhance user satisfaction. Through the above anal-ysis, this algorithm currently has many aspects to beimproved, and it is worth encouraging to achieve the input

10 20 30 40 50 60 70

10

15

20

25

30

35

40

Alg

orith

m effi

cien

cy

Evolution of different algorithms

212223242

52

Efficiency

Figure 6: Comparison of the efficiency of the 3D reconstruction of Poisson’s equation.

10

20

30

Effici

ent

40

50

020

40A

60 2

4

68

10

Degree of efficiency56.00

50.80

45.60

40.40

35.20

30.00

24.80

19.60

14.40

9.200

4.000

Different algorithms

10

Figure 7: Efficiency analysis chart.


of face images under unconstrained conditions and recon-struct the corresponding 3D model of the face quickly andaccurately, with the efficiency shown in Figure 8.

With the increasing demand for face 3D models, there isbound to be huge room for the development of applicationssimilar to this system, which will become an indispensablepart of people’s life and entertainment in the future. In thefuture improvement, this system will add more new methodswith excellent performance in time to meet the user’sincreasingly strong personalized demand for face 3D recon-struction and make more intelligent and humanized applica-tion products in this technology field.

5. Conclusion

With the rapid development of computer vision technology,people’s research on many aspects of animation images hasgradually matured. For example, face recognition, from theprevious traditional image processing methods based onSIFT features, to today’s deep learning methods, there hasbeen a whole set of research systems for the 3D reconstruc-tion of animated images. However, images limited to thetwo-dimensional plane are increasingly unable to deal withmany of the problems encountered today. Therefore, manyresearch scholars have turned their research direction to3D models of faces. However, the primary problem is thataccess to models is very difficult, and the number of modelsis low, which sets a big obstacle in the research process.Moreover, the process of 3D reconstruction often implies ahuge amount of computation, which is a serious challenge

for both the reconstruction algorithm and the hardwareimplementation. To overcome the current problems thatexist, an effective and convenient 3D reconstruction methodfor faces is urgently needed. In this paper, after studyingexisting methods and combining computer graphics anddeep learning contents, we propose a 3D reconstructionmethod for faces based on deformation model, which trans-forms the traditional parameter solving the problem ofdeformation model into an image learning problem anddesign and produce a simple and effective animation imagereconstruction system.

Since previous 3D reconstruction methods have certainshortcomings, the main content of this paper is the 3Dreconstruction of a human face with a deformation modelas the research object. Firstly, the deformation model isintroduced, and the generalization ability is partiallyimproved by the two classical deformation model datasetsthat are combined. This is followed by feature point detec-tion on the sample images using cascade regression. Theoriginal 2D image and the corresponding 3D data informa-tion are transformed into UV space based on the inheritanceof previous scholars’ work to obtain the position map underUV space. By designing a network structure containing twomain parts of convolution and deconvolution and selectingappropriate loss functions, a new location map is generated,and a very realistic 3D model of the face is finally generatedby meshing and thus achieving a 3D reconstruction of theface with texture mapping. It is demonstrated by several setsof comparative experiments that the proposed method isvery realistic for the shape-shifting model of the human face.

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Figure 8: Modeling efficiency diagram.


The effectiveness of the face 3D reconstruction methodin terms of face alignment, face alignment in large anglecases, face 3D model reconstruction, and computationalconsumption, its shown through experimental results thatthe Poisson equation based 3D reconstruction method foranimated images, is efficient and effective. At the end ofthe article, Poisson’s equation can play a certain role in animage restoration algorithm in the future. A new image res-toration algorithm based on gradients can remove influentialobjects from pictures or photos. This is the Poissonequation.

Data Availability

The data used to support the findings of this study are avail-able from the corresponding author upon request.

Conflicts of Interest

The author declares that no known competing financialinterests or personal relationships could have appeared toinfluence the work reported in this paper.

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