# 3D Shape Generation via Variational Autoencoder with
## 1 Introduction  
The advancements in Artificial Intelligence (AI) technology have enabled the use of 3D shapes in the computer vision (CV) and computer graphics (CG) domains. The demand for fine details and accurate 3D shapes in robotics, 3D games, medical imaging, and virtual and augmented reality applications has necessitated the development of robust deep learning algorithms such as generative adversarial networks (GAN), variational autoencoder (VAE) and so on, to generate 3D mesh-based shapes. Recent advancements in unsupervised learning with deep neural networks, trained on robust 3D datasets [1, 2, 3, 4], have led to more detailed representations of shape space in deep learning models. These advancements enable realistic 3D shape generation for applications such as 3D scene understanding, data augmentation, visualization, simulation, analysis, and interpretation in the field of AI technology [5, 6, 7]. However, convolutional neural network (CNN)-based generative models that use voxels [6, 7, 8] and point-cloud [9, 10, 11] data representations as inputs often yield unsatisfactory results due to low resolution and high memory usage with voxels and the unstructured nature of point-clouds. On theother hand, using mesh data representation as input for 3D-GANs and 3D-VAEs leads to the generation of high-resolution, compact, and computationally efficient 3D shapes. However, the irregular and complex organization of mesh data representation [12; 13; 14] makes it challenging for it to be used as input for learning in a CNN. Recently, several variations of the 3D-VAE-GAN [6; 8; 15] have been proposed to address the problem of 3D shape generation. These models have the ability to produce accurate 3D shape samples, classify 3D shapes, and reconstruct 3D shapes from 2D images. They are based on the well-known original GAN architecture [16] and have a tendency towards unstable training. While these models excel in creating 3D shapes from single classes, their outputs tend to be coarse, and training on complex data distributions with multiple object classes in various poses can be challenging. This paper aims to improve upon existing VAE-GAN models by generating 3D mesh-based shapes using a 3D-CNN framework and jointly training over multiple classes to achieve fully data-driven modeling. To generate high-quality 3D mesh-based shapes from 2D images and capture a more complex distribution, this paper presents an innovative deep learning approach by integrating a relativistic average loss function with a signed distance function-based VAE-GAN framework, referred to as 3D-VAE-SDFRaGAN. The 3D-VAE-SDFRaGAN method is a novel approach for generating 3D shapes that combines the strengths of VAEs and GANs. By using a signed distance function (SDF), this method can overcome the limitations of traditional methods for generating 3D shapes and creating more realistic shapes with smoother surfaces. The relativistic average GAN (RaGAN) is also employed to enhance the stability and quality of the generated shapes. This approach has broad potential applications in several fields, including computer vision, computer graphics, gaming, robotics, medical imaging, simulation, and virtual reality. The generated 3D shapes offer improved communication, better design, and an enhanced understanding of complex structures and concepts. The proposed 3D-VAE-SDFRaGAN aims to produce a signed distance function field on a gridded domain by utilizing a deconvolution layer. In doing so, it helps to produce a polygon mesh surface reconstruction with a higher quality 3D shape generation and stabilizes the GAN training process through the proposed relativistic average GAN loss function. The main contributions of this work are as follows:  
* VAE and GAN are integrated to enable the simultaneous learning of encoding, generating, and comparing data. The proposed method explicitly learns the latent spaces of 2D images, which are then used to generate corresponding signed distance functions of objects and reconstruct them into high-quality 3D mesh-based shapes.