## 1 Introduction  
In recent years, rapid advancements in optical-based industries have significantly influenced sectors such as microelectronics [1; 2; 3] and automotive design [4; 5; 6], leading to a strong demand for highly accurate methods for measuring specular surfaces [7; 8]. Deflectometry [9; 10; 11] is an optical measurement technique which relies on the reflection of structured light patterns from an unknown surface to infer its shape. Deflectometry is highly effective for measuring specular surfaces due to its accuracy, speed, and non-contact approach, making it particularly suitable for capturing detailed surface variations [12]. Additionally, free-form surface reconstruction [13; 14] is crucial for modeling complex, irregular shapes found in advanced optical systems [15]. Together, these methods enhance precision and innovation in optical manufacturing. The more specific context of single-shot deflectometry [16; 17; 18], performing deflectometry based on only a single image, is crucial for several reasons, particularly for high-speed imaging of fast dynamically changing surfaces. This includes applications such as fast-moving or morphing specular surfaces like liquids (water, metals), where capturing multiple images is impractical due to the rapidly changing nature of the surface. Additionally, in industrial scanning, single-shot deflectometry enhances efficiency by eliminating the need for multiple images, important for quality control of reflective objects [9]. For example, in General Fusion's Magnetized Target Fusion technology scheme, measurement of the 3D shape of an imploding specular liquid metal vortex to compress plasma to generate nuclear fusion with microsecond precision is required [19]. Traditional methods [10] require multiple captures and complex setups to obtain surface-gradient information in perpendicular directions, which can be time-consuming and problematically inefficient for dynamic applications. To further enhance the efficiency and accuracy of single-shot deflectometry, recent research has explored the use of composite fringe patterns in deep learning-based 3D surface measurement techniques [17; 20]. These composite patterns are particularly effective in addressing the challenges associated with single-shot deflectometry, such as encoding complex surface details in a single capture. Historically, in the use of fringe patterns for surface measurement, vertical and horizontal patterns are often combined into a single composite pattern. The separation of these orthogonal fringe patterns into their individual components is crucial for accurate analysis and has traditionally been achieved through methods like the 2D Fourier algorithm [21] which, despite its utility, often results in phase errors and spectrum aliasing [22]. In contrast, in this paper, we propose a novel deep neural network (DNN)-based approach designed for the end-to-end 3D reconstruction of free-form surfaces. Our network, VUDNet, processes 2D images captured from a surface onto which an orthogonal fringe pattern is projected (Figure 1). This discriminative-generative hybrid network excels in depth estimation. To the best of our knowledge, VUDNet represents the first DNN-based approach in deflectometry, pioneering the use of generative methods for depth estimation. The integration of both generative and discriminative components in our network is pivotal for achieving high-fidelity 3D surface reconstruction. The discriminative part of the network is essential for accurately identifying and interpreting the orthogonal fringe patterns projected onto the specular surface. This involves extracting meaningful features from an image, which are crucial for determining the gradient information necessary for depth estimation. The discriminative model's ability to parse complex visual patterns with precision directly impacts the accuracy of the subsequent 3D reconstruction. On the other hand, the generative component plays a crucial role in translating the extracted gradient information into a coherent 3D representation.