UDiFF is a 3D diffusion model that generates textured 3D shapes with open and closed surfaces from text conditions or unconditionally. The model uses an optimal wavelet transformation to generate unsigned distance fields (UDFs) in the spatial-frequency domain, which provides a compact representation space for UDF generation. Instead of manually selecting an appropriate wavelet transformation, which can lead to significant information loss, UDiFF employs a data-driven approach to learn the optimal wavelet transformation for UDFs. This approach minimizes unsigned distance errors during self-reconstruction through wavelet transformation, especially near the zero-level set of UDFs, preserving geometry details and enabling high-fidelity generation of 3D geometries. UDiFF is evaluated on widely used benchmarks, including DeepFashion3D and ShapeNet, demonstrating superior performance compared to existing state-of-the-art methods in both qualitative and quantitative evaluations. The model incorporates conditions from CLIP models to enable text and image signal control of 3D generation. UDiFF also introduces a fine predictor to generate fine coefficient volumes from coarse ones, enhancing the generation of UDFs. The model is capable of generating diverse 3D real-world contents containing open surfaces, which previous 3D implicit diffusion models could not achieve due to their limitation to closed shapes. UDiFF's data-driven wavelet optimization significantly reduces information loss during transformation, leading to more accurate and diverse generations. The model is evaluated on both open-surface and closed-surface shape generation, showing its effectiveness in generating high-fidelity watertight geometries. The results demonstrate that UDiFF achieves superior performance in generating 3D shapes with both open and closed surfaces, outperforming existing methods in terms of generation quality and fidelity.UDiFF is a 3D diffusion model that generates textured 3D shapes with open and closed surfaces from text conditions or unconditionally. The model uses an optimal wavelet transformation to generate unsigned distance fields (UDFs) in the spatial-frequency domain, which provides a compact representation space for UDF generation. Instead of manually selecting an appropriate wavelet transformation, which can lead to significant information loss, UDiFF employs a data-driven approach to learn the optimal wavelet transformation for UDFs. This approach minimizes unsigned distance errors during self-reconstruction through wavelet transformation, especially near the zero-level set of UDFs, preserving geometry details and enabling high-fidelity generation of 3D geometries. UDiFF is evaluated on widely used benchmarks, including DeepFashion3D and ShapeNet, demonstrating superior performance compared to existing state-of-the-art methods in both qualitative and quantitative evaluations. The model incorporates conditions from CLIP models to enable text and image signal control of 3D generation. UDiFF also introduces a fine predictor to generate fine coefficient volumes from coarse ones, enhancing the generation of UDFs. The model is capable of generating diverse 3D real-world contents containing open surfaces, which previous 3D implicit diffusion models could not achieve due to their limitation to closed shapes. UDiFF's data-driven wavelet optimization significantly reduces information loss during transformation, leading to more accurate and diverse generations. The model is evaluated on both open-surface and closed-surface shape generation, showing its effectiveness in generating high-fidelity watertight geometries. The results demonstrate that UDiFF achieves superior performance in generating 3D shapes with both open and closed surfaces, outperforming existing methods in terms of generation quality and fidelity.