SplineGen: a generative model for B-spline approximation of unorganized points

TL;DR

This paper presents a learning-based method to solve the traditional parameterization and knot placement problems in B-spline approximation, and demonstrates a notable improvement over existing methods, with a one to two orders of magnitude increase in approximation accuracy on test data.

Abstract

This paper presents a learning-based method to solve the traditional parameterization and knot placement problems in B-spline approximation. Different from conventional heuristic methods or recent AI-based methods, the proposed method does not assume ordered or fixed-size data points as input. There is also no need for manually setting the number of knots. It casts the parameterization and knot placement problems as a sequence-to-sequence translation problem, a generative process automatically determining the number of knots, their placement, parameter values, and their ordering. Once trained, SplineGen demonstrates a notable improvement over existing methods, with a one to two orders of magnitude increase in approximation accuracy on test data.

Figures (9)

• Figure 1: Influence of parameters, knots, and their alignment on approximation accuracy: (a) results from two different parameters; (b) results from two different knots; and (c) results of matched (top) and unmatched (bottom) cases.
• Figure 2: The overall architecture of SplineGen.
• Figure 3: The network for learning point embeddings. The right grayed part is only used in the training phrase, not in the inference phrase.
• Figure 4: The network for generating knots and parameters. Left: The pipeline of knots and decoder generation. After random masking, the input points are sent to a transformer encoder, yielding the point embeddings. Both the knot decoder and parameter decoders take the embeddings for generation. Besides, point embeddings are gathered as input of the parameter decoder. For each layer of the parameter decoder, internal cross-attention is applied. Right: The details of the internal cross attention. The key and value in the self-attention module at the last layer of the knot decoder are used for the internal cross-attention with query vectors converted from $X_P^i$. Here, $X_k^{n}$ denotes the output of the last layer of the knot decoder, and $X_{P}^i$ denotes the output of $i^{th}$ layer of parameter decoder.
• Figure 5: Results of 2D parametrization, knot placement, and B-spline approximation.