Existing Large Multimodal Models (LMMs) struggle with mathematical geometric reasoning due to a lack of high-quality image-text paired data. Current geometric data generation approaches, which apply preset templates to generate geometric data or use Large Language Models (LLMs) to rephrase questions and answers (Q&A), unavoidably limit data accuracy and diversity. To synthesize higherquality data, we propose a two-stage Reverse Chain-of-Thought (R-CoT) geometry problem generation pipeline. First, we introduce GeoChain to produce highfidelity geometric images and corresponding descriptions highlighting relations among geometric elements. We then design a Reverse A&Q method that reasons step-by-step based on the descriptions and generates questions in reverse from the reasoning results. Experiments demonstrate that the proposed method brings significant and consistent improvements on multiple LMM baselines, achieving new performance records in the 2B, 7B, and 8B settings. Notably, R-CoT-8B significantly outperforms previous state-of-the-art open-source mathematical models by 16.6% on MathVista and 9.2% on GeoQA, while also surpassing the closedsource model GPT-4o by an average of 13% across both datasets. The code is available at https://github.com/dle666/R-CoT. Large Language Models (LLMs) exhibit excellent reasoning capabilities and draw extensive attention from the artificial intelligence research community (Lu et al., 2023b) to mathematical problemsolving in textual form (Chen et al., 2024b; Liao et al., 2024; Zhou et al., 2024; Zhao et al., 2024b; Zhou & Zhao, 2024; Kim et al., 2024). However, LLMs still struggle to solve mathematical problems involving images that require visual comprehension. Geometry problems, as typical mathematical problems with images, play an important role in evaluating mathematical reasoning skills (Zhang et al., 2023c), requiring a high level of visual comprehension. Besides, even though some problems are not related to geometry on the surface, they require the same skills for models (e.g., fine-grained image comprehension skills and multi-step reasoning skills). With the appearance of o1 (OpenAI, 2024), GPT-4o (Islam & Moushi, 2024), Gemini (Team et al., 2023), and numerous Large Multimodal Models (LMMs) (Li et al., 2024a; Liu et al., 2024; Chen et al., 2024d; Bai et al., 2023), recent researches progressively investigate using LMMs to solve mathematical geometry problems. Although LMMs show impressive results in general visual question-answering (VQA) tasks (Fan et al., 2024; Liu et al., 2024), they still face challenges in solving mathematical geometry problems. Adjust values in the question and generate answers.

With the rapid advancement of large language models, there has been a growing interest in their capabilities in mathematical reasoning. However, existing research has primarily focused on text-based algebra problems, neglecting the study of geometry due to the lack of high-quality geometric datasets. To address this gap, this paper introduces AutoGeo, a novel approach for automatically generating mathematical geometric images to fulfill the demand for large-scale and diverse geometric datasets. AutoGeo facilitates the creation of AutoGeo-100k, an extensive repository comprising 100k high-quality geometry image-text pairs. By leveraging precisely defined geometric clauses, AutoGeo-100k contains a wide variety of geometric shapes, including lines, polygons, circles, and complex spatial relationships, etc. Furthermore, this paper demonstrates the efficacy of AutoGeo-100k in enhancing the performance of multimodal large language models through fine-tuning. Experimental results indicate significant improvements in the model's ability in handling geometric images, as evidenced by enhanced accuracy in tasks such as geometric captioning and mathematical reasoning. This research not only fills a critical gap in the availability of geometric datasets but also paves the way for the advancement of sophisticated AI-driven tools in education and research. Project page: https://autogeo-official.github.io/.

Convolutional neural networks and their ilk have been very successful for many learning tasks involving images. These methods assume that the input is a scalar image representing the intensity in each pixel, possibly in multiple channels for color images. In natural-science domains however, image-like data sets might have vectors (velocity, say), tensors (polarization, say), pseudovectors (magnetic field, say), or other geometric objects in each pixel. Treating the components of these objects as independent channels in a CNN neglects their structure entirely. Our formulation -- the GeometricImageNet -- combines a geometric generalization of convolution with outer products, tensor index contractions, and tensor index permutations to construct geometric-image functions of geometric images that use and benefit from the tensor structure. The framework permits, with a very simple adjustment, restriction to function spaces that are exactly equivariant to translations, discrete rotations, and reflections. We use representation theory to quantify the dimension of the space of equivariant polynomial functions on 2-dimensional vector images. We give partial results on the expressivity of GeometricImageNet on small images. In numerical experiments, we find that GeometricImageNet has good generalization for a small simulated physics system, even when trained with a small training set. We expect this tool will be valuable for scientific and engineering machine learning, for example in cosmology or ocean dynamics.

Recent progress in the field of 3D human shape estimation enables the efficient and accurate modeling of naked body shapes, but doesn't do so well when tasked with displaying the geometry of clothes. A team of researchers from Institut de Robòtica i Informàtica Industrial and Harvard University recently introduced 3DPeople, a large-scale comprehensive dataset with specific geometric shapes of clothes that is suitable for many computer vision tasks involving clothed humans. In addition to the new dataset, researchers also developed a novel shape parameterization algorithm and a multi-resolution end-to-end deep generative network for predicting dressed body shape. Researchers used four cameras to capture each subject's action sequence. In addition to providing textured 3D meshes for the clothing and bodies, researchers also annotated the dataset with RGB (one of the most widely used color systems, including almost all colors that can be perceived by human vision), 3D skeleton, depth, optical flow, and semantic information (body parts and cloth labels).