arXiv subjects

Georg Bökman

Publications and source records attributed to Georg Bökman.

At least 19 recordsLinked to original sources

arXiv2026

Quick ViTs: Speeding up Vision Transformers through Equivariance

Natural images exhibit strong geometric regularities: local structures, such as edges, corners, and textures, appear in many orientations and mirror configurations. Since Vision Transformers (ViTs) operate on square image patches, these transformations naturally correspond to the dihedral symmetry group $\mathrm{D}_8$, also known as the octic group. Recent work has shown that ViTs can be made reflection equivariant and more efficient than standard ViTs simultaneously by implementing the linear layers in the Fourier domain of the reflection group. In this work, we extend the equivariance to reflections and rotations and analyze the scalability of the resulting networks. Our Quick ViTs, based on octic equivariant linear layers, achieve 5.33x reductions in FLOPs and up to 8x reductions in memory compared to ordinary linear layers. By analyzing the arithmetic intensity of these layers, we identify theoretical limits on how much the FLOP savings translate into throughput improvements on modern GPUs. However, these limitations disappear as the embedding dimensions increase. Enabled by their computational efficiency, we conduct a broader empirical evaluation of equivariant ViTs than in previous work. Upon training supervised (DeiT-III) and self-supervised (DINOv2) on ImageNet-1K, we find that our Quick ViTs match or exceed baseline accuracy while at the same time providing substantial efficiency gains.

cs.CV↗

arXiv2026

RoMa v2: Harder Better Faster Denser Feature Matching

Dense feature matching aims to estimate all correspondences between two images of a 3D scene and has recently been established as the gold standard due to its high accuracy and robustness. However, existing dense matchers still fail or perform poorly for many hard real-world scenarios, and high-precision models are often slow, limiting their applicability. In this paper, we attack these weaknesses on a wide front through a series of systematic improvements that together yield a significantly better model. In particular, we construct a novel matching architecture and loss, which, combined with a curated diverse training distribution, enables our model to solve many complex matching tasks. We further make training faster through a decoupled two-stage matching-then-refinement pipeline, and at the same time, significantly reduce refinement memory usage through a custom CUDA kernel. Finally, we leverage the recent DINOv3 foundation model along with multiple other insights to make the model more robust and unbiased. In our extensive set of experiments, we show that the resulting novel matcher sets a new state-of-the-art, being significantly more accurate than its predecessors. Code is available at https://github.com/Parskatt/romav2

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arXiv2026

LoMa: Local Feature Matching Revisited

Local feature matching has long been a fundamental component of 3D vision systems such as Structure-from-Motion (SfM), yet progress has lagged behind the rapid advances of modern data-driven approaches. The newer approaches, such as feed-forward reconstruction models, have benefited extensively from scaling dataset sizes, whereas local feature matching models are still only trained on a few mid-sized datasets. In this paper, we revisit local feature matching from a data-driven perspective. In our approach, which we call LoMa, we combine large and diverse data mixtures, modern training recipes, scaled model capacity, and scaled compute, resulting in remarkable gains in performance. Since current standard benchmarks mainly rely on collecting sparse views from successful 3D reconstructions, the evaluation of progress in feature matching has been limited to relatively easy image pairs. To address the resulting saturation of benchmarks, we collect 1000 highly challenging image pairs from internet data into a new dataset called HardMatch. Ground truth correspondences for HardMatch are obtained via manual annotation by the authors. In our extensive benchmarking suite, we find that LoMa makes outstanding progress across the board, outperforming the state-of-the-art method ALIKED+LightGlue by +18.6 mAA on HardMatch, +29.5 mAA on WxBS, +21.4 (1m, 10$^\circ$) on InLoc, +24.2 AUC on RUBIK, and +12.4 mAA on IMC 2022. We release our code and models publicly at https://github.com/davnords/LoMa.

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arXiv2026

A Unified Framework for Vision Transformers Equivariant to Discrete Subgroups of $\mathrm{O}(2)$

Vision transformers have become a dominant architecture for visual recognition. However, standard models do not explicitly encode the planar symmetries that arise in many vision domains. We introduce a family of vision transformers equivariant to arbitrary discrete subgroups of $\mathrm{O}(2)$, providing a unified framework that generalizes prior flipping- and $D_4$-equivariant transformer architectures. Our construction yields equivariant analogues of the core transformer components, together with expressivity guarantees for the resulting layers. In particular, we show that whenever $H \le G$, the class of $G$-equivariant ViTs embeds naturally into the class of $H$-equivariant ViTs. We also prove that, in the single-head setting, the corresponding equivariant self-attention layer realizes every $G$-equivariant self-attention map representable by ordinary self-attention. We further construct a $D_6$-equivariant model based on hexagonal patches, making the architecture compatible with six-fold rotational symmetries. We evaluate the resulting models on the PatternNet aerial image dataset in artificially data-scarce regimes across subgroups of $D_4$ and $D_6$. Our experiments compare two equivariant attention mechanisms and analyze how the choice of homogeneous-space configurations used in the nonlinearities affects performance. Preliminary results under matched parameter budgets indicate that equivariance can improve recognition accuracy, motivating further study of how discrete symmetry groups shape transformer-based visual recognition models.

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arXiv2026

Platonic Transformers: A Solid Choice For Equivariance

While widespread, Transformers lack inductive biases for geometric symmetries common in science and computer vision. Existing equivariant methods often sacrifice the efficiency and flexibility that make Transformers so effective through complex, computationally intensive designs. We introduce the Platonic Transformer to resolve this trade-off. By defining attention relative to reference frames from the Platonic solid symmetry groups, our method induces a principled weight-sharing scheme. This enables combined equivariance to continuous translations and Platonic symmetries, while preserving the exact architecture and computational cost of a standard Transformer. Furthermore, we show that this attention is formally equivalent to a dynamic group convolution, which reveals that the model learns adaptive geometric filters and enables a highly scalable, linear-time convolutional variant. Across diverse benchmarks in computer vision (CIFAR-10), 3D point clouds (ScanObjectNN), and molecular property prediction (QM9, OMol25), the Platonic Transformer achieves competitive performance by leveraging these geometric constraints at no additional cost.

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arXiv2026

Identifiable Equivariant Networks are Layerwise Equivariant

We investigate the relation between end-to-end equivariance and layerwise equivariance in deep neural networks. We prove the following: For a network whose end-to-end function is equivariant with respect to group actions on the input and output spaces, there is a parameter choice yielding the same end-to-end function such that its layers are equivariant with respect to some group actions on the latent spaces. Our result assumes that the parameters of the model are identifiable in an appropriate sense. This identifiability property has been established in the literature for a large class of networks, to which our results apply immediately, while it is conjectural for others. The theory we develop is grounded in an abstract formalism, and is therefore architecture-agnostic. Overall, our results provide a mathematical explanation for the emergence of equivariant structures in the weights of neural networks during training -- a phenomenon that is consistently observed in practice.

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arXiv2026

Who Handles Orientation? Investigating Invariance in Feature Matching

Finding matching keypoints between images is a core problem in 3D computer vision. However, modern matchers struggle with large in-plane rotations. A straightforward mitigation is to learn rotation invariance via data augmentation. However, it remains unclear at which stage rotation invariance should be incorporated. In this paper, we study this in the context of a modern sparse matching pipeline. We perform extensive experiments by training on a large collection of 3D vision datasets and evaluating on popular image matching benchmarks. Surprisingly, we find that incorporating rotation invariance already in the descriptor yields similar performance to handling it in the matcher. However, rotation invariance is achieved earlier in the matcher when it is learned in the descriptor, allowing for a faster rotation-invariant matcher. Further, we find that enforcing rotation invariance does not hurt upright performance when trained at scale. Finally, we study the emergence of rotation invariance through scale and find that increasing the training data size substantially improves generalization to rotated images. We release two matchers robust to in-plane rotations that achieve state-of-the-art performance on e.g. multi-modal (WxBS), extreme (HardMatch), and satellite image matching (SatAst). Code is available at https://github.com/davnords/loma.

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arXiv2025

MuM: Multi-View Masked Image Modeling for 3D Vision

Self-supervised learning on images seeks to extract meaningful visual representations from unlabeled data. When scaled to large datasets, this paradigm has achieved state-of-the-art performance and the resulting trained models such as DINOv3 have seen widespread adoption. However, most prior efforts are optimized for semantic understanding rather than geometric reasoning. One important exception is Cross-View Completion, CroCo, which is a form of masked autoencoding (MAE) tailored for 3D understanding. In this work, we continue on the path proposed by CroCo and focus on learning features tailored for 3D vision. In a nutshell, we extend MAE to arbitrarily many views of the same scene. By uniformly masking all views and employing a lightweight decoder with inter-frame attention, our approach is inherently simpler and more scalable than CroCo. We evaluate the resulting model, MuM, extensively on downstream tasks including feedforward reconstruction, dense image matching and relative pose estimation, finding that it outperforms the state-of-the-art visual encoders DINOv3 and CroCo v2.

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arXiv2025

Flopping for FLOPs: Leveraging equivariance for computational efficiency

Incorporating geometric invariance into neural networks enhances parameter efficiency but typically increases computational costs. This paper introduces new equivariant neural networks that preserve symmetry while maintaining a comparable number of floating-point operations (FLOPs) per parameter to standard non-equivariant networks. We focus on horizontal mirroring (flopping) invariance, common in many computer vision tasks. The main idea is to parametrize the feature spaces in terms of mirror-symmetric and mirror-antisymmetric features, i.e., irreps of the flopping group. This decomposes the linear layers to be block-diagonal, requiring half the number of FLOPs. Our approach reduces both FLOPs and wall-clock time, providing a practical solution for efficient, scalable symmetry-aware architectures.

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arXiv2025

DaD: Distilled Reinforcement Learning for Diverse Keypoint Detection

Keypoints are what enable Structure-from-Motion (SfM) systems to scale to thousands of images. However, designing a keypoint detection objective is a non-trivial task, as SfM is non-differentiable. Typically, an auxiliary objective involving a descriptor is optimized. This however induces a dependency on the descriptor, which is undesirable. In this paper we propose a fully self-supervised and descriptor-free objective for keypoint detection, through reinforcement learning. To ensure training does not degenerate, we leverage a balanced top-K sampling strategy. While this already produces competitive models, we find that two qualitatively different types of detectors emerge, which are only able to detect light and dark keypoints respectively. To remedy this, we train a third detector, DaD, that optimizes the Kullback-Leibler divergence of the pointwise maximum of both light and dark detectors. Our approach significantly improve upon SotA across a range of benchmarks. Code and model weights are publicly available at https://github.com/parskatt/dad

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arXiv2025

A Framework for Reducing the Complexity of Geometric Vision Problems and its Application to Two-View Triangulation with Approximation Bounds

In this paper, we present a new framework for reducing the computational complexity of geometric vision problems through targeted reweighting of the cost functions used to minimize reprojection errors. Triangulation - the task of estimating a 3D point from noisy 2D projections across multiple images - is a fundamental problem in multiview geometry and Structure-from-Motion (SfM) pipelines. We apply our framework to the two-view case and demonstrate that optimal triangulation, which requires solving a univariate polynomial of degree six, can be simplified through cost function reweighting reducing the polynomial degree to two. This reweighting yields a closed-form solution while preserving strong geometric accuracy. We derive optimal weighting strategies, establish theoretical bounds on the approximation error, and provide experimental results on real data demonstrating the effectiveness of the proposed approach compared to standard methods. Although this work focuses on two-view triangulation, the framework generalizes to other geometric vision problems.

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arXiv2024

Affine steerers for structured keypoint description

We propose a way to train deep learning based keypoint descriptors that makes them approximately equivariant for locally affine transformations of the image plane. The main idea is to use the representation theory of GL(2) to generalize the recently introduced concept of steerers from rotations to affine transformations. Affine steerers give high control over how keypoint descriptions transform under image transformations. We demonstrate the potential of using this control for image matching. Finally, we propose a way to finetune keypoint descriptors with a set of steerers on upright images and obtain state-of-the-art results on several standard benchmarks. Code will be published at github.com/georg-bn/affine-steerers.

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arXiv2024

From 2D to 3D: AISG-SLA Visual Localization Challenge

Research in 3D mapping is crucial for smart city applications, yet the cost of acquiring 3D data often hinders progress. Visual localization, particularly monocular camera position estimation, offers a solution by determining the camera's pose solely through visual cues. However, this task is challenging due to limited data from a single camera. To tackle these challenges, we organized the AISG-SLA Visual Localization Challenge (VLC) at IJCAI 2023 to explore how AI can accurately extract camera pose data from 2D images in 3D space. The challenge attracted over 300 participants worldwide, forming 50+ teams. Winning teams achieved high accuracy in pose estimation using images from a car-mounted camera with low frame rates. The VLC dataset is available for research purposes upon request via vlc-dataset@aisingapore.org.

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arXiv2024

DeDoDe v2: Analyzing and Improving the DeDoDe Keypoint Detector

In this paper, we analyze and improve into the recently proposed DeDoDe keypoint detector. We focus our analysis on some key issues. First, we find that DeDoDe keypoints tend to cluster together, which we fix by performing non-max suppression on the target distribution of the detector during training. Second, we address issues related to data augmentation. In particular, the DeDoDe detector is sensitive to large rotations. We fix this by including 90-degree rotations as well as horizontal flips. Finally, the decoupled nature of the DeDoDe detector makes evaluation of downstream usefulness problematic. We fix this by matching the keypoints with a pretrained dense matcher (RoMa) and evaluating two-view pose estimates. We find that the original long training is detrimental to performance, and therefore propose a much shorter training schedule. We integrate all these improvements into our proposed detector DeDoDe v2 and evaluate it with the original DeDoDe descriptor on the MegaDepth-1500 and IMC2022 benchmarks. Our proposed detector significantly increases pose estimation results, notably from 75.9 to 78.3 mAA on the IMC2022 challenge. Code and weights are available at https://github.com/Parskatt/DeDoDe

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arXiv2024

Steerers: A framework for rotation equivariant keypoint descriptors

Image keypoint descriptions that are discriminative and matchable over large changes in viewpoint are vital for 3D reconstruction. However, descriptions output by learned descriptors are typically not robust to camera rotation. While they can be made more robust by, e.g., data augmentation, this degrades performance on upright images. Another approach is test-time augmentation, which incurs a significant increase in runtime. Instead, we learn a linear transform in description space that encodes rotations of the input image. We call this linear transform a steerer since it allows us to transform the descriptions as if the image was rotated. From representation theory, we know all possible steerers for the rotation group. Steerers can be optimized (A) given a fixed descriptor, (B) jointly with a descriptor or (C) we can optimize a descriptor given a fixed steerer. We perform experiments in these three settings and obtain state-of-the-art results on the rotation invariant image matching benchmarks AIMS and Roto-360. We publish code and model weights at https://github.com/georg-bn/rotation-steerers.

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arXiv2023

In Search of Projectively Equivariant Networks

Equivariance of linear neural network layers is well studied. In this work, we relax the equivariance condition to only be true in a projective sense. We propose a way to construct a projectively equivariant neural network through building a standard equivariant network where the linear group representations acting on each intermediate feature space are "multiplicatively modified lifts" of projective group representations. By theoretically studying the relation of projectively and linearly equivariant linear layers, we show that our approach is the most general possible when building a network out of linear layers. The theory is showcased in two simple experiments.

cs.CV↗

arXiv2023

DeDoDe: Detect, Don't Describe -- Describe, Don't Detect for Local Feature Matching

Keypoint detection is a pivotal step in 3D reconstruction, whereby sets of (up to) K points are detected in each view of a scene. Crucially, the detected points need to be consistent between views, i.e., correspond to the same 3D point in the scene. One of the main challenges with keypoint detection is the formulation of the learning objective. Previous learning-based methods typically jointly learn descriptors with keypoints, and treat the keypoint detection as a binary classification task on mutual nearest neighbours. However, basing keypoint detection on descriptor nearest neighbours is a proxy task, which is not guaranteed to produce 3D-consistent keypoints. Furthermore, this ties the keypoints to a specific descriptor, complicating downstream usage. In this work, we instead learn keypoints directly from 3D consistency. To this end, we train the detector to detect tracks from large-scale SfM. As these points are often overly sparse, we derive a semi-supervised two-view detection objective to expand this set to a desired number of detections. To train a descriptor, we maximize the mutual nearest neighbour objective over the keypoints with a separate network. Results show that our approach, DeDoDe, achieves significant gains on multiple geometry benchmarks. Code is provided at https://github.com/Parskatt/DeDoDe

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arXiv2023

RoMa: Robust Dense Feature Matching

Feature matching is an important computer vision task that involves estimating correspondences between two images of a 3D scene, and dense methods estimate all such correspondences. The aim is to learn a robust model, i.e., a model able to match under challenging real-world changes. In this work, we propose such a model, leveraging frozen pretrained features from the foundation model DINOv2. Although these features are significantly more robust than local features trained from scratch, they are inherently coarse. We therefore combine them with specialized ConvNet fine features, creating a precisely localizable feature pyramid. To further improve robustness, we propose a tailored transformer match decoder that predicts anchor probabilities, which enables it to express multimodality. Finally, we propose an improved loss formulation through regression-by-classification with subsequent robust regression. We conduct a comprehensive set of experiments that show that our method, RoMa, achieves significant gains, setting a new state-of-the-art. In particular, we achieve a 36% improvement on the extremely challenging WxBS benchmark. Code is provided at https://github.com/Parskatt/RoMa

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