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With their meaningful geometry and their omnipresence in the 3D world, edges are extremely useful primitives in computer vision. 3D edges comprise of lines and curves, and methods to reconstruct them use either multi-view images or point clouds as input. State-of-the-art image-based methods first learn a 3D edge point cloud then fit 3D edges to it. The edge point cloud is obtained by learning a 3D neural implicit edge field from which the 3D edge points are sampled on a specific level set (0 or 1). However, such methods present two important drawbacks: i) it is not realistic to sample points on exact level sets due to float imprecision and training inaccuracies. Instead, they are sampled within a range of levels so the points do not lie accurately on the 3D edges and require further processing. ii) Such implicit representations are computationally expensive and require long training times. In this paper, we address these two limitations and propose a 3D edge mapping that is simpler, more efficient, and preserves accuracy. Our method learns explicitly the 3D edge points and their edge direction hence bypassing the need for point sampling. It casts a 3D edge point as the center of a 3D Gaussian and the edge direction as the principal axis of the Gaussian. Such a representation has the advantage of being not only geometrically meaningful but also compatible with the efficient training optimization defined in Gaussian Splatting. Results show that the proposed method produces edges as accurate and complete as the state-of-the-art while being an order of magnitude faster. Code is released at https://github.com/kunalchelani/EdgeGaussians.

Rapid growth in the popularity of AR/VR/MR applications and cloud-based visual localization systems has given rise to an increased focus on the privacy of user content in the localization process. This privacy concern has been further escalated by the ability of deep neural networks to recover detailed images of a scene from a sparse set of 3D or 2D points and their descriptors - the so-called inversion attacks. Research on privacy-preserving localization has therefore focused on preventing these inversion attacks on both the query image keypoints and the 3D points of the scene map. To this end, several geometry obfuscation techniques that lift points to higher-dimensional spaces, i.e., lines or planes, or that swap coordinates between points % have been proposed. In this paper, we point to a common weakness of these obfuscations that allows to recover approximations of the original point positions under the assumption of known neighborhoods. We further show that these neighborhoods can be computed by learning to identify descriptors that co-occur in neighborhoods. Extensive experiments show that our approach for point recovery is practically applicable to all existing geometric obfuscation schemes. Our results show that these schemes should not be considered privacy-preserving, even though they are claimed to be privacy-preserving. Code will be available at \url{https://github.com/kunalchelani/RecoverPointsNeighborhood}.

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., Comment: To be presented at ECCV 2024

In open-set semi-supervised learning (OSSL), we consider unlabeled datasets that may contain unknown classes. Existing OSSL methods often use the softmax confidence for classifying data as in-distribution (ID) or out-of-distribution (OOD). Additionally, many works for OSSL rely on ad-hoc thresholds for ID/OOD classification, without considering the statistics of the problem. We propose a new score for ID/OOD classification based on angles in feature space between data and an ID subspace. Moreover, we propose an approach to estimate the conditional distributions of scores given ID or OOD data, enabling probabilistic predictions of data being ID or OOD. These components are put together in a framework for OSSL, termed \emph{ProSub}, that is experimentally shown to reach SOTA performance on several benchmark problems. Our code is available at https://github.com/walline/prosub., Comment: ECCV2024

Few-shot image classification involves classifying images using very few training examples. Recent vision foundation models show excellent few-shot transfer abilities, but are large and slow at inference. Using knowledge distillation, the capabilities of high-performing but slow models can be transferred to tiny, efficient models. However, common distillation methods require a large set of unlabeled data, which is not available in the few-shot setting. To overcome this lack of data, there has been a recent interest in using synthetic data. We expand on this work by presenting a novel diffusion model inversion technique (TINT) combining the diversity of textual inversion with the specificity of null-text inversion. Using this method in a few-shot distillation pipeline leads to state-of-the-art accuracy among small student models on popular benchmarks, while being significantly faster than prior work. This allows us to push even tiny models to high accuracy using only a tiny application-specific dataset, albeit relying on extra data for pre-training. Popular few-shot benchmarks involve evaluation over a large number of episodes, which is computationally cumbersome for methods involving synthetic data generation. Therefore, we also present a theoretical analysis on how the variance of the accuracy estimator depends on the number of episodes and query examples, and use these results to lower the computational effort required for method evaluation. In addition, to further motivate the use of generative models in few-shot distillation, we demonstrate that our method performs better compared to training on real data mined from the dataset used to train the diffusion model. Source code will be made available at https://github.com/pixwse/tiny2., Comment: 21 pages (9 main pages + references and appendix)

Object detection applied to LiDAR point clouds is a relevant task in robotics, and particularly in autonomous driving. Single frame methods, predominant in the field, exploit information from individual sensor scans. Recent approaches achieve good performance, at relatively low inference time. Nevertheless, given the inherent high sparsity of LiDAR data, these methods struggle in long-range detection (e.g. 200m) which we deem to be critical in achieving safe automation. Aggregating multiple scans not only leads to a denser point cloud representation, but it also brings time-awareness to the system, and provides information about how the environment is changing. Solutions of this kind, however, are often highly problem-specific, demand careful data processing, and tend not to fulfil runtime requirements. In this context we propose TimePillars, a temporally-recurrent object detection pipeline which leverages the pillar representation of LiDAR data across time, respecting hardware integration efficiency constraints, and exploiting the diversity and long-range information of the novel Zenseact Open Dataset (ZOD). Through experimentation, we prove the benefits of having recurrency, and show how basic building blocks are enough to achieve robust and efficient results., Comment: 10 pages, 5 figures

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., Comment: CVPR 2024 Camera ready

In this paper we tackle the problem of learning Structure-from-Motion (SfM) through the use of graph attention networks. SfM is a classic computer vision problem that is solved though iterative minimization of reprojection errors, referred to as Bundle Adjustment (BA), starting from a good initialization. In order to obtain a good enough initialization to BA, conventional methods rely on a sequence of sub-problems (such as pairwise pose estimation, pose averaging or triangulation) which provide an initial solution that can then be refined using BA. In this work we replace these sub-problems by learning a model that takes as input the 2D keypoints detected across multiple views, and outputs the corresponding camera poses and 3D keypoint coordinates. Our model takes advantage of graph neural networks to learn SfM-specific primitives, and we show that it can be used for fast inference of the reconstruction for new and unseen sequences. The experimental results show that the proposed model outperforms competing learning-based methods, and challenges COLMAP while having lower runtime. Our code is available at https://github.com/lucasbrynte/gasfm/., Comment: CVPR camera-ready updates

We present a generalizable novel view synthesis method which enables modifying the visual appearance of an observed scene so rendered views match a target weather or lighting condition without any scene specific training or access to reference views at the target condition. Our method is based on a pretrained generalizable transformer architecture and is fine-tuned on synthetically generated scenes under different appearance conditions. This allows for rendering novel views in a consistent manner for 3D scenes that were not included in the training set, along with the ability to (i) modify their appearance to match the target condition and (ii) smoothly interpolate between different conditions. Experiments on real and synthetic scenes show that our method is able to generate 3D consistent renderings while making realistic appearance changes, including qualitative and quantitative comparisons. Please refer to our project page for video results: https://ava-nvs.github.io/

Many data symmetries can be described in terms of group equivariance and the most common way of encoding group equivariances in neural networks is by building linear layers that are group equivariant. In this work we investigate whether equivariance of a network implies that all layers are equivariant. On the theoretical side we find cases where equivariance implies layerwise equivariance, but also demonstrate that this is not the case generally. Nevertheless, we conjecture that CNNs that are trained to be equivariant will exhibit layerwise equivariance and explain how this conjecture is a weaker version of the recent permutation conjecture by Entezari et al. [2022]. We perform quantitative experiments with VGG-nets on CIFAR10 and qualitative experiments with ResNets on ImageNet to illustrate and support our theoretical findings. These experiments are not only of interest for understanding how group equivariance is encoded in ReLU-networks, but they also give a new perspective on Entezari et al.'s permutation conjecture as we find that it is typically easier to merge a network with a group-transformed version of itself than merging two different networks., Comment: NeurIPS camera ready

Visual localization is the task of estimating the camera pose from which a given image was taken and is central to several 3D computer vision applications. With the rapid growth in the popularity of AR/VR/MR devices and cloud-based applications, privacy issues are becoming a very important aspect of the localization process. Existing work on privacy-preserving localization aims to defend against an attacker who has access to a cloud-based service. In this paper, we show that an attacker can learn about details of a scene without any access by simply querying a localization service. The attack is based on the observation that modern visual localization algorithms are robust to variations in appearance and geometry. While this is in general a desired property, it also leads to algorithms localizing objects that are similar enough to those present in a scene. An attacker can thus query a server with a large enough set of images of objects, \eg, obtained from the Internet, and some of them will be localized. The attacker can thus learn about object placements from the camera poses returned by the service (which is the minimal information returned by such a service). In this paper, we develop a proof-of-concept version of this attack and demonstrate its practical feasibility. The attack does not place any requirements on the localization algorithm used, and thus also applies to privacy-preserving representations. Current work on privacy-preserving representations alone is thus insufficient.

Open-set semi-supervised learning (OSSL) embodies a practical scenario within semi-supervised learning, wherein the unlabeled training set encompasses classes absent from the labeled set. Many existing OSSL methods assume that these out-of-distribution data are harmful and put effort into excluding data belonging to unknown classes from the training objective. In contrast, we propose an OSSL framework that facilitates learning from all unlabeled data through self-supervision. Additionally, we utilize an energy-based score to accurately recognize data belonging to the known classes, making our method well-suited for handling uncurated data in deployment. We show through extensive experimental evaluations that our method yields state-of-the-art results on many of the evaluated benchmark problems in terms of closed-set accuracy and open-set recognition when compared with existing methods for OSSL. Our code is available at https://github.com/walline/ssl-tf2-sefoss., Comment: WACV2024

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., Comment: v3: Another significant rewrite. Accepted for publication in TMLR. v2: Significant rewrite. The title has been changed: "neural network" -> "network". More general description of projectively equivariant linear layers, with new proposed architectures, and a completely new accompanying experiment section, as a result

Following the success of supervised learning, semi-supervised learning (SSL) is now becoming increasingly popular. SSL is a family of methods, which in addition to a labeled training set, also use a sizable collection of unlabeled data for fitting a model. Most of the recent successful SSL methods are based on pseudo-labeling approaches: letting confident model predictions act as training labels. While these methods have shown impressive results on many benchmark datasets, a drawback of this approach is that not all unlabeled data are used during training. We propose a new SSL algorithm, DoubleMatch, which combines the pseudo-labeling technique with a self-supervised loss, enabling the model to utilize all unlabeled data in the training process. We show that this method achieves state-of-the-art accuracies on multiple benchmark datasets while also reducing training times compared to existing SSL methods. Code is available at https://github.com/walline/doublematch., Comment: ICPR2022

The aim of this paper is to demonstrate that a state of the art feature matcher (LoFTR) can be made more robust to rotations by simply replacing the backbone CNN with a steerable CNN which is equivariant to translations and image rotations. It is experimentally shown that this boost is obtained without reducing performance on ordinary illumination and viewpoint matching sequences., Comment: CVPRW 2022, updated version

We characterize the class of image plane transformations which realize rigid camera motions and call these transformations `rigidity preserving'. In particular, 2D translations of pinhole images are not rigidity preserving. Hence, when using CNNs for 3D inference tasks, it can be beneficial to modify the inductive bias from equivariance towards translations to equivariance towards rigidity preserving transformations. We investigate how equivariance with respect to rigidity preserving transformations can be approximated in CNNs, and test our ideas on both 6D object pose estimation and visual localization. Experimentally, we improve on several competitive baselines., Comment: v2: Substantially revised version. Among other things, experiments with the PixLoc model added

In this paper, we are concerned with rotation equivariance on 2D point cloud data. We describe a particular set of functions able to approximate any continuous rotation equivariant and permutation invariant function. Based on this result, we propose a novel neural network architecture for processing 2D point clouds and we prove its universality for approximating functions exhibiting these symmetries. We also show how to extend the architecture to accept a set of 2D-2D correspondences as indata, while maintaining similar equivariance properties. Experiments are presented on the estimation of essential matrices in stereo vision., Comment: CVPR 2022 camera ready

Visual localization is the problem of estimating the position and orientation from which a given image (or a sequence of images) is taken in a known scene. It is an important part of a wide range of computer vision and robotics applications, from self-driving cars to augmented/virtual reality systems. Visual localization techniques should work reliably and robustly under a wide range of conditions, including seasonal, weather, illumination and man-made changes. Recent benchmarking efforts model this by providing images under different conditions, and the community has made rapid progress on these datasets since their inception. However, they are limited to a few geographical regions and often recorded with a single device. We propose a new benchmark for visual localization in outdoor scenes, using crowd-sourced data to cover a wide range of geographical regions and camera devices with a focus on the failure cases of current algorithms. Experiments with state-of-the-art localization approaches show that our dataset is very challenging, with all evaluated methods failing on its hardest parts. As part of the dataset release, we provide the tooling used to generate it, enabling efficient and effective 2D correspondence annotation to obtain reference poses.

Camera pose estimation in known scenes is a 3D geometry task recently tackled by multiple learning algorithms. Many regress precise geometric quantities, like poses or 3D points, from an input image. This either fails to generalize to new viewpoints or ties the model parameters to a specific scene. In this paper, we go Back to the Feature: we argue that deep networks should focus on learning robust and invariant visual features, while the geometric estimation should be left to principled algorithms. We introduce PixLoc, a scene-agnostic neural network that estimates an accurate 6-DoF pose from an image and a 3D model. Our approach is based on the direct alignment of multiscale deep features, casting camera localization as metric learning. PixLoc learns strong data priors by end-to-end training from pixels to pose and exhibits exceptional generalization to new scenes by separating model parameters and scene geometry. The system can localize in large environments given coarse pose priors but also improve the accuracy of sparse feature matching by jointly refining keypoints and poses with little overhead. The code will be publicly available at https://github.com/cvg/pixloc., Comment: Accepted to CVPR 2021

Visual localization is the problem of estimating the camera pose of a given image with respect to a known scene. Visual localization algorithms are a fundamental building block in advanced computer vision applications, including Mixed and Virtual Reality systems. Many algorithms used in practice represent the scene through a Structure-from-Motion (SfM) point cloud and use 2D-3D matches between a query image and the 3D points for camera pose estimation. As recently shown, image details can be accurately recovered from SfM point clouds by translating renderings of the sparse point clouds to images. To address the resulting potential privacy risks for user-generated content, it was recently proposed to lift point clouds to line clouds by replacing 3D points by randomly oriented 3D lines passing through these points. The resulting representation is unintelligible to humans and effectively prevents point cloud-to-image translation. This paper shows that a significant amount of information about the 3D scene geometry is preserved in these line clouds, allowing us to (approximately) recover the 3D point positions and thus to (approximately) recover image content. Our approach is based on the observation that the closest points between lines can yield a good approximation to the original 3D points. Code is available at https://github.com/kunalchelani/Line2Point., Comment: Computer Vision and Pattern Recognition (CVPR) 2021

Why is it that semidefinite relaxations have been so successful in numerous applications in computer vision and robotics for solving non-convex optimization problems involving rotations? In studying the empirical performance we note that there are few failure cases reported in the literature, in particular for estimation problems with a single rotation, motivating us to gain further theoretical understanding. A general framework based on tools from algebraic geometry is introduced for analyzing the power of semidefinite relaxations of problems with quadratic objective functions and rotational constraints. Applications include registration, hand-eye calibration and rotation averaging. We characterize the extreme points, and show that there exist failure cases for which the relaxation is not tight, even in the case of a single rotation. We also show that some problem classes are always tight given an appropriate parametrization. Our theoretical findings are accompanied with numerical simulations, providing further evidence and understanding of the results., Comment: Accepted for the Journal of Mathematical Imaging and Vision (JMIV)

Good local features improve the robustness of many 3D re-localization and multi-view reconstruction pipelines. The problem is that viewing angle and distance severely impact the recognizability of a local feature. Attempts to improve appearance invariance by choosing better local feature points or by leveraging outside information, have come with pre-requisites that made some of them impractical. In this paper, we propose a surprisingly effective enhancement to local feature extraction, which improves matching. We show that CNN-based depths inferred from single RGB images are quite helpful, despite their flaws. They allow us to pre-warp images and rectify perspective distortions, to significantly enhance SIFT and BRISK features, enabling more good matches, even when cameras are looking at the same scene but in opposite directions., Comment: 14 pages, 7 figures, accepted for publication at the European conference on computer vision (ECCV) 2020

In recent years, considerable progress has been made for the task of rigid object pose estimation from a single RGB-image, but achieving robustness to partial occlusions remains a challenging problem. Pose refinement via rendering has shown promise in order to achieve improved results, in particular, when data is scarce. In this paper we focus our attention on pose refinement, and show how to push the state-of-the-art further in the case of partial occlusions. The proposed pose refinement method leverages on a simplified learning task, where a CNN is trained to estimate the reprojection error between an observed and a rendered image. We experiment by training on purely synthetic data as well as a mixture of synthetic and real data. Current state-of-the-art results are outperformed for two out of three metrics on the Occlusion LINEMOD benchmark, while performing on-par for the final metric., Comment: Added acknowledgements

Long-term visual localization is the problem of estimating the camera pose of a given query image in a scene whose appearance changes over time. It is an important problem in practice, for example, encountered in autonomous driving. In order to gain robustness to such changes, long-term localization approaches often use segmantic segmentations as an invariant scene representation, as the semantic meaning of each scene part should not be affected by seasonal and other changes. However, these representations are typically not very discriminative due to the limited number of available classes. In this paper, we propose a new neural network, the Fine-Grained Segmentation Network (FGSN), that can be used to provide image segmentations with a larger number of labels and can be trained in a self-supervised fashion. In addition, we show how FGSNs can be trained to output consistent labels across seasonal changes. We demonstrate through extensive experiments that integrating the fine-grained segmentations produced by our FGSNs into existing localization algorithms leads to substantial improvements in localization performance., Comment: Accepted to ICCV 2019

Three-dimensional object detection from a single view is a challenging task which, if performed with good accuracy, is an important enabler of low-cost mobile robot perception. Previous approaches to this problem suffer either from an overly complex inference engine or from an insufficient detection accuracy. To deal with these issues, we present SS3D, a single-stage monocular 3D object detector. The framework consists of (i) a CNN, which outputs a redundant representation of each relevant object in the image with corresponding uncertainty estimates, and (ii) a 3D bounding box optimizer. We show how modeling heteroscedastic uncertainty improves performance upon our baseline, and furthermore, how back-propagation can be done through the optimizer in order to train the pipeline end-to-end for additional accuracy. Our method achieves SOTA accuracy on monocular 3D object detection, while running at 20 fps in a straightforward implementation. We argue that the SS3D architecture provides a solid framework upon which high performing detection systems can be built, with autonomous driving being the main application in mind., Comment: For associated videos file, see http://tinyurl.com/SS3D-YouTube ; without supplementary material

In this paper, we present a method to utilize 2D-2D point matches between images taken during different image conditions to train a convolutional neural network for semantic segmentation. Enforcing label consistency across the matches makes the final segmentation algorithm robust to seasonal changes. We describe how these 2D-2D matches can be generated with little human interaction by geometrically matching points from 3D models built from images. Two cross-season correspondence datasets are created providing 2D-2D matches across seasonal changes as well as from day to night. The datasets are made publicly available to facilitate further research. We show that adding the correspondences as extra supervision during training improves the segmentation performance of the convolutional neural network, making it more robust to seasonal changes and weather conditions., Comment: In Proc. CVPR 2019

Visual localization enables autonomous vehicles to navigate in their surroundings and augmented reality applications to link virtual to real worlds. Practical visual localization approaches need to be robust to a wide variety of viewing condition, including day-night changes, as well as weather and seasonal variations, while providing highly accurate 6 degree-of-freedom (6DOF) camera pose estimates. In this paper, we introduce the first benchmark datasets specifically designed for analyzing the impact of such factors on visual localization. Using carefully created ground truth poses for query images taken under a wide variety of conditions, we evaluate the impact of various factors on 6DOF camera pose estimation accuracy through extensive experiments with state-of-the-art localization approaches. Based on our results, we draw conclusions about the difficulty of different conditions, showing that long-term localization is far from solved, and propose promising avenues for future work, including sequence-based localization approaches and the need for better local features. Our benchmark is available at visuallocalization.net., Comment: Accepted to CVPR 2018 as a spotlight

In this paper we explore the role of duality principles within the problem of rotation averaging, a fundamental task in a wide range of computer vision applications. In its conventional form, rotation averaging is stated as a minimization over multiple rotation constraints. As these constraints are non-convex, this problem is generally considered challenging to solve globally. We show how to circumvent this difficulty through the use of Lagrangian duality. While such an approach is well-known it is normally not guaranteed to provide a tight relaxation. Based on spectral graph theory, we analytically prove that in many cases there is no duality gap unless the noise levels are severe. This allows us to obtain certifiably global solutions to a class of important non-convex problems in polynomial time. We also propose an efficient, scalable algorithm that out-performs general purpose numerical solvers and is able to handle the large problem instances commonly occurring in structure from motion settings. The potential of this proposed method is demonstrated on a number of different problems, consisting of both synthetic and real-world data.

Are we using the right potential functions in the Conditional Random Field models that are popular in the Vision community? Semantic segmentation and other pixel-level labelling tasks have made significant progress recently due to the deep learning paradigm. However, most state-of-the-art structured prediction methods also include a random field model with a hand-crafted Gaussian potential to model spatial priors, label consistencies and feature-based image conditioning. In this paper, we challenge this view by developing a new inference and learning framework which can learn pairwise CRF potentials restricted only by their dependence on the image pixel values and the size of the support. Both standard spatial and high-dimensional bilateral kernels are considered. Our framework is based on the observation that CRF inference can be achieved via projected gradient descent and consequently, can easily be integrated in deep neural networks to allow for end-to-end training. It is empirically demonstrated that such learned potentials can improve segmentation accuracy and that certain label class interactions are indeed better modelled by a non-Gaussian potential. In addition, we compare our inference method to the commonly used mean-field algorithm. Our framework is evaluated on several public benchmarks for semantic segmentation with improved performance compared to previous state-of-the-art CNN+CRF models., Comment: Presented at EMMCVPR 2017 conference

We study the relation between the target functions of low-resolution and high-resolution intensity-based registration for the class of rigid transformations. Our results show that low resolution target values can tightly bound the high-resolution target function in natural images. This can help with analyzing and better understanding the process of multiresolution image registration. It also gives a guideline for designing multiresolution algorithms in which the search space in higher resolution registration is restricted given the fitness values for lower resolution image pairs. To demonstrate this, we incorporate our multiresolution technique into a Lipschitz global optimization framework. We show that using the multiresolution scheme can result in large gains in the efficiency of such algorithms. The method is evaluated by applying to 2D and 3D registration problems as well as the detection of reflective symmetry in 2D and 3D images.

We present the first method to handle curvature regularity in region-based image segmentation and inpainting that is independent of initialization. To this end we start from a new formulation of length-based optimization schemes, based on surface continuation constraints, and discuss the connections to existing schemes. The formulation is based on a \emph{cell complex} and considers basic regions and boundary elements. The corresponding optimization problem is cast as an integer linear program. We then show how the method can be extended to include curvature regularity, again cast as an integer linear program. Here, we are considering pairs of boundary elements to reflect curvature. Moreover, a constraint set is derived to ensure that the boundary variables indeed reflect the boundary of the regions described by the region variables. We show that by solving the linear programming relaxation one gets quite close to the global optimum, and that curvature regularity is indeed much better suited in the presence of long and thin objects compared to standard length regularity.