Image tokenizers map images to sequences of discrete tokens, and are a crucial component of autoregressive transformer-based image generation. The tokens are typically associated with spatial locations in the input image, arranged in raster scan order, which is not ideal for autoregressive modeling. In this paper, we propose to tokenize the image spectrum instead, obtained from a discrete wavelet transform (DWT), such that the sequence of tokens represents the image in a coarse-to-fine fashion. Our tokenizer brings several advantages: 1) it leverages that natural images are more compressible at high frequencies, 2) it can take and reconstruct images of different resolutions without retraining, 3) it improves the conditioning for next-token prediction -- instead of conditioning on a partial line-by-line reconstruction of the image, it takes a coarse reconstruction of the full image, 4) it enables partial decoding where the first few generated tokens can reconstruct a coarse version of the image, 5) it enables autoregressive models to be used for image upsampling. We evaluate the tokenizer reconstruction metrics as well as multiscale image generation, text-guided image upsampling and editing.

Classical light field rendering for novel view synthesis can accurately reproduce view-dependent effects such as reflection, refraction, and translucency, but requires a dense view sampling of the scene. Methods based on geometric reconstruction need only sparse views, but cannot accurately model non-Lambertian effects. We introduce a model that combines the strengths and mitigates the limitations of these two directions. By operating on a four-dimensional representation of the light field, our model learns to represent view-dependent effects accurately. By enforcing geometric constraints during training and inference, the scene geometry is implicitly learned from a sparse set of views. Concretely, we introduce a two-stage transformer-based model that first aggregates features along epipolar lines, then aggregates features along reference views to produce the color of a target ray. Additionally, we propose modifications that allow the model to generalize to scenes without any fine-tuning. Our model outperforms the state-of-the-art on multiple forward-facing and 360$^\circ$∘ datasets, with larger margins on scenes with severe view-dependent variations.

We introduce a framework for intrinsic latent diffusion models operating directly on the surfaces of 3D shapes, with the goal of synthesizing high-quality textures. Our approach is underpinned by two contributions: Field Latents, a latent representation encoding textures as discrete vector fields on the mesh vertices, and Field Latent Diffusion Models, which learn to denoise a diffusion process in the learned latent space on the surface. We consider a single-textured-mesh paradigm, where our models are trained to generate variations of a given texture on a mesh. We show the synthesized textures are of superior fidelity compared those from existing single-textured-mesh generative models. Our models can also be adapted for user-controlled editing tasks such as inpainting and label-guided generation. The efficacy of our approach is due in part to the equivariance of our proposed framework under isometries, allowing our models to seamlessly reproduce details across locally similar regions and opening the door to a notion of generative texture transfer. Code and visualizations are available at https://single-mesh-diffusion.github.io/.

We present a method for joint alignment of sparse in-the-wild image collections of an object category. Most prior works assume either ground-truth keypoint annotations or a large dataset of images of a single object category. However, neither of the above assumptions hold true for the long-tail of the objects present in the world. We present a self-supervised technique that directly optimizes on a sparse collection of images of a particular object/object category to obtain consistent dense correspondences across the collection. We use pairwise nearest neighbors obtained from deep features of a pre-trained vision transformer (ViT) model as noisy and sparse keypoint matches and make them dense and accurate matches by optimizing a neural network that jointly maps the image collection into a learned canonical grid. Experiments on CUB, SPair-71k and PF-Willow benchmarks demonstrate that our method can produce globally consistent and higher quality correspondences across the image collection when compared to existing self-supervised methods. Code and other material will be made available at https://kampta.github.io/asic.

A critical obstacle preventing NeRF models from being deployed broadly in the wild is their reliance on accurate camera poses. Consequently, there is growing interest in extending NeRF models to jointly optimize camera poses and scene representation, which offers an alternative to off-the-shelf SfM pipelines which have well-understood failure modes. Existing approaches for unposed NeRF operate under limiting assumptions, such as a prior pose distribution or coarse pose initialization, making them less effective in a general setting. In this work, we propose a novel approach, LU-NeRF, that jointly estimates camera poses and neural radiance fields with relaxed assumptions on pose configuration. Our approach operates in a local-to-global manner, where we first optimize over local subsets of the data, dubbed "mini-scenes." LU-NeRF estimates local pose and geometry for this challenging few-shot task. The mini-scene poses are brought into a global reference frame through a robust pose synchronization step, where a final global optimization of pose and scene can be performed. We show our LU-NeRF pipeline outperforms prior attempts at unposed NeRF without making restrictive assumptions on the pose prior. This allows us to operate in the general SE(3) pose setting, unlike the baselines. Our results also indicate our model can be complementary to feature-based SfM pipelines as it compares favorably to COLMAP on low-texture and low-resolution images.

Computer vision research has long aimed to build systems that are robust to transformations found in natural data. Traditionally, this is done using data augmentation or hard-coding invariances into the architecture. However, too much or too little invariance can hurt, and the correct amount is unknown a priori and dependent on the instance. Ideally, the appropriate invariance would be learned from data and inferred at test-time.We treat invariance as a prediction problem. Given any image, we predict a distribution over transformations can and average over them to make invariant predictions. Combined with a graphical model approach, this distribution forms a flexible, generalizable, and adaptive form of invariance. Our experiments show that it can be used to align datasets and discover prototypes, adapt to out-of-distribution poses, and generalize invariance across classes. When as data augmentation, our method shows accuracy and robustness gains on CIFAR 10, CIFAR10-LT, and TinyImageNet.

Spherical CNNs generalize CNNs to functions on the sphere, by using spherical convolutions as the main linear operation. The most accurate and efficient way to compute spherical convolutions is in the spectral domain (via the convolution theorem), which is still costlier than the usual planar convolutions. For this reason, applications of spherical CNNs have so far been limited to small problems that can be approached with low model capacity. In this work, we show how spherical CNNs can be scaled for much larger problems. To achieve this, we make critical improvements including novel variants of common model components, an implementation of core operations to exploit hardware accelerator characteristics, and application-specific input representations that exploit the properties of our model. Experiments show our larger spherical CNNs reach state-of-the-art on several targets of the QM9 molecular benchmark, which was previously dominated by equivariant graph neural networks, and achieve competitive performance on multiple weather forecasting tasks. Our code is available at https://github.com/google-research/spherical-cnn.

Recent advances in neural reconstruction enable high-quality 3D object reconstruction from casually captured image collections. Current techniques mostly analyze their progress on relatively simple image collections where SfM techniques can provide ground-truth (GT) camera poses. We note that SfM techniques tend to fail on in-the-wild image collections such as image search results with varying backgrounds and illuminations. To enable systematic research progress on 3D reconstruction from casual image captures, we propose `NAVI': a new dataset of category-agnostic image collections of objects with high-quality 3D scans along with per-image 2D-3D alignments providing near-perfect GT camera parameters. These 2D-3D alignments allow us to extract accurate derivative annotations such as dense pixel correspondences, depth and segmentation maps. We demonstrate the use of NAVI image collections on different problem settings and show that NAVI enables more thorough evaluations that were not possible with existing datasets. We believe NAVI is beneficial for systematic research progress on 3D reconstruction and correspondence estimation.

Representational learning forms the backbone of most deep learning applications, and the value of a learned representation is intimately tied to its information content regarding different factors of variation. Finding good representations depends on the nature of supervision and the learning algorithm. We propose a novel algorithm that utilizes a weak form of supervision where the data is partitioned into sets according to certain inactive (common) factors of variation which are invariant across elements of each set. Our key insight is that by seeking correspondence between elements of different sets, we learn strong representations that exclude the inactive factors of variation and isolate the active factors that vary within all sets. As a consequence of focusing on the active factors, our method can leverage a mix of setsupervised and wholly unsupervised data, which can even belong to a different domain. We tackle the challenging problem of synthetic-to-real object pose transfer, without pose annotations on anything, by isolating pose information which generalizes to the category level and across the synthetic/real domain gap. The method can also boost performance in supervised settings, by strengthening intermediate representations, as well as operate in practically attainable scenarios with set-supervised natural images, where quantity is limited and nuisance factors of variation are more plentiful. Accompanying code may be found on github.

Classical light field rendering for novel view synthesis can accurately reproduce view-dependent effects such as reflection, refraction, and translucency, but requires a dense view sampling of the scene. Methods based on geometric reconstruction need only sparse views, but cannot accurately model non-Lambertian effects. We introduce a model that combines the strengths and mitigates the limitations of these two directions. By operating on a four-dimensional representation of the light field, our model learns to represent view-dependent effects accurately. By enforcing geometric constraints during training and inference, the scene geometry is implicitly learned from a sparse set of views. Concretely, we introduce a two-stage transformer-based model that first aggregates features along epipolar lines, then aggregates features along reference views to produce the color of a target ray. Our model outperforms the state-of-the-art on multiple forward-facing and 360° datasets, with larger margins on scenes with severe view-dependent variations. Code and results can be found at light-field-neural-rendering.github. io.

We present a system for accurately predicting stable orientations for diverse rigid objects. We propose to overcome the critical issue of modelling multimodality in the space of rotations by using a conditional generative model to accurately classify contact surfaces. Our system is capable of operating from noisy and partially-observed pointcloud observations captured by real world depth cameras. Our method substantially outperforms the current state-of-the-art systems on a simulated stacking task requiring highly accurate rotations, and demonstrates strong sim2real zero-shot transfer results across a variety of unseen objects on a real world reorientation task.

Modern deep learning techniques that regress the relative camera pose between two images have difficulty dealing with challenging scenarios, such as large camera motions resulting in occlusions and significant changes in perspective that leave little overlap between images. These models continue to struggle even with the benefit of large supervised training datasets. To address the limitations of these models, we take inspiration from techniques that show regressing keypoint locations in 2D and 3D can be improved by estimating a discrete distribution over keypoint locations. Analogously, in this paper we explore improving camera pose regression by instead predicting a discrete distribution over camera poses. To realize this idea, we introduce DirectionNet, which estimates discrete distributions over the 5D relative pose space using a novel parameterization to make the estimation problem tractable. Specifically, DirectionNet factorizes relative camera pose, specified by a 3D rotation and a translation direction, into a set of 3D direction vectors. Since 3D directions can be identified with points on the sphere, DirectionNet estimates discrete distributions on the sphere as its output. We evaluate our model on challenging synthetic and real pose estimation datasets constructed from Matterport3D and InteriorNet. Promising results show a near 50% reduction in error over direct regression methods.

We introduce KeypointDeformer, a novel unsupervised method for shape control through automatically discovered 3D keypoints. We cast this as the problem of aligning a source 3D object to a target 3D object from the same object category. Our method analyzes the difference between the shapes of the two objects by comparing their latent representations. This latent representation is in the form of 3D keypoints that are learned in an unsupervised way. The difference between the 3D keypoints of the source and the target objects then informs the shape deformation algorithm that deforms the source object into the target object. The whole model is learned end-to-end and simultaneously discovers 3D keypoints while learning to use them for deforming object shapes. Our approach produces intuitive and semantically consistent control of shape deformations. Moreover, our discovered 3D keypoints are consistent across object category instances despite large shape variations. As our method is unsupervised, it can be readily deployed to new object categories without requiring annotations for 3D keypoints and deformations. Project page: http://tomasjakab.github.io/KeypointDeformer.

Recent works have shown exciting results in unsupervised image de-rendering—learning to decompose 3D shape, appearance, and lighting from single-image collections without explicit supervision. However, many of these assume simplistic material and lighting models. We propose a method, termed RADAR, that can recover environment illumination and surface materials from real single-image collections, relying neither on explicit 3D supervision, nor on multi-view or multi-light images. Specifically, we focus on rotationally symmetric artefacts that exhibit challenging surface properties including specular reflections, such as vases. We introduce a novel self-supervised albedo discriminator, which allows the model to recover plausible albedo without requiring any ground-truth during training. In conjunction with a shape reconstruction module exploiting rotational symmetry, we present an end-to-end learning framework that is able to de-render the world’s revolutionary artefacts. We conduct experiments on a real vase dataset and demonstrate compelling decomposition results, allowing for applications including free-viewpoint rendering and relighting. More results and code at: https://sorderender.github.io/.

We introduce the problem of perpetual view generation— long-range generation of novel views corresponding to an arbitrarily long camera trajectory given a single image. This is a challenging problem that goes far beyond the capabilities of current view synthesis methods, which quickly degenerate when presented with large camera motions. Methods for video generation also have limited ability to produce long sequences and are often agnostic to scene geometry. We take a hybrid approach that integrates both geometry and image synthesis in an iterative ‘render, refine and repeat’ framework, allowing for long-range generation that cover large distances after hundreds of frames. Our approach can be trained from a set of monocular video sequences. We propose a dataset of aerial footage of coastal scenes, and compare our method with recent view synthesis and conditional video generation baselines, showing that it can generate plausible scenes for much longer time horizons over large camera trajectories compared to existing methods. Project page at https://infinite-nature.github.io.

In the deep learning era, the vast majority of methods to predict pose from a single image are trained to classify or regress to a single given ground truth pose per image. Such methods have two main shortcomings, i) they cannot represent uncertainty about the predictions, and ii) they cannot handle symmetric objects, where multiple (potentially infinite) poses may be correct. Only recently these shortcomings have been addressed, but current approaches as limited in that they cannot express the full rich space of distributions on the rotation manifold. To this end, we introduce a method to estimate arbitrary, non-parametric distributions on SO(3). Our key idea is to represent the distributions implicitly, with a neural network that estimates the probability density, given the input image and a candidate pose. At inference time, grid sampling or gradient ascent can be used to find the most likely pose, but it is also possible to evaluate the density at any pose, enabling reasoning about symmetries and uncertainty. This is the most general way of representing distributions on manifolds, and to demonstrate its expressive power we introduce a new dataset containing symmetric and nearly-symmetric objects. Our method also shows advantages on the popular object pose estimation benchmarks ModelNet10-SO(3) and T-LESS. Code, data, and visualizations may be found at implicit-pdf.github.io.

Shape priors learned from data are commonly used to reconstruct 3D objects from partial or noisy data. Yet no such shape priors are available for indoor scenes, since typical 3D autoencoders cannot handle their scale, complexity, or diversity. In this paper, we introduce Local Implicit Grid Representations, a new 3D shape representation designed for scalability and generality. The motivating idea is that most 3D surfaces share geometric details at some scale - i.e., at a scale smaller than an entire object and larger than a small patch. We train an autoencoder to learn an embedding of local crops of 3D shapes at that size. Then, we use the decoder as a component in a shape optimization that solves for a set of latent codes on a regular grid of overlapping crops such that an interpolation of the decoded local shapes matches a partial or noisy observation. We demonstrate the value of this proposed approach for 3D surface reconstruction from sparse point observations, showing significantly better results than alternative approaches.

We present a benchmark for online, video-based depth estimation, a problem that is not covered by the current set of benchmarks for evaluating 3D reconstruction, which focus on offline, batch reconstruction. Online depth estimation from video captured by a moving camera is a key enabling technology for compelling applications in robotics and augmented reality. Inspired by progress in many aspects of robotics due to benchmarks and datasets, we propose a new benchmark called NBVC for evaluating methods for online depth estimation from video. Our benchmark is composed of short video sequences with corresponding high-quality ground truth depth maps, derived from the recent Tanks and Temples dataset. We are hopeful that our work will be instrumental in the development of learning-based algorithms for online depth estimation from video clips, and will also lead to improvements in conventional approaches. In addition to the benchmark, we present a superpixel-based plane sweeping stereo algorithm and use it to investigate various aspects of the problem. The paper contains our initial findings and conclusions.

Learning equivariant representations is a promising way to reduce sample and model complexity and improve the generalization performance of deep neural networks. The spherical CNNs are successful examples, producing SO(3)-equivariant representations of spherical inputs. There are two main types of spherical CNNs. The first type lifts the inputs to functions on the rotation group SO(3) and applies convolutions on the group, which are computationally expensive since SO(3) has one extra dimension. The second type applies convolutions directly on the sphere, which are limited to zonal (isotropic) filters, and thus have limited expressivity. In this paper, we present a new type of spherical CNN that allows anisotropic filters in an efficient way, without ever leaving the spherical domain. The key idea is to consider spin-weighted spherical functions, which were introduced in physics in the study of gravitational waves. These are complex-valued functions on the sphere whose phases change upon rotation. We define a convolution between spin-weighted functions and build a CNN based on it. The spin-weighted functions can also be interpreted as spherical vector fields, allowing applications to tasks where the inputs or outputs are vector fields. Experiments show that our method outperforms previous methods on tasks like classification of spherical images, classification of 3D shapes and semantic segmentation of spherical panoramas.

Symmetric orthogonalization via SVD, and closely related procedures, are well-known techniques for projecting matrices onto O(n) or SO(n). These tools have long been used for applications in computer vision, for example optimal 3D alignment problems solved by orthogonal Procrustes, rotation averaging, or Essential matrix decomposition. Despite its utility in different settings, SVD orthogonalization as a procedure for producing rotation matrices is typically overlooked in deep learning models, where the preferences tend toward classic representations like unit quaternions, Euler angles, and axis-angle, or more recently-introduced methods. Despite the importance of 3D rotations in computer vision and robotics, a single universally effective representation is still missing. Here, we explore the viability of SVD orthogonalization for 3D rotations in neural networks. We present a theoretical analysis of SVD as used for projection onto the rotation group. Our extensive quantitative analysis shows simply replacing existing representations with the SVD orthogonalization procedure obtains state of the art performance in many deep learning applications covering both supervised and unsupervised training.

Spherical convolutional networks have been introduced recently as tools to learn powerful feature representations of 3D shapes. Spherical CNNs are equivariant to 3D rotations making them ideally suited to applications where 3D data may be observed in arbitrary orientations. In this paper we learn 2D image embeddings with a similar equivariant structure: embedding the image of a 3D object should commute with rotations of the object. We introduce a cross-domain embedding from 2D images into a spherical CNN latent space. This embedding encodes images with 3D shape properties and is equivariant to 3D rotations of the observed object. The model is supervised only by target embeddings obtained from a spherical CNN pretrained for 3D shape classification. We show that learning a rich embedding for images with appropriate geometric structure is sufficient for tackling varied applications, such as relative pose estimation and novel view synthesis, without requiring additional task-specific supervision.

The availability of affordable and portable depth sensors has made scanning objects and people simpler than ever. However, dealing with occlusions and missing parts is still a significant challenge. The problem of reconstructing a (possibly non-rigidly moving) 3D object from a single or multiple partial scans has received increasing attention in recent years. In this work, we propose a novel learning-based method for the completion of partial shapes. Unlike the majority of existing approaches, our method focuses on objects that can undergo non-rigid deformations. The core of our method is a variational autoencoder with graph convolutional operations that learns a latent space for complete realistic shapes. At inference, we optimize to find the representation in this latent space that best fits the generated shape to the known partial input. The completed shape exhibits a realistic appearance on the unknown part. We show promising results towards the completion of synthetic and real scans of human body and face meshes exhibiting different styles of articulation and partiality.

For large collections of 3D models, the ability to detect and localize parts of interest is necessary to provide search and visualization enhancements beyond simple high-level categorization. While current 3D labeling approaches rely on learning from fully labeled meshes, such training data is difficult to acquire at scale. In this work we explore learning to detect object parts from sparsely labeled data, i.e. we operate under the assumption that for any object part we have only one labeled vertex rather than a full region segmentation. Similarly, we also learn to output a single representative vertex for each detected part. Such localized predictions are useful for applications where visualization is important. Our approach relies heavily on exploiting the spatial configuration of parts on a model to drive the detection. Inspired by structured multi-class object detection models for images, we develop an algorithm that combines independently trained part classifiers with a structured SVM model, and show promising results on real-world textured 3D data.

We present a novel co-segmentation method for textured 3D shapes. Our algorithm takes a collection of textured shapes belonging to the same category and sparse annotations of foreground segments, and produces a joint dense segmentation of the shapes in the collection. We model the segments by a collectively trained Gaussian mixture model. The final model segmentation is formulated as an energy minimization across all models jointly, where intra-model edges control the smoothness and separation of model segments, and inter-model edges impart global consistency. We show promising results on two large real-world datasets, and also compare with previous shape-only 3D segmentation methods using publicly available datasets.

We consider the case of ranking a very large set of labels, items, or documents, which is common to information retrieval, recommendation, and large-scale annotation tasks. We present a general approach for converting an algorithm which has linear time in the size of the set to a sublinear one via label partitioning. Our method consists of learning an input partition and a label assignment to each partition of the space such that precision at k is optimized, which is the loss function of interest in this setting. Experiments on large-scale ranking and recommendation tasks show that our method not only makes the original linear time algorithm computationally tractable, but can also improve its performance.

In this paper we address the problem of recognizing moving objects in videos by utilizing synthetic 3D models. We use only the silhouette space of the synthetic models making thus our approach independent of appearance. To deal with the decrease in discriminability in the absence of appearance, we align sequences of object masks from video frames to paths in silhouette space. We extract object silhouettes from video by an integration of feature tracking, motion grouping of tracks, and co-segmentation of successive frames. Subsequently, the object masks from the video are matched to 3D model silhouettes in a robust matching and alignment phase. The result is a matching score for every 3D model to the video, along with a pose alignment of the model to the video. Promising experimental results indicate that a purely shape-based matching scheme driven by synthetic 3D models can be successfully applied for object recognition in videos.

We propose a novel technique for the registration of 3D point clouds which makes very few assumptions: we avoid any manual rough alignment or the use of landmarks, displacement can be arbitrarily large, and the two point sets can have very little overlap. Crude alignment is achieved by estimation of the 3D-rotation from two Extended Gaussian Images even when the data sets inducing them have partial overlap. The technique is based on the correlation of the two EGIs in the Fourier domain and makes use of the spherical and rotational harmonic transforms. For pairs with low overlap which fail a critical verification step, the rotational alignment can be obtained by the alignment of constellation images generated from the EGIs. Rotationally aligned sets are matched by correlation using the Fourier transform of volumetric functions. A fine alignment is acquired in the final step by running Iterative Closest Points with just few iterations.

This paper addresses the problem of rotation estimation directly from images defined on the sphere and without correspondence. The method is particularly useful for the alignment of large rotations and has potential impact on 3D shape alignment. The foundation of the method lies in the fact that the spherical harmonic coefficients undergo a unitary mapping when the original image is rotated. The correlation between two images is a function of rotations and we show that it has an SO(3)-Fourier transform equal to the pointwise product of spherical harmonic coefficients of the original images. The resolution of the rotation space depends on the bandwidth we choose for the harmonic expansion and the rotation estimate is found through a direct search in this 3D discretized space. A refinement of the rotation estimate can be obtained from the conservation of harmonic coefficients in the rotational shift theorem. A novel decoupling of the shift theorem with respect to the Euler angles is presented and exploited in an iterative scheme to refine the initial rotation estimates. Experiments show the suitability of the method for large rotations and the dependence of the method on bandwidth and the choice of the spherical harmonic coefficients.

We present a novel approach for the estimation of 3D-motion directly from two images using the Radon transform. We assume a similarity function defined on the cross-product of two images which assigns a weight to all feature pairs. This similarity function is integrated over all feature pairs that satisfy the epipolar constraint. This integration is equivalent to filtering the similarity function with a Dirac function embedding the epipolar constraint. The result of this convolution is a function of the five unknown motion parameters with maxima at the positions of compatible rigid motions. The breakthrough is in the realization that the Radon transform is a filtering operator: If we assume that images are defined on spheres and the epipolar constraint is a group action of two rotations on two spheres, then the Radon transform is a convolution/correlation integral. We propose a new algorithm to compute this integral from the spherical harmonics of the similarity and Dirac functions. The resulting resolution in the motion space depends on the bandwidth we keep from the spherical transform. The strength of the algorithm is in avoiding a commitment to correspondences, thus being robust to erroneous feature detection, outliers, and multiple motions. The algorithm has been tested in sequences of real omnidirectional images and it outperforms correspondence-based structure from motion.

Mobile robots can be easily equipped with numerous sensors which can aid in the tasks of localization and ego-motion estimation. Two such examples are Inertial Measurement Units (IMU), which provide a gravity vector via pitch and roll angular velocities, and wide-angle or panoramic imaging devices. As the number of powerful devices on a single robot increases, an important problem arises in how to fuse the information coming from multiple sources to obtain an accurate and efficient motion estimate. The IMU provides real-time readings which can be employed in orientation estimation, while in principle an Omnidirectional camera provides enough information to estimate the full rigid motion (up to translational scale). However, in addition to being computationally overwhelming, such an estimation is traditionally based on the sensitive search for feature correspondences between image frames. In this paper we present a novel algorithm that exploits information from an IMU to reduce the five parameter motion search to a three-parameter estimation. For this task we formulate a generalized Hough transform which processes image features directly to avoid searching for correspondences. The Hough space is computed rapidly by re-treating the transform as a convolution of spherical images.

Omnidirectional images arising from 3D-motion of a camera contain persistent structures over a large variation of motions because of their large field of view. This persistence made appearance-based methods attractive for robot localization given reference views. Assuming that central omnidirectional images can be mapped to the sphere, the question is what are the underlying mappings of the sphere that can reflect a rotational camera motion. Given such a mapping, we propose a systematic way for finding invariance and the mapping parameters themselves based on the generalization of the Fourier transform. Using results from representation theory, we can generalize the Fourier transform to any homogeneous space with a transitively acting group. Such a case is the sphere with rotation as the acting group. The spherical harmonics of an image pair are related to each other through a shift theorem involving the irreducible representation of the rotation group. We show how to extract Euler angles using this theorem. We study the effect of the number of spherical harmonic coefficients as well as the effect of violation of appearance persistence in real imagery.