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Rahul Garg

Publications and source records attributed to Rahul Garg.

At least 37 records · Page 2Linked to original sources
arXiv2022

On extension of Calderón-Zygmund type singular integrals and their commutators

Motivated by the recent works [Huan Yu, Quansen Jiu, and Dongsheng Li, 2021] and [Yanping Chen and Zihua Guo, 2021], we study the following extension of Calderón-Zygmund type singular integrals $$ T_βf (x) = p.v. \int_{\mathbb{R}^n} \frac{Ω(y)}{|y|^{n-β}} f(x-y) \, dy, $$ for $0 < β< n$, and their commutators. We establish estimates of these singular integrals on Lipschitz spaces, Hardy spaces and Muckenhoupt $A_p$-weighted $L^p$-spaces. We also establish Lebesgue and Hardy space estimates of their commutators. Our estimates are uniform in small $β$, and therefore one can pass onto the limits as $β\to 0$ to deduce analogous estimates for the classical Calderón-Zygmund type singular integrals and their commutators.

math.CA↗
arXiv2022

Sparse bounds for pseudo-multipliers associated to Grushin operators, II

In this article, we establish pointwise sparse domination results for Grushin pseudo-multipliers corresponding to various symbol classes, as a continuation of our investigation initiated in [BBGG21]. As a consequence, we deduce quantitative weighted estimates for these pseudo-multipliers.

math.AP↗
arXiv2021

Defocus Map Estimation and Deblurring from a Single Dual-Pixel Image

We present a method that takes as input a single dual-pixel image, and simultaneously estimates the image's defocus map -- the amount of defocus blur at each pixel -- and recovers an all-in-focus image. Our method is inspired from recent works that leverage the dual-pixel sensors available in many consumer cameras to assist with autofocus, and use them for recovery of defocus maps or all-in-focus images. These prior works have solved the two recovery problems independently of each other, and often require large labeled datasets for supervised training. By contrast, we show that it is beneficial to treat these two closely-connected problems simultaneously. To this end, we set up an optimization problem that, by carefully modeling the optics of dual-pixel images, jointly solves both problems. We use data captured with a consumer smartphone camera to demonstrate that, after a one-time calibration step, our approach improves upon prior works for both defocus map estimation and blur removal, despite being entirely unsupervised.

cs.CV↗
arXiv2021

How to Train Neural Networks for Flare Removal

When a camera is pointed at a strong light source, the resulting photograph may contain lens flare artifacts. Flares appear in a wide variety of patterns (halos, streaks, color bleeding, haze, etc.) and this diversity in appearance makes flare removal challenging. Existing analytical solutions make strong assumptions about the artifact's geometry or brightness, and therefore only work well on a small subset of flares. Machine learning techniques have shown success in removing other types of artifacts, like reflections, but have not been widely applied to flare removal due to the lack of training data. To solve this problem, we explicitly model the optical causes of flare either empirically or using wave optics, and generate semi-synthetic pairs of flare-corrupted and clean images. This enables us to train neural networks to remove lens flare for the first time. Experiments show our data synthesis approach is critical for accurate flare removal, and that models trained with our technique generalize well to real lens flares across different scenes, lighting conditions, and cameras.

eess.IV↗
arXiv2020

Localisation of Spectral Sums corresponding to the sub-Laplacian on the Heisenberg Group

In this article we study localisation of spectral sums $\{S_R\}_{R > 0}$ associated to the sub-Laplacian $\mathcal{L}$ on the Heisenberg Group $\mathbb{H}^d$ where $S_R f := \int_0^R dE_{λ}f$, with $\mathcal{L} = \int_0^{\infty} λ\, dE_λ$ being the spectral resolution of $\mathcal{L}.$ We prove that for any compactly supported function $f \in L^2(\mathbb{H}^d)$, and for any $γ< \frac{1}{2}$, $R^γ S_R f \to 0$ as $ R \to \infty$, almost everywhere off $supp (f)$.

math.AP↗
arXiv2020

Learned Dual-View Reflection Removal

Traditional reflection removal algorithms either use a single image as input, which suffers from intrinsic ambiguities, or use multiple images from a moving camera, which is inconvenient for users. We instead propose a learning-based dereflection algorithm that uses stereo images as input. This is an effective trade-off between the two extremes: the parallax between two views provides cues to remove reflections, and two views are easy to capture due to the adoption of stereo cameras in smartphones. Our model consists of a learning-based reflection-invariant flow model for dual-view registration, and a learned synthesis model for combining aligned image pairs. Because no dataset for dual-view reflection removal exists, we render a synthetic dataset of dual-views with and without reflections for use in training. Our evaluation on an additional real-world dataset of stereo pairs shows that our algorithm outperforms existing single-image and multi-image dereflection approaches.

cs.CV↗
arXiv2020

Learning to Autofocus

Autofocus is an important task for digital cameras, yet current approaches often exhibit poor performance. We propose a learning-based approach to this problem, and provide a realistic dataset of sufficient size for effective learning. Our dataset is labeled with per-pixel depths obtained from multi-view stereo, following "Learning single camera depth estimation using dual-pixels". Using this dataset, we apply modern deep classification models and an ordinal regression loss to obtain an efficient learning-based autofocus technique. We demonstrate that our approach provides a significant improvement compared with previous learned and non-learned methods: our model reduces the mean absolute error by a factor of 3.6 over the best comparable baseline algorithm. Our dataset and code are publicly available.

cs.CV↗
arXiv2020

Du$^2$Net: Learning Depth Estimation from Dual-Cameras and Dual-Pixels

Computational stereo has reached a high level of accuracy, but degrades in the presence of occlusions, repeated textures, and correspondence errors along edges. We present a novel approach based on neural networks for depth estimation that combines stereo from dual cameras with stereo from a dual-pixel sensor, which is increasingly common on consumer cameras. Our network uses a novel architecture to fuse these two sources of information and can overcome the above-mentioned limitations of pure binocular stereo matching. Our method provides a dense depth map with sharp edges, which is crucial for computational photography applications like synthetic shallow-depth-of-field or 3D Photos. Additionally, we avoid the inherent ambiguity due to the aperture problem in stereo cameras by designing the stereo baseline to be orthogonal to the dual-pixel baseline. We present experiments and comparisons with state-of-the-art approaches to show that our method offers a substantial improvement over previous works.

cs.CV↗
arXiv2019

Learning Single Camera Depth Estimation using Dual-Pixels

Deep learning techniques have enabled rapid progress in monocular depth estimation, but their quality is limited by the ill-posed nature of the problem and the scarcity of high quality datasets. We estimate depth from a single camera by leveraging the dual-pixel auto-focus hardware that is increasingly common on modern camera sensors. Classic stereo algorithms and prior learning-based depth estimation techniques under-perform when applied on this dual-pixel data, the former due to too-strong assumptions about RGB image matching, and the latter due to not leveraging the understanding of optics of dual-pixel image formation. To allow learning based methods to work well on dual-pixel imagery, we identify an inherent ambiguity in the depth estimated from dual-pixel cues, and develop an approach to estimate depth up to this ambiguity. Using our approach, existing monocular depth estimation techniques can be effectively applied to dual-pixel data, and much smaller models can be constructed that still infer high quality depth. To demonstrate this, we capture a large dataset of in-the-wild 5-viewpoint RGB images paired with corresponding dual-pixel data, and show how view supervision with this data can be used to learn depth up to the unknown ambiguities. On our new task, our model is 30% more accurate than any prior work on learning-based monocular or stereoscopic depth estimation.

cs.CV↗
arXiv2019

Wireless Software Synchronization of Multiple Distributed Cameras

We present a method for precisely time-synchronizing the capture of image sequences from a collection of smartphone cameras connected over WiFi. Our method is entirely software-based, has only modest hardware requirements, and achieves an accuracy of less than 250 microseconds on unmodified commodity hardware. It does not use image content and synchronizes cameras prior to capture. The algorithm operates in two stages. In the first stage, we designate one device as the leader and synchronize each client device's clock to it by estimating network delay. Once clocks are synchronized, the second stage initiates continuous image streaming, estimates the relative phase of image timestamps between each client and the leader, and shifts the streams into alignment. We quantitatively validate our results on a multi-camera rig imaging a high-precision LED array and qualitatively demonstrate significant improvements to multi-view stereo depth estimation and stitching of dynamic scenes. We release as open source 'libsoftwaresync', an Android implementation of our system, to inspire new types of collective capture applications.

cs.CV↗
arXiv2018

Synthetic Depth-of-Field with a Single-Camera Mobile Phone

Shallow depth-of-field is commonly used by photographers to isolate a subject from a distracting background. However, standard cell phone cameras cannot produce such images optically, as their short focal lengths and small apertures capture nearly all-in-focus images. We present a system to computationally synthesize shallow depth-of-field images with a single mobile camera and a single button press. If the image is of a person, we use a person segmentation network to separate the person and their accessories from the background. If available, we also use dense dual-pixel auto-focus hardware, effectively a 2-sample light field with an approximately 1 millimeter baseline, to compute a dense depth map. These two signals are combined and used to render a defocused image. Our system can process a 5.4 megapixel image in 4 seconds on a mobile phone, is fully automatic, and is robust enough to be used by non-experts. The modular nature of our system allows it to degrade naturally in the absence of a dual-pixel sensor or a human subject.

cs.CV↗
arXiv2018

Aperture Supervision for Monocular Depth Estimation

We present a novel method to train machine learning algorithms to estimate scene depths from a single image, by using the information provided by a camera's aperture as supervision. Prior works use a depth sensor's outputs or images of the same scene from alternate viewpoints as supervision, while our method instead uses images from the same viewpoint taken with a varying camera aperture. To enable learning algorithms to use aperture effects as supervision, we introduce two differentiable aperture rendering functions that use the input image and predicted depths to simulate the depth-of-field effects caused by real camera apertures. We train a monocular depth estimation network end-to-end to predict the scene depths that best explain these finite aperture images as defocus-blurred renderings of the input all-in-focus image.

cs.CV↗
arXiv2014

The lattice point counting problem on the Heisenberg groups

We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{α,A}((z,t)) = \left(|z|^α+ A |t|^{α/2}\right)^{1/α}$, for $α\ge 2$ and $A>0$. This natural family includes the canonical Cygan-Korányi norm, corresponding to $α=4$. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius $R$. The exponent we establish for the error in the case $α=2$ is the best possible, in all dimensions.

math.NT↗
arXiv2014

On the role of Riesz potentials in Poisson's equation and Sobolev embeddings

In this paper, we study the mapping properties of the classical Riesz potentials acting on $L^p$-spaces. In the supercritical exponent, we obtain new "almost" Lipschitz continuity estimates for these and related potentials (including, for instance, the logarithmic potential). Applications of these continuity estimates include the deduction of new regularity estimates for distributional solutions to Poisson's equation, as well as a proof of the supercritical Sobolev embedding theorem first shown by Brezis and Wainger in 1980.

math.FA↗
arXiv2014

On regularity of solutions to Poisson's equation

In this note, we announce new regularity results for some locally integrable distributional solutions to Poisson's equation. This includes, for example, the standard solutions obtained by convolution with the fundamental solution. In particular, our results show that there is no qualitative difference in the regularity of these solutions in the plane and in higher dimensions.

math.AP↗
arXiv2012

Variations on a theorem of Beurling

We consider functions satisfying the subcritical Beurling's condition, viz., $$\int_{\R^n}\int_{\R^n} |f(x)| |\hat{f}(y)| e^{a |x \cdot y|} \, dx \, dy < \infty$$ for some $ 0 < a < 1.$ We show that such functions are entire vectors for the Schrödinger representations of the Heisenberg group. If an eigenfunction $f$ of the Fourier transform satisfies the above condition we show that the Hermite coefficients of $f$ have certain exponential decay which depends on $a$.

math.CA↗
arXiv2010

On the structure of analytic vectors for the schrodinger representation

This article deals with the structure of analytic and entire vectors for the Schrödinger representations of the Heisenberg group. Using refined versions of Hardy's theorem and their connection with Hermite expansions we obtain very precise representation theorems for analytic and entire vectors.

math.FA↗