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Depth Image Inpainting: Improving Low Rank Matrix Completion with Low Gradient Regularization
Hongyang Xue, Shengming Zhang, Deng Cai
TL;DR
Single-depth-image inpainting is challenging, and low-rank completion does not fully use depth-image properties. The paper combines a low-gradient regularizer with a low-rank prior; its experiments report better results than low rank alone and sparse-gradient comparison approaches.
Problem
Single-depth-image inpainting is challenging, and low-rank completion does not fully use depth-image properties; sparse-gradient regularization also misses the prevalence of small nonzero depth gradients.
Source: §I · p. 1Method
The paper reduces the penalty for gradient 1 in a low-gradient regularizer and combines it with a low-rank prior in LRL0ψ.
Source: §I · p. 1Results
The proposed LRL0ψ approach outperforms low-rank, LRTV, and LRL0 approaches in the reported experiments.
Source: §VI · pp. 8–9Takeaways & Limitations
The reported results support using low-gradient regularization to better utilize depth-image gradient properties in single-image inpainting.
Source: D. Results and Analysis · p. 8, §VI · p. 9
Abstract
from arXiv · showhide
We consider the case of inpainting single depth images. Without corresponding color images, previous or next frames, depth image inpainting is quite challenging. One natural solution is to regard the image as a matrix and adopt the low rank regularization just as inpainting color images. However, the low rank assumption does not make full use of the properties of depth images. A shallow observation may inspire us to penalize the non-zero gradients by sparse gradient regularization. However, statistics show that though most pixels have zero gradients, there is still a non-ignorable part of pixels whose gradients are equal to 1. Based on this specific property of depth images , we propose a low gradient regularization method in which we reduce the penalty for gradient 1 while penalizing the non-zero gradients to allow for gradual depth changes. The proposed low gradient regularization is integrated with the low rank regularization into the low rank low gradient approach for depth image inpainting. We compare our proposed low gradient regularization with sparse gradient regularization. The experimental results show the effectiveness of our proposed approach.
I. INTRODUCTION
Single-depth-image inpainting is less studied than inpainting natural or medical images, and low-rank completion can produce noisy results on textureless depth images. The paper proposes a low-gradient regularizer, combined with a low-rank prior, to account for depth-gradient statistics and allow gradual changes.
I. INTRODUCTION
- Low-rank completion can leave noisy inpainted depth maps, while textureless depth images make that approach difficult.The paper illustrates the noise in a low-rank completion result and motivates regularizing gradients alongside the low-rank prior.Source: §I · p. 1, Figure 1 · p. 2
- Depth-image gradients are often zero but can also be small and nonzero, making a sparse-gradient assumption insufficiently accurate.The paper describes this low-gradient property as a statistically common feature of depth images.Source: §I · p. 1
- The proposed low-gradient regularizer reduces the penalty for gradient 1, allowing gradual depth changes while penalizing nonzero gradients.The paper integrates this regularizer with the low-rank assumption in its LRL0ψ approach.Source: §I · p. 1
- The paper compares its approach with low-rank total variation and low-rank L0 gradient methods that enforce sparse-gradient constraints.The introduction identifies LRTV and LRL0 as comparison approaches.Source: §I · pp. 1–2
II. BACKGROUND
This section reviews low-rank completion, depth-inpainting settings, and gradient regularization, then introduces a low-rank approach with sparse-gradient regularization. It motivates replacing total variation with direct L0-gradient minimization before presenting the paper’s low-gradient method in the next section.
B. Depth Inpainting
- Most prior depth-inpainting approaches use color guidance or address specific missing-region settings, whereas this paper considers a single depth image.The cited examples include RGB-D, stereoscopic, sensor-defect, and object-removal settings.Source: B. Depth Inpainting · p. 2
C. TV and L0 Gradient Regularization
- Total variation can smooth real depth edges, so the paper reviews direct L0-gradient minimization as an alternative sparse-gradient regularizer.The related-work discussion also notes that TV penalizes large gradient magnitudes; L0 minimization is non-convex, while relaxing it to TV makes the problem convex.Source: L0 gradient · p. 2, §III · p. 3
III. LOW RANK SPARSE GRADIENT APPROACH
- The conventional low-rank inpainting formulation uses the nuclear norm as a tractable low-rank prior and seeks a reconstruction matching observed pixels.The problem is framed as recovering U from corrupted image D and mask Ω, with U|Ω = D|Ω.Source: §III · p. 3
- The low-rank sparse-gradient approach combines a nuclear-norm low-rank prior with a gradient regularizer for depth recovery.The section reviews LRTV and then replaces TV regularization with L0-gradient regularization.Source: §III · p. 3
A. Total Variation
The section describes solving the TV-regularized low-rank inpainting problem with ADMM and iterative subproblems. Although TV improves results over low rank alone, the authors report that it smooths real depth edges and does not adequately optimize the L0 gradient.
A. Total Variation
- The TV-regularized problem is solved with ADMM by iteratively updating three subproblems.The U update uses Bregman iteration, followed by updates to M and Y.Source: A. Total Variation · p. 3
- The experiments initialize U with low-rank completion results and set M and Y to zero.Source: A. Total Variation · p. 3
- TV regularization improves inpainting results compared with using only low rank.The authors also note that TV can smooth real depth edges and that a lower-than-groundtruth TV norm can still leave visually noisy results.Source: A. Total Variation · p. 3
B. L0 Gradient
The section motivates replacing total variation with direct L0-gradient regularization and adapts region fusion to optimize it. Its example shows that lower TV need not mean gradients closer to ground truth.
B. L0 Gradient
- The method replaces the TV term with an L0-gradient penalty over four-connected neighboring pixels.The optimization uses a pairwise region cost and a fusion criterion to decide whether neighboring regions should merge.Source: B. L0 Gradient · pp. 3–4
- LRTV has PSNR 28.1319 versus 27.7596 for low-rank inpainting, but its L0 gradient norm remains above ground truth despite lower TV.The TV norms are 622276 for LRTV and 642186 for ground truth; their L0 norms are 81644 and 63765, respectively.Source: Figure 2 · p. 4
- The region-fusion procedure modifies Nguyen et al.’s method by updating regions to their optimal values when the fusion criterion rejects a merge.The approach is presented as an efficient method for L0-gradient minimization, while its solution remains an approximation.Source: B. L0 Gradient · pp. 3–4
- L0-gradient regularization often improves depth inpainting, but it does not consistently outperform LRTV and treats every nonzero gradient equally.The authors identify a non-ignorable share of unit gradients as motivation for reducing their penalty.Source: B. L0 Gradient · p. 4
IV. LOW RANK LOW GRADIENT APPROACH
This section presents the LRL0ψ algorithm as the paper’s main contribution, comprising a low-gradient definition and a low-rank low-gradient regularization algorithm.
IV. LOW RANK LOW GRADIENT APPROACH
- The section introduces LRL0ψ as the paper’s main contribution.It first defines low gradients and then describes the low-rank low-gradient regularization algorithm.Source: §IV · p. 4
A. Integral Gradient
The method defines low gradients on integral-valued gradients and reduces the penalty for unit gradients. It incorporates that measure into low-rank inpainting and uses a modified region-fusion procedure.
A. Integral Gradient
- The integral-gradient definition treats gradients as integer-valued; truncated magnitude 1 corresponds to eight directional gradient patterns.These include (±1, ±1), (0, ±1), and (±1, 0).Source: A. Integral Gradient · pp. 4–5
- LRL0 performs better than LRTV on Adirondack (PSNR 28.6198 versus 28.1319) but not on Jadeplant (22.7730 versus 22.8231).Figure 3 presents original, corrupted, LRTV, and LRL0 images in columns.Source: Figure 3 · p. 5
- The Adirondack disparity-map histogram has mostly zero gradients and a non-ignorable share of unit gradients, motivating low-gradient regularization.The method defines its low gradient on integral gradients.Source: Figure 4 · p. 5, A. Integral Gradient · p. 5
B. Low Gradient
- The Lψ0 measure reduces the penalty for small elements, particularly gradient 1, allowing gradual depth changes.The measure uses 0 < α < 1, with α = 0.75 in the reported experiments.Source: B. Low Gradient · p. 5
- The authors extend region-fusion minimization to the Lψ0 gradient objective using a pairwise cost and a three-condition fusion criterion.The conditions compare solutions for equal values, values differing by one, and values differing by more than one.Source: B. Low Gradient · pp. 5–6
- When the fusion criterion rejects a merge, the algorithm updates regions to the optimal {B, B±1} or {Ci, Cj} values.This is a modification of the region-fusion iterations for Lψ0 minimization.Source: B. Low Gradient · p. 6
A. Dataset
The authors create an evaluation dataset from Middlebury disparity maps, with random and textual missing masks, and provide data and code for generating masks and inpainting results.
A. Dataset
- The dataset converts 14 Middlebury stereo disparity maps into grayscale depth images, with unknown ground-truth values represented as zeros.It includes images such as Adirondack, Jadeplant, Motorcycle, and Teddy.Source: A. Dataset · p. 6
- The damaged images use random missing masks and textual masks, both illustrated in Figure 5.The masks are provided with the dataset.Source: A. Dataset · p. 6, Figure 5 · p. 7
- The authors provide results and code that can generate new random masks and process new depth images.The dataset and code are made available through the cited project resources.Source: A. Dataset · p. 6
- For Adirondack, the reported LRL0ψ result is best among the compared approaches when its regularization parameter is sufficiently large.Figure 7 selects the LRL0ψ result with the highest PSNR across parameter values and reports the best results of other methods.Source: Figure 7 · p. 7
B. Experiments
The experiments compare low-rank completion, LRTV, LRL0, and LRL0ψ on depth-image inpainting, evaluating PSNR only in missing areas.
B. Experiments
- The study evaluates inpainted depth images using PSNR computed only over missing areas.The compared methods are low-rank completion, LRTV, LRL0, and LRL0ψ.Source: B. Experiments · p. 7
- LRL0 generally outperforms LRTV but fails to do so on some depth images, motivating evaluation of LRL0ψ.The authors report that LRL0ψ is based on depth-gradient statistics.Source: B. Experiments · p. 7
- The experimental sequence applies low-rank completion, LRTV, LRL0, and then LRL0ψ to the inpainting results.LRTV reduces noise from low-rank completion, while LRL0 improves on LRTV in most cases.Source: B. Experiments · p. 7
C. Parameters and Setup
The setup uses fixed regularization weights across the algorithms, and the reported dataset results are PSNR scores for random and textual masks.
C. Parameters and Setup
- All algorithms use λ_r = 10 for the low-rank term.The iteration limit is 30, or stopping occurs when relative solution error falls below 1 × 10^-3.Source: C. Parameters and Setup · p. 7
- Table I reports PSNR for different inpainting methods on the authors’ dataset under random and textual masks.The caption states that LRL0ψ achieves the best PSNR results.Source: Table I · p. 7
- For the reported experiments, LRTV uses a total-variation weight λtv = 40, while LRL0ψ uses weight λl0ψ = 100; these are the stated settings for the two algorithms.Source: C. Parameters and Setup · p. 7
D. Results and Analysis
LRL0ψ achieves the best PSNR results in the reported comparisons, and the parameter analysis reports improvement over low rank as its regularization weight increases. The conclusion attributes its advantage over sparse-gradient approaches to better use of depth-image gradient properties.
D. Results and Analysis
- LRL0ψ always achieves the best PSNR on the dataset, while LRL0 is generally better than LRTV but worse on MotorE and Jadeplant.The authors report close results between LRL0 and LRL0ψ.Source: D. Results and Analysis · pp. 7–8
- For textual masks, LRL0ψ has PSNR 29.2523 on Adirondack and 17.1101 on Teddy, the highest values among the four shown methods.The comparison includes low rank, LRTV, and LRL0.Source: Figure 6 · p. 8
E. Parameter Analysis
- As λ_l0ψ increases, results improve over LR; PSNR stabilizes once the weight exceeds 80, and LRL0ψ outperforms the other approaches when the weight is large enough.At λ_l0ψ = 0, LRL0ψ degenerates to LR.Source: E. Parameter Analysis · p. 8
VI. CONCLUSIONS
- The conclusion states that LRL0ψ combines low-rank and low-gradient regularization and outperforms low rank, LRTV, and LRL0 in experiments.The authors attribute the comparison with LRTV and LRL0 to better use of the gradient property.Source: §VI · pp. 8–9