Decoupling Noises and Features via Weighted l1-analysis Compressed Sensing

Ruimin Wang     Zhouwang Yang     Ligang Liu     Jiansong Deng     Falai Chen 
  University of Science and Technology of China  

ACM Transactions on Graphics, 33(2), Article 18: 1-12, 2014.

Teaser: Our approach is able to faithfully recover sharp features corrupted by noise. (a) The octa-flower model as a ground truth; (b) the model artificially corrupted by independent and identically distributed (i.i.d.) noise (with zero mean and standard deviation 1% of the diagonal of the bounding box of the model); (c) the error map of (b); (d) the denoising result by our approach; (e) the error map of (d). The colored models (c,e) visualize the errors between the processed models and the ground truth model (a). Note that sharp features such as creases and corners are well retained in our result (d).



Many geometry processing applications are sensitive to noises and sharp features. Although there are a number of works on detecting noises and sharp features in the literature, they are heuristic. On one hand, traditional denoising methods use filtering operators to remove noises, however, they may blur sharp features and shrink the object. On the other hand, noises make detection of features, which relies on computation of differential properties, unreliable and unstable. Therefore, detecting noises and features on discrete surfaces still remains challenging.

In this paper, we present an approach for simultaneously decoupling noises and features on 3D shapes. Our approach consists of two phases. In the first phase, an estimated base mesh is generated to approximate the true underlying surface of the input noisy mesh by a global Laplacian regularization denoising scheme. The base mesh is guaranteed to asymptotically converge to the underlying surface with probability one as the sample size goes to infinity. In the second phase, an
l1-analysis compressed sensing optimization is proposed to recover sharp features from the residual between the base mesh and the input mesh. This is based on our discovery that sharp features can be sparsely represented in some coherent dictionary which is constructed by the pseudo-inverse matrix of the Laplacian of the shape. The features are recovered from the residual in a progressive way. Theoretical analysis and experimental results show that our approach can reliably and robustly remove noises and extract sharp features on 3D shapes.

Keywords Denoising, asymptotic optimality, sharp feature, l1-analysis compressed sensing

Decoupling feature and noise is a chicken-and-egg problem:

  • Feature detection is unreliable in the presence of noise

  • Denoising (filtering) operations would blur sharp features

Observation and basic idea:

  • Sharp features are sparse in geometry

  • The tool of compressed sensing enables extract sparse signals


Phase I: Decouple the Base Smooth Surface and the Residual (denoising the mesh)

Figure 2. Our approach computes the optimal smoothness parameter to denoise the mesh. (a) the ground truth 8-like model which is a C2 smooth surface; (b) the model artificially corrupted by severe synthetic noise; (c)-(g) denoised results by the global smoothing approach with various parameters \lambda in the Laplacian smoothing model. Our approach obtains the best smoothing result (e) via the automatically chosen optimal parameter \lambda=2.02.

Figure 3. Our approach is robust to different levels of noise. For the cube model with sharp features, the first row shows the ground truth model followed by the noisy models with different levels of noise (the standard deviation is shown below each model), the second row shows the denoised models, and the third row visualizes the differences between the denoised models and the ground truth model.


Phase II: Decouple Feature and Noise from Residual

Figure 4. (a) The input noisy mesh model; (b) phase I: denoising the model; (c) phase II: recovering sharp features (shown in red dots) progressively based on L1-analysis compressed sensing; (d) the final denoised result with recovered features.

Figure 5. (a) the ground truth model; (b) the noisy model; (c) phase I: denoising the model; (d)-(e) phase II: recovering sharp features (shown in red dots) progressively based on L1-analysis compressed sensing; (f) the final denoised result with recovered features.


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(at SIGGRAPH 2014)

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We would like to thank the anonymous reviewers for their constructive comments. The work is supported by the 973 Program 2011CB302400, the NSF of China (nos. 11031007, 11171322, 61222206), One Hundred Talent Project of the Chinese Academy of Sciences, the 111 Project (no. b07033), and the Program for New Century Excellent Talents in University (no. NCET-11-0881).

BibTex @article {Wang:TOG2014,
    title = {Decoupling Noises and Features via Weighted L1-analysis Compressed Sensing},
    author = {Ruimin Wang and Zhouwang Yang
and Ligang Liu and Jiansong Deng and Falai Chen}
    journal = {ACM Transactions on Graphics},
    pages={Article 18: 1-12},   
    year = {2014}

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