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Bilateral Normal Integration

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Computer Vision – ECCV 2022 (ECCV 2022)


This paper studies the discontinuity preservation problem in recovering a surface from its surface normal map. To model discontinuities, we introduce the assumption that the surface to be recovered is semi-smooth, i.e., the surface is one-sided differentiable (hence one-sided continuous) everywhere in the horizontal and vertical directions. Under the semi-smooth surface assumption, we propose a bilaterally weighted functional for discontinuity preserving normal integration. The key idea is to relatively weight the one-sided differentiability at each point’s two sides based on the definition of one-sided depth discontinuity. As a result, our method effectively preserves discontinuities and alleviates the under- or over-segmentation artifacts in the recovered surfaces compared to existing methods. Further, we unify the normal integration problem in the orthographic and perspective cases in a new way and show effective discontinuity preservation results in both cases (Source code is available at

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  1. 1.

    This requirement is stricter than jump discontinuity, which requires the one-sided limits exist but are unequal at a point’s two sides. Figure 2(b), (c), and (e) are jump discontinuity examples, but a semi-smooth surface allows only Fig. 2(b) and (c).

  2. 2.

    See experiments in the supplementary material.

  3. 3.

    The method we call robust estimator is called non-convex estimator in [25].

  4. 4.


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This work was supported by JSPS KAKENHI Grant Number JP19H01123, and National Natural Science Foundation of China under Grant No. 62136001, 61872012.

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Correspondence to Xu Cao .

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Cao, X., Santo, H., Shi, B., Okura, F., Matsushita, Y. (2022). Bilateral Normal Integration. In: Avidan, S., Brostow, G., Cissé, M., Farinella, G.M., Hassner, T. (eds) Computer Vision – ECCV 2022. ECCV 2022. Lecture Notes in Computer Science, vol 13661. Springer, Cham.

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