Second-Order Algebraic Surfaces and Two Image Photometric Stereo

Kozera, Ryszard; Prokopenya, Alexander N.

doi:10.1007/978-3-030-00692-1_21

Cited by 3 publications

(5 citation statements)

References 21 publications

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“…Then, the same arguments as those used around Eq. ( 41) yield A 12 = A 13 = A 14 = 0, and the form (16) of Am implies A 11 = α, which concludes the proof.…”

Section: Perspective Casesupporting

confidence: 54%

“…Using the expression of Theorem 2 to modify the genuine m-field, and decomposing the latter field into albedo and normals according to (16), we observe that the transformation has the effect depicted in (27). That is to say, albedo values are globally scaled, while all concavities and convexities are simultaneously inverted.…”

Section: Orthographic Casementioning

confidence: 94%

“…Such a simple least-squares approach may be replaced by robust variational or learning-based strategies to ensure robustness [13,17,33,34]. There also exist numerical solutions for handling non-Lambertian reflectance models [7,15,21,40], non-distant light sources [18,22,29,32], or the ill-posed cases where m = 2 [16,20,24,31] or m = 1 [5,8,39,45]. Such issues are not adressed in the present paper which rather focuses on the theoretical foundations of uncalibrated photometric stereo.…”

Section: Calibrated Photometric Stereo Under Directional Lightingmentioning

confidence: 99%

“…field with the form of Equation (16). The underlying normal field is integrable iff the following relationship holds over Ω:…”

Section: Appendix 1: Proof Of Theoremmentioning

confidence: 99%

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On the well-posedness of uncalibrated photometric stereo under general lighting

Brahimi¹,

Quéau²,

Haefner³

et al. 2019

Preprint

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Uncalibrated photometric stereo aims at estimating the 3D-shape of a surface, given a set of images captured from the same viewing angle, but under unknown, varying illumination. While the theoretical foundations of this inverse problem under directional lighting are well-established, there is a lack of mathematical evidence for the uniqueness of a solution under general lighting. On the other hand, stable and accurate heuristical solutions of uncalibrated photometric stereo under such general lighting have recently been proposed. The quality of the results demonstrated therein tends to indicate that the problem may actually be well-posed, but this still has to be established. The present paper addresses this theoretical issue, considering first-order spherical harmonics approximation of general lighting. Two important theoretical results are established. First, the orthographic integrability constraint ensures uniqueness of a solution up to a global concave-convex ambiguity, which had already been conjectured, yet not proven. Second, the perspective integrability constraint makes the problem well-posed, which generalizes a previous result limited to directional lighting. Eventually, a closed-form expression for the unique least-squares solution of the problem under perspective projection is provided, allowing numerical simulations on synthetic data to empirically validate our findings.

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“…Then, the same arguments as those used around Eq. ( 41) yield A 12 = A 13 = A 14 = 0, and the form (16) of Am implies A 11 = α, which concludes the proof.…”

Section: Perspective Casesupporting

confidence: 54%

Section: Orthographic Casementioning

confidence: 94%

Section: Calibrated Photometric Stereo Under Directional Lightingmentioning

confidence: 99%

“…field with the form of Equation (16). The underlying normal field is integrable iff the following relationship holds over Ω:…”

Section: Appendix 1: Proof Of Theoremmentioning

confidence: 99%

See 2 more Smart Citations

On the well-posedness of uncalibrated photometric stereo under general lighting

Brahimi¹,

Quéau²,

Haefner³

et al. 2019

Preprint

View full text Add to dashboard Cite

show abstract

“…However, this apparent motion is close to being planar, and thus the rank of the set of illumination vectors is equal or close to 2 [45] (see also [23] for additional discussion on the stability of single-day photometric stereo). This situation is thus similar to the two-image case, which is known to be ill-posed since the early 90s [28,37,51], although it is still an active research area [29]. In order to limit the instabilities due to this issue, one possibility is to consider images acquired over many seasons as in [2,3], or to resort to deep neural networks [22].…”

Section: Outputmentioning

confidence: 99%

Variational Uncalibrated Photometric Stereo Under General Lighting

Haefner

Gao³

et al. 2019

2019 IEEE/CVF International Conference on Computer Vision (ICCV)

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Photometric stereo (PS) techniques nowadays remain constrained to an ideal laboratory setup where modeling and calibration of lighting is amenable. To eliminate such restrictions, we propose an efficient principled variational approach to uncalibrated PS under general illumination. To this end, the Lambertian reflectance model is approximated through a spherical harmonic expansion, which preserves the spatial invariance of the lighting. The joint recovery of shape, reflectance and illumination is then formulated as a single variational problem. There the shape estimation is carried out directly in terms of the underlying perspective depth map, thus implicitly ensuring integrability and bypassing the need for a subsequent normal integration. To tackle the resulting nonconvex problem numerically, we undertake a two-phase procedure to initialize a balloon-like perspective depth map, followed by a "lagged" block coordinate descent scheme. The experiments validate efficiency and robustness of this approach. Across a variety of evaluations, we are able to reduce the mean angular error consistently by a factor of 2-3 compared to the state-of-the-art. * Authors contributed equally. 1 https://github.com/zhenzhangye/general_ups

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