Matlab codes for shape-from-shading. Several variants are implemented:

• ADMM-based variational shape from shading with general camera (orthographic or perspective) and lighting (spherical harmonics), see [1]
• Lax-Friedriechs solving of the eikonal case (orthographic camera, frontal directional lighting), cf. Equation (8) in [2]
• Semi-Lagrangian solver for the eikonal case (orthographic camera, frontal directional lighting), see [3]
• Semi-Lagrangian solver for the perspective eikonal case (perspective camera, frontal directional lighting), see [4]

These codes can be used to solve the shape-from-shading (SfS) problem (estimate shape, given a single image). Main features:

• possibility to add a shape prior in order to guide the solution (useful for instance in RGB-D sensing)
• minimal surface regularization to smooth out the residual noise
• handles second-order spherical harmonics lighting
• handles orthographic or perspective camera
• handles grey or RGB images

Note: the classic eikonal SfS can also be achieved as a special case.

The following two demo files accompanying [1] are provided:

• demo_1_lena_eikonal.m : classic SfS (greylevel image, orthographic camera, frontal lighting) applied to the standard Lena image

• demo_2_vase_SH2.m : refinement of the depth map obtained with a RGB-D sensor. Source of the dataset: https://github.com/pengsongyou/SRmeetsPS

The three other demo files illustrate alternative PDE-based methods based semi-Lagrangian schemes, when lighting is frontal and directional, see [2,3,4] for details.

The main fuctions for ref [1] are in the Toolbox/ folder:

• generic_sfs.m: main SfS code
• theta_fun.m: cost function with respect to the surface gradient values
• estimate_lighting.m: can be used to estimate spherical harmonics lighting, given an image and a shape estimate
• make_gradient.m: finite differences stencils on a non-rectangular grid
• export_obj2.m: to produce a .obj file readable with meshlab

• minFunc: need first be compiled: go to Toolbox/minFunc and run mexAll.m script (source: https://www.cs.ubc.ca/~schmidtm/Software/minFunc.html)

• CMG (recommended for faster results): http://www.cs.cmu.edu/~jkoutis/cmg.html

[1] "A Variational Approach to Shape-from-shading Under Natural Illumination", Y. Quéau et al., EMMCVPR 2017

[2] "A comprehensive introduction to photometric 3D-reconstruction", J.-D. Durou et al., 2020

[3] "An algorithm for the global solution of the Shape-fromShading model", M. Falcone and M. Sagona, ICIAP 1997

[4] "Some remarks on perspective shape-from-shading models", E. Cristiani et al., SSVM 2007