Specular optimization

We aim at recovering specular albedo and shininess in a photometric stereo procedure.

The adopted camera model is pinhole.

The image model for $I$ is $I = I_{\text{diffuse}} + I_{\text{specular}}$, where we have $I_{\text{specular}}=\rho_s(c+2)*\max(0, h \cdot n)^c \max(0, s\cdot n)\phi$, where $\rho_s$ is the specular albedo, $c$ the shininess, $h$ the half-way vector at the surface point, $n$ the normal vector, $s$ is the source (normalized) direction, $\phi$ its intensity.

We run an alternating optimization procedure, aiming at solving:

$$\text{given } I,I_{\text{diffuse}}, h, n, s, \phi, \quad \text{find } \rho_s, c \text{ with}\quad I - I_{\text{diffuse}}=\rho_s(c+2)*\max(0, h \cdot n)^c \max(0, s\cdot n)\phi $$

Applying the logarithm on both sides, this is equivalent to solving:

$$\delta -(\eta + c \alpha) = 0 $$

where $\delta = \log( I - I_{\text{diffuse}})-\beta$, $\beta = \log(\max(0, s\cdot n)\phi)$, $\alpha = \log(\max(0, h \cdot n))$, $\eta= \log(c+2)\rho_s$.

To the $\max$ terms, a spherical gaussian approximation can be applied, yielding a modification of this standard Phong BRDF.

Moreover, calling the residual $r:=\delta -(\eta + c \alpha)$, we can use the theory of robust estimators to develop an alternatively reweighted least squares algorithm. In fact our model, can be cast in a regression-like form:

$$s=h(b)+\epsilon$$

where abstractly, $s$ is data, $b$ are the parameters to be estimated from the knowledge of $s$, $\epsilon$ is noise. For $\Phi$ a robust estimator, and $r(b)=\log(s)-\log(h(b))$, the logarthmic optimization reads:

$$\min_b \Phi(s(1-e^{-r(b)}))$$

The corresponding weight for reweighted optimization is:

$$w(r) = \frac{\Phi'(s(1-e^{-r(b)}))se^{-r}}{r}$$

so that for a fixed $w_i=w(r_i^-)$, we aim at solving:

$$\min_b\sum_i w(r_i^-)r_i(b)^2$$

The best performing robust estimator is the Cauchy one.

Limitations

• wherever a part of the image is free from observations of specularities, there the optimization is ill posed, so that the specular albedo explodes

Improvements

• connect this purely specular optimization in an alternating fashion to a full photometric stereo pipeline

Notes

• the algorithm performance can vary substantially according to the choice of the initial specular albedo, and the variance parameter in the Cauchy estimator

Code

It is in Matlab, and mainly contained in launch.m, which the user can run (just by pressing the "Run" button in Matlab) to perform the optimization. The various options for the optimization are described at the beginning of the script.

Two ready datasets are provided in the folder Datasets. Best parameters configurations for both of them are also indicated in launch.m.

These are generated using OpenGL and the script Datasets\opengl2matlab.m. To generate additional data, see the next sections.

Data

The user should generate his/her data with the script opengl2matlab.m, that can be just run using the "Run" button of Matlab.

In turn, this file also needs some input before being run: we now explain how to generate such input.

Beforehands, we explain how to setup the OpenGL data generation, with Microsoft Visual Studio 2022. The instructions work on Windows 10. We are all the time working in the folder OpenGL.

Configuring the Visual Studio Project

• set up glad, khr, glfw, assimp, glm as in https://learnopengl.com/Getting-started/Creating-a-window, https://learnopengl.com/Model-Loading/Assimp, https://learnopengl.com/Getting-started/Transformations. Populate lib with the files indicated in lib\list of libs that should be here.txt, and include with the necessary header files.
• create a new project in the folder OpenGL, call it e.g. data_generation
• add src\glad.c, src\rgbd_generator.cpp as source files for the data_generation solution
• add the lib and include directories in the solution properties > configuration properties > VC++ directories > general > library directory and inclusion directories. The two folders must be in front of the other
• prepend assimp-vc143-mtd.lib;opengl32.lib;glfw3.lib; to solution properties > configuration properties > linker > additional dependencies
• add the lib folder to solution properties > configuration properties > linker > general > additional library directories
• add the bin path to solution properties > configuration properties > debug > environment: use the syntax PATH=C:\path\to\bin;%PATH%
• add mesh.h, conf.h, model.h in the header files of the solution

Running the C++ code

• the user has to provide a textured geometric model. For the two aforementioned examples, we provide all the needed files. For other models apart from the provided ones, it is important to set up the .mtl file correctly, a guide on how to do this is avaiable here: https://www.youtube.com/watch?v=4DQquG_o-Ac&ab_channel=Code%2CTech%2CandTutorials. The model folder has to be specified in OpenGL\conf.h through the string model
• with reference to OpenGL\conf.h, manually the user has specify an out_folder, which must exist in the system. Everything will be saved there
• then, run the main file OpenGL\depth_map.cpp. A window will pop up. The navigation is locked, press u or l to unlock or lock the navigation. With the keys a,w,s,d the user can translate the camera, with the mouse position the orientation can be changed, a zoom effect is available with the mouse wheel. With the keys, the light can be modified, just for visualization purposes. Once enter is pressed, a series of pictures of the object are rendered, see OpenGL\conf.h for more details. Press esc to exit
• the shaders can be changed in OpenGL\conf.h. There the user can implement any BRDF function. Phong is the default one

What is happening down the line: the object is illuminated at different angles with directional lighting, and HDR RGBD images (with normals etc) are saved. HDR is crucial in specular optimization, because specular observation sometimes ecceed the usual range 0-255.