📘

Surface Integration from Normals

MediumCV

Implement a method to integrate surface normals and recover a depth map, a crucial step in 3D Reconstruction using Photometric Stereo. This process involves calculating the surface gradient from given normals.

The surface gradient is a fundamental concept in Computer Vision, representing the rate of change of the surface depth in the x and y directions. Given the surface normals $n_x$ , $n_y$ , $n_z$ , the surface gradient can be calculated as $p = -n_x/n_z$ and $q = -n_y/n_z$ , which are essential for integrating the surface.

To integrate the surface, follow these steps:

Calculate the surface gradient $p$ and $q$ from the given normals.
Use the calculated $p$ and $q$ in a Poisson equation solver to obtain the depth $Z$ .

p = -n_x/n_z, \quad q = -n_y/n_z

This technique is widely used in 3D scanning and object recognition applications.

Example:

Input:

Normal map

Output:

Depth map

Reasoning:

Compute gradients from normals, integrate using Poisson

Constraints:

Input normals: 2D numpy array of shape (height, width, 3) with dtype float32, representing the surface normals $n_x$ , $n_y$ , $n_z$
Valid ranges: $n_z$ values are non-zero, and all normal values are normalized to have a length of 1
Output depth map: 2D numpy array of shape (height, width) with dtype float32, representing the recovered depth $Z$ with precision up to 6 decimal places
Special conditions: The input normals are assumed to be noise-free and the surface is assumed to be Lambertian and convex
Boundary conditions: The depth values at the boundary of the image are assumed to be zero, and the Poisson equation solver should handle the boundary conditions accordingly

Editor

Python 3.13.1

Test Results

0/0

Run code to see test results.

📘