Plane Sweep Stereo

hardPROMulti-view StereoCVCV: Depth Estimation

Implement a plane sweep stereo algorithm for multi-view stereo, a technique used in depth estimation to calculate the depth of a scene from multiple images. This task involves sweeping a series of depth planes through the scene to estimate the depth at each point.

The concept of plane sweep stereo relies on the idea of homography, which describes the transformation between two images of the same planar scene. For a given depth plane, the homography matrix $H$ can be used to warp all images to a reference view. The photo-consistency of the warped images is then computed, typically using the variance of the pixel values.

Inputs

images: a list of $N$ images of the same size; the first one is the reference view.
poses: the $N$ camera pose matrices $T_i$ ( $4 \times 4$ ); the first one belongs to the reference view.
K: the $3 \times 3$ intrinsic matrix shared by all views.
depths: the candidate plane depths $d$ to sweep.

For view $i$ , write the relative pose $T_0^{-1} T_i$ as a rotation $R$ and translation $t$ , and use the fronto-parallel plane normal $n = (0, 0, 1)^\top$ . The plane-induced homography at depth $d$ is

H_i(d) = K \left( R + \frac{t\, n^\top}{d} \right) K^{-1}

The warped view $i$ at reference pixel $(x, y)$ takes the pixel of view $i$ at $H_i(d)\,(x, y, 1)^\top$ after dividing by the third coordinate and truncating to integer pixel coordinates. Locations that fall outside the image read 0. The reference image is used as is.

Here are the general steps to follow:

Define a series of depth planes to sweep through the scene
For each depth plane, compute the homography matrix $H$ and warp all images to the reference view
Compute the photo-consistency of the warped images: the per-pixel variance across the $N$ images
For every pixel, return the depth whose variance is smallest (the first such depth on ties)

\sigma^2 = \frac{1}{N} \sum_{i=1}^{N} (I_i - \bar{I})^2

This technique is widely used in computer vision applications, such as 3D reconstruction and robotics.

Example

Input

Multiple images and poses

Output

Depth map

Reasoning

Inputs (illustrative): two $1 \times 6$ grayscale views, reference $(10, 20, 30, 40, 50, 60)$ and second view $(0, 0, 10, 20, 30, 40)$ . Focal length $f = 100$ , principal point 0, second camera $R = I$ , $t = (0.04, 0, 0)$ , depths $1, 2, 4$ .
Homography: with $R = I$ and $n = (0,0,1)$ , $H$ is a horizontal shift of $f\,t_x / d$ : $4$ , $2$ , $1$ pixels for $d = 1, 2, 4$ .
Cost at $d = 2$ : the warped view is $(10, 20, 30, 40, 0, 0)$ , since the last two samples fall outside the image. With two views $\sigma^2 = \left(\tfrac{I_0 - I_1}{2}\right)^2$ , giving $(0, 0, 0, 0, 625, 900)$ .
Other planes: $d = 1$ gives $(100, 100, 225, 400, 625, 900)$ ; $d = 4$ gives $(25, 25, 25, 25, 25, 900)$ .
Output: each pixel takes its lowest-variance depth, so the depth map is $(2, 2, 2, 2, 4, 1)$ , same height and width as the images.
Notice: only interior pixels recover the true depth 2. Pixel 6 ties at 900 everywhere, so the first depth wins; pixel 5 drifts to 4 because its match left the image.

Constraints

Input images: 3D numpy arrays (color or grayscale) with shape (height, width, channels) and pixel values in [0, 255]
Input poses: 2D numpy arrays with shape (num_views, 4, 4) representing the camera transformation matrices
Valid depth range: [0, 100] meters, with a minimum and maximum depth plane distance of 0.1 and 10 meters respectively
Output depth map: 2D numpy array with shape (height, width) and pixel values representing the estimated depth in meters, returned as float32 numpy array
Assumption: All input images are already undistorted and have the same intrinsic camera parameters

Learn this in

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Background Knowledge

The plane sweep algorithm is a popular method for multi-view stereo, which is a technique used in computer vision to estimate the depth of a scene from multiple images taken from different viewpoints. The key concept behind this algorithm is to sweep a series of depth planes through the scene and compute the photo-consistency of each plane. Photo-consistency measures how well the images match each other when warped to a specific depth plane. The plane with the best consistency is chosen as the estimated depth.

The homography between two images is a fundamental concept in this algorithm. Homography is a transformation that maps one image to another, and it can be used to warp an image to a different viewpoint. In the context of plane sweep stereo, the homography is used to warp all images to the reference view using a specific depth plane. The variance of warped pixels is then computed to measure the photo-consistency of each plane. This process is repeated for multiple depth planes, and the plane with the lowest variance is chosen as the estimated depth.

The plane sweep algorithm has several advantages, including its ability to handle complex scenes and its robustness to noise and outliers. However, it can be computationally expensive and requires careful tuning of parameters. Understanding the underlying concepts of homography, photo-consistency, and depth estimation is crucial to implementing this algorithm effectively.

Algorithm/Approach

The general approach to solving this problem involves the following steps:

Initialize a range of depth planes to sweep through the scene
For each depth plane, compute the homography between the reference view and each of the other views
Warp each image to the reference view using the computed homography
Compute the photo-consistency of each plane by calculating the variance of warped pixels
Choose the plane with the best consistency as the estimated depth

This approach can be implemented using a variety of techniques, including linear algebra for computing homographies and image processing for warping and computing variance.

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