# Dense stereo processing

## Table of Contents

## Overview

The tour of mrcal shows an example of dense stereo processing done with mrcal; details about that computation appear here.

Given a pair of calibrated (both intrinsics and extrinsics) cameras, mrcal can
perform stereo processing to produce a dense stereo map. This is relatively
slow, and is often overkill for what is actually needed. But sometimes it is
useful, and the resulting depth images look *really* nice.

On a high level, mrcal stereo processing is the usual epipolar geometry technique:

- Ingest
- Two camera models, each containing the intrinsics
*and*extrinsics (the relative pose between the two cameras) - A pair of images captured by these two cameras

- Two camera models, each containing the intrinsics
- Compute a "rectified" system: a pair of models where each corresponding row
of pixels in the two cameras all represent observation rays that lie in the
same
*epipolar*plane - Reproject the images to these rectified models to produce
*rectified*images - Perform "stereo matching". For each pixel in the left rectified image we try
to find the corresponding pixel in the same row of the right rectified image.
The difference in columns is written to a
*disparity*image. This is the most computationally-intensive part of the process. - Convert the
*disparity*image to a*range*image using the geometry defined by the rectified system

The epipolar constraint (all pixels in the same row in both rectified images represent the same plane in space) allows for one-dimensional stereo matching, which is a massive computational win over the two-dimensional matching that would be required with another formulation.

The rectified coordinate system looks like this:

The code and documentatio refers to two angles:

- \(\theta\): the "azimuth"; the lateral angle inside the epipolar plane. Related directly to the \(x\) pixel coordinate in the rectified images
- \(\phi\): the "elevation"; the tilt of the epipolar plane. Related directly to the \(y\) pixel coordinate in the rectified images

## Rectification models

A rectified system satisfies the epipolar constraint (see above). mrcal supports
two models that can have this property, selected with the `rectification_model`

argument to `mrcal.rectified_system()`

or with the `--rectification`

commandline
argument to `mrcal-stereo`

.

`LENSMODEL_PINHOLE`

: this is the traditional rectification model, used in most existing tools. It works decently well for small fields of view (as with a long lens), but fails with large fields of view (as with a wide lens). The issues stem from the uneven angular resolution across the image, which shoots out to \(\infty \frac{\mathrm{pixels}}{\mathrm{deg}}\) as \(\theta \rightarrow \pm 90^\circ\). This produces highly distorted rectified images, which affects stereo matching adversely, since areas of disparate resolution are being compared. This is supported by mrcal purely for compatibility with other tools; there's little reason to use this representation otherwise`LENSMODEL_LATLON`

: this is a "transverse equirectangular projection". It is defined with even angle spacing in both directions, so \(x - x_0 = k_x \theta\) and \(y - y_0 = k_y \phi\) where \(x\) and \(y\) are pixel coordinates in the rectified images, \(x_0\) and \(y_0\) are the centers of projection of the rectified system and \(k_x\) and \(k_y\) are the angular resolution in the two directions. This is the recommended rectification model, and is the default in mrcal

Let's demonstrate the two rectification models. In the tour of mrcal we showed a
dense stereo processing sequence. Let's re-rectify those same images and models
with `LENSMODEL_PINHOLE`

and `LENSMODEL_LATLON`

. This is a demo, so I ask for
the same pixels/angle resolution at the center of the image in both cases, and
demonstrate how this affects the resolution at other parts of the image.

for model (LENSMODEL_PINHOLE LENSMODEL_LATLON) { mrcal-stereo \ --az-fov-deg 160 \ --el-fov-deg 140 \ --pixels-per-deg -0.05 \ --rectification $model \ [01].cameramodel \ [01].jpg }

The left image rectified with `LENSMODEL_LATLON`

(resolution at the center is
1/20 of the resolution of the original image):

The left image rectified with `LENSMODEL_PINHOLE`

(resolution at the center is
still 1/20 of the resolution of the original image):

These are the actual, unscaled rectified images. Note the identical resolution
at the center of the image. And note how `LENSMODEL_PINHOLE`

rectification
causes the image to expand dramatically as we approach the edges.

Using `LENSMODEL_PINHOLE`

with wide lenses like this introduces an unwinnable
trade-off. If you choose the pixels/angle resolution to keep all your
information at the center, you'll get a huge image, with low-information pixels
at the edges. But if you want to reduce this expansion by lowering the
resolution, you'll lose data at the center. Or you can cut off data at the
edges. No matter what you do, you either lose information or you're stuck with a
huge image.

And even if this was all fine, with `LENSMODEL_PINHOLE`

the stereo-matching
algorithm has to match image patches with widely different resolutions.
`LENSMODEL_LATLON`

solves all these issues with no down sides, and is thus the
recommended rectification function.

## Interfaces

Currently stereo processing is available via the `mrcal-stereo`

tool. This tool
implements the usual stereo processing for a single frame.

More complex usages are available via the Python APIs and the C functions in
`stereo.h`

. A sequence of images captured with a stereo pair can be processed
like this:

`mrcal.rectified_system()`

to construct the rectified system defined by the stereo pair`mrcal.rectification_maps()`

to construct the pixel mappings needed to transform captured images into rectified images. This is relatively slow, but it depends on the relative stereo geometry only, so this can be computed once, and applied to*all*the subsequent images captured by the stereo pair- For each pair of captured images
`mrcal.transform_image()`

to generate rectified images- stereo matching to compute disparities. mrcal does not provide its own
method, and the
`mrcal-stereo`

tool uses the OpenCV SGBM stereo matcher. Any stereo matcher can be used. The result is a*disparity*image, where each pixel in the first rectified image is mapped to a corresponding pixel offset from the same feature in the second rectified image `mrcal.stereo_range()`

to convert the disparities to ranges, which can then be used to produce a point cloud

A demo of the process if shown in the tour of mrcal.