Computer stereo vision

Applications

3D stereo displays find many applications in entertainment, information transfer and automated systems. Stereo vision is highly important in fields such as robotics to extract information about the relative position of 3D objects in the vicinity of autonomous systems. Other applications for robotics include object recognition, where depth information allows for the system to separate occluding image components, such as one chair in front of another, which the robot may otherwise not be able to distinguish as a separate object by any other criteria.

Scientific applications for digital stereo vision include the extraction of information from aerial surveys, for calculation of contour maps or even geometry extraction for 3D building mapping, photogrammetric satellite mapping, or calculation of 3D heliographical information such as obtained by the NASA STEREO project.

Detailed definition

A pixel records color at a position. The position is identified by position in the grid of pixels (x, y) and depth to the pixel z.

Stereoscopic vision gives two images of the same scene, from different positions. In the adjacent diagram light from the point A is transmitted through the entry points of pinhole cameras at B and D, onto image screens at E and H.

In the attached diagram the distance between the centers of the two camera lens is BD = BC + CD. The triangles are similar,

• ACB and BFE
• ACD and DGH

${\displaystyle \begin{array}{cc}\text{Therefore displacement }d& =EF+GH\\ & =BF(\frac{EF}{BF}+\frac{GH}{BF})\\ & =BF(\frac{EF}{BF}+\frac{GH}{DG})\\ & =BF(\frac{BC+CD}{AC})\\ & =BF\frac{BD}{AC}\\ & =\frac{k}{z}\text{, where}\end{array}}$

• k = BD BF
• z = AC is the distance from the camera plane to the object.

So assuming the cameras are level, and image planes are flat on the same plane, the displacement in the y axis between the same pixel in the two images is,

${\displaystyle d=\frac{k}{z}}$

Where k is the distance between the two cameras times the distance from the lens to the image.

The depth component in the two images are ${\displaystyle z_1}$ and ${\displaystyle z_2}$, given by,

${\displaystyle z_2(x,y)=min\left\{v:v=z_1(x,y-\frac{k}{z_1(x,y)})\right\}}$

${\displaystyle z_1(x,y)=min\left\{v:v=z_2(x,y+\frac{k}{z_2(x,y)})\right\}}$

These formulas allow for the occlusion of voxels, seen in one image on the surface of the object, by closer voxels seen in the other image, on the surface of the object.

Image rectification

Where the image planes are not co-planar, image rectification is required to adjust the images as if they were co-planar. This may be achieved by a linear transformation.

The images may also need rectification to make each image equivalent to the image taken from a pinhole camera projecting to a flat plane.

Smoothness

Smoothness is a measure of the similarity of colors. Given the assumption that a distinct object has a small number of colors, similarly-colored pixels are more likely to belong to a single object than to multiple objects.

The method described above for evaluating smoothness is based on information theory, and an assumption that the influence of the color of a voxel influences the color of nearby voxels according to the normal distribution on the distance between points. The model is based on approximate assumptions about the world.

Another method based on prior assumptions of smoothness is auto-correlation.

Smoothness is a property of the world rather than an intrinsic property of an image. An image comprising random dots would have no smoothness, and inferences about neighboring points would be useless.

In principle, smoothness, as with other properties of the world, should be learned. This appears to be what the human vision system does.

Information measure

Least squares information measure

The normal distribution is

${\displaystyle P(x,\mu ,\sigma )=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\sigma }^2}}}$

Probability is related to information content described by message length L,

${\displaystyle P(x)=2^{-L(x)}}$

${\displaystyle L(x)=-{\log }_{2}P(x)}$

so,

${\displaystyle L(x,\mu ,\sigma )={\log }_2(\sigma \sqrt{2\pi })+\frac{(x-\mu {)}^2}{2{\sigma }^2}{\log }_{2}e}$

For the purposes of comparing stereoscopic images, only the relative message length matters. Based on this, the information measure I, called the Sum of Squares of Differences (SSD) is,

${\displaystyle I(x,\mu ,\sigma )=\frac{(x-\mu {)}^2}{{\sigma }^2}}$

where,

${\displaystyle L(x,\mu ,\sigma )={\log }_2(\sigma \sqrt{2\pi })+I(x,\mu ,\sigma )\frac{{\log }_{2}e}{2}}$

Because of the cost in processing time of squaring numbers in SSD, many implementations use Sum of Absolute Difference (SAD) as the basis for computing the information measure. Other methods use normalized cross correlation (NCC).

Information measure for stereoscopic images

The least squares measure may be used to measure the information content of the stereoscopic images, given depths at each point ${\displaystyle z(x,y)}$. Firstly the information needed to express one image in terms of the other is derived. This is called ${\displaystyle I_m}$.

A color difference function should be used to fairly measure the difference between colors. The color difference function is written cd in the following. The measure of the information needed to record the color matching between the two images is,

${\displaystyle I_m(z_1,z_2)=\frac{1}{{\sigma }_m^2}\sum _{x,y}\mathrm{cd}({\mathrm{color}}_1(x,y+\frac{k}{z_1(x,y)}),{\mathrm{color}}_2(x,y){)}^2}$

An assumption is made about the smoothness of the image. Assume that two pixels are more likely to be the same color, the closer the voxels they represent are. This measure is intended to favor colors that are similar being grouped at the same depth. For example, if an object in front occludes an area of sky behind, the measure of smoothness favors the blue pixels all being grouped together at the same depth.

The total measure of smoothness uses the distance between voxels as an estimate of the expected standard deviation of the color difference,

${\displaystyle I_s(z_1,z_2)=\frac{1}{2{\sigma }_h^2}\sum _{i:\{1,2\}}\sum _{x_1,y_1}\sum _{x_2,y_2}\frac{\mathrm{cd}({\mathrm{color}}_i(x_1,y_1),{\mathrm{color}}_i(x_2,y_2){)}^2}{(x_1-x_2{)}^2+(y_1-y_2{)}^2+(z_i(x_1,y_1)-z_i(x_2,y_2){)}^2}}$

The total information content is then the sum,

${\displaystyle I_t(z_1,z_2)=I_m(z_1,z_2)+I_s(z_1,z_2)}$

The z component of each pixel must be chosen to give the minimum value for the information content. This will give the most likely depths at each pixel. The minimum total information measure is,

${\displaystyle I_{\min }=min\{i:i=I_t(z_1,z_2)\}}$

The depth functions for the left and right images are the pair,

${\displaystyle (z_1,z_2)\in \{(z_1,z_2):I_t(z_1,z_2)=I_{\min }\}}$

Methods of implementation

The minimization problem is NP-complete. This means a general solution to this problem will take a long time to reach. However methods exist for computers based on heuristics that approximate the result in a reasonable amount of time. Also methods exist based on neural networks. Efficient implementation of stereoscopic vision is an area of active research.