3D Stereogram

Each eye of the viewer is fed a different image of the 3D scene to have an illusion of depth.

Red/green stereograms
Single image random dot stereograms (SIRDS)
Active shutter systems (e.g. TV)
Polarization systems (e.g. cinemas)

Red/Green 3D Stereogram

Store the two images in different color channels, then filter them by wearing spectacles.Parallel image planes can be ensured either by using highly calibrated stereoscopic cameras, or recovering from any stereo pair of images.

Recovering Parallel Image Planes

Recover Camera Extrinsics

Assume camera calibration matrix $$K$$ is known.

Take two pictures
Estimate fundamental matrix $$F$$
1. Corresponding features
Recover essential matrix $$E$$
1. $$E = K^T F K$$
Recover relative rotation $$R$$ & relative translation $$T$$
1. $$E = [T]_xR$$

Rectification

$$R_r$$
- Align $$X_c$$ with $$T$$
- Align $$Y_c$$ with $$V_I$$ parallel to the intersection of the two image planes
- Align $$Z_c$$ with $$T \times V_I$$

$$ R_r = \begin{bmatrix} V_x^T \ V_y^T \ V_z^T \end{bmatrix}\ \begin{aligned} V_x &= \frac{-R^TT}{|T|}\ V_y &= (R^T \begin{bmatrix}0\0\1\end{bmatrix}) \times \begin{bmatrix}0\0\1\end{bmatrix}\ V_z &= V_x \times V_y\ \end{aligned}

Relate Raw & Rectified Images

Relate ray $$p$$ in the raw image to ray $$p_r$$ in the rectified image:

$$ p_r = R_r p

With $$K$$:

$$ K^{-1}\tilde{w}_r = R_r K^{-1}\tilde{w}\ KR_r^{T}K^{-1}\tilde{w}_r = \tilde{w}

Finally, associate the grey level $$I(w)$$ with each pixel $$w_r$$ in the rectified image.

Single Image Random Dot Stereogram

Does not require any special equipment.

A pattern of dots repeats across the width of the SIRDS with subtle distortion in accordance with the disparities from the 3D scene.

By solving the correspondence problem within the same image, the resulting disparity map would enable the perception of depth; but our visual system can only solve the correspondence problem from the left & right images but not the same image.

Diverge the eyes to look at adjacent repeats of the pattern:

Construction

Place an image plane between the 3D scene & the two viewpoints
Assign an arbitrary color to a given point $$X$$
Project $$X$$ onto the image as $$x_l$$ and $$x_r$$
Locate & assign the same color to points that will project as $$x_l$$ or $$x_r$$
Repeat 3.4. with the new points
1. All the points lie on the same epipolar plane, and all the image points lie on the same epipolar line
Repeat 2.-5. with a new color
Assign a random color to unfilled pixels

Stereo Vision IV