Weak-Perspective Projection

In some images, parallel lines in scene appear as parallel lines in image, and length ratios along parallel lines are preserved. This is best modelled by weak-perspective projection.

• $$\Delta Z$$: depth variations
• $$Z_{avg}$$: average distance of objects from camera

If $$Z{avg} > c \Delta Z$$, perspective projection can be approximated by weak-perspective projection. All points assumed to lie on the same depth $$Z{avg}$$.

• $$x = \frac{f Xc}{Z{avg}}$$
• $$y = \frac{f Yc}{Z{avg}}$$
• $$\frac{f}{Z_{avg}}$$: scale factor a orthographic projection is up to

$$ \begin{aligned} \tilde{w} &= P{wp} \tilde{X} = P_c P{pll} Pr \tilde{X}\ \begin{bmatrix} su \ sv \ s \end{bmatrix} &= \begin{bmatrix} k_u & 0 & u_0 \ 0 & k_v & v_0 \ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} f & 0 & 0 & 0 \ 0 & f & 0 & 0 \ 0 & 0 & 0 & Z{avg} \end{bmatrix}\begin{bmatrix} R & T \ 0_3^T & 1 \end{bmatrix}\begin{bmatrix} X \ Y \ Z \ 1 \end{bmatrix} \end{aligned}

$$

where $$P{wp}$$ is the projection matrix for a weak-perspective camera, $$P{pll}$$ is the weak-perspective projection matrix.

Affine Camera

Relax the constraints:

$$ \tilde{w} = P{aff} \tilde{X}\ P{aff} = \begin{bmatrix} p{00} & p{01} & p{02} & p{03}\ p{10} & p{11} & p{12} & p{13}\ 0 & 0 & 0 & p_{23}\ \end{bmatrix}

$$

where $$P_{aff}$$ is the projection matrix for an affine camera.

Degree of freedom: 8 (scale does not matter)

Planar Scenes

$$ \begin{aligned} \tilde{w} &= P{wp}^p \tilde{X}^p = P_c P{pll} Pr^p \tilde{X}^p\ \begin{bmatrix} su \ sv \ s \end{bmatrix} &= \begin{bmatrix} k_u & 0 & u_0 \ 0 & k_v & v_0 \ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix} f & 0 & 0 & 0 \ 0 & f & 0 & 0 \ 0 & 0 & 0 & Z{avg} \end{bmatrix}\begin{bmatrix} r{00} & r{01} & Tx \ r{10} & r{11} & T_y \ r{20} & r_{21} & T_z \ 0 & 0 & 1 \ \end{bmatrix}\begin{bmatrix} X \ Y \ 1 \end{bmatrix}\end{aligned}

$$

Relaxing on the constraints,

$$ \tilde{w} = P^p{aff} \tilde{X}^p\ P^p{aff} = \begin{bmatrix} p{00} & p{01} & p{02}\ p{10} & p{11} & p{12}\ 0 & 0 & p_{22}\ \end{bmatrix}

$$

$$ \begin{bmatrix} su \ sv \ s \end{bmatrix} = \begin{bmatrix} p{00} & p{01} & p{02} \ p{10} & p{11} & p{12} \ 0 & 0 & 1 \end{bmatrix}\begin{bmatrix}X \ Y \ 1\end{bmatrix}\ \begin{bmatrix} u \ v \end{bmatrix} = \begin{bmatrix} p{00} & p{01} & p{02} \ p{10} & p{11} & p{12} \end{bmatrix}\begin{bmatrix}X \ Y \ 1\end{bmatrix}

$$

Degrees of freedom: 6 (scale does not matter)

Linear Scenes

$$ \tilde{w} = P^l{aff} \tilde{X}^p\ P^l{aff} = \begin{bmatrix} p{00} & p{01}\ p{10} & p{11}\ 0 & p_{21}\ \end{bmatrix}

$$

$$ \begin{bmatrix} u \ v \end{bmatrix} = \begin{bmatrix} p{00} & p{01} \ p{10} & p{11} \end{bmatrix}\begin{bmatrix}X \ 1\end{bmatrix}

$$

Degrees of freedom: 4 (scale does not matter)

Error Analysis

Image error:

$$ u - u' = (u - u0) \frac{\Delta Z}{Z{avg}}\ v - v' = (v - v0) \frac{\Delta Z}{Z{avg}}

$$

• The depth variations of objects $$\Delta Z$$ need to be small compared to viewing distance $$Z_{avg}$$
• The image points need to be close to the principal point

Invariants

A measurement made in an image which does not change across different viewpoints of the same object (can change across different objects). Dependent on the type of camera used.

Planar Scene

Euclidean Camera

Image plane in parallel. Fixed distance from world plane.

• Invariants
  • Length
  • Area

Degrees of freedom: 3 (2 for translation, 1 for rotation)

Similarity Camera

Image plan in parallel. Distance not fixed.

• Invariants
  • Angles
  • Ratio of length

Degrees of freedom: 4 (2 for translation, 1 for rotation, 1 for scaling)

Affine Camera

Little depth variation.

• Invariants
  • Parallelism
  • Ratio of length along collinear or parallel lines
  • Ratio of area

Degrees of freedom: 6 (2 for translation, 1 for rotation, 1 for scaling, 1 for stretching, 1 for shearing)

Projective Camera

Viewing conditions completely unrestricted.

• Invariants
  • Concurrency (cross points)
  • Collinearity
  • Cross-ratio (ratio of ratio)

Degrees of freedom: 8 (2 for translation, 1 for rotation, 1 for scaling, 1 for stretching, 1 for shearing, 2 for equation of vanishing line)

Cross-Ratio

$$ \begin{bmatrix} su \ s \end{bmatrix} = \begin{bmatrix} p{00} & p{01} \ p_{10} & 1 \end{bmatrix}\begin{bmatrix} X \ 1 \end{bmatrix}\ \begin{bmatrix} sl_i \ s \end{bmatrix} = \begin{bmatrix} \frac{1}{cos \theta} & -\frac{u_o}{cos \theta} \ 0 & 1 \end{bmatrix}\begin{bmatrix} s u_i \ s \end{bmatrix} = \begin{bmatrix} p & q \ r & 1 \end{bmatrix}\begin{bmatrix} X_i \ 1 \end{bmatrix}

$$ Hence,

$$ l_i = \frac{pX_i + q}{rX_i + 1}

$$ Cross ratio:

$$ \frac{\frac{l_c - l_a}{l_c - l_b}}{\frac{l_d - l_a}{l_d - l_b}} = \frac{(l_c - l_a)(l_d - l_b)}{(l_c - l_b)(l_d - l_a)} = \frac{(X_c - X_a)(X_d - X_b)}{(X_c - X_b)(X_d - X_a)}

$$ Planar imaging situations, need 5 distinguished points: