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This is a simple task that demonstrates working with homogeneous planar points and lines.

- Let the image area has an extent
`[1, 1]`

to`[800, 600]`

. Draw its boundary. Use the image coordinate system, i.e., x-axis pointing**right**and y-axis pointing**down**. - In your solution, allow entering two pairs of points within this area and display them.
- Calculate the straight line passing through the first pair and the straight line passing through the second pair. Use homogeneous representation. Display the intersection of each line with the image area.
- Calculate the intersection of both lines and draw it,
**if it is inside the image area**. - Apply the following homography to all entities (points, lines, image boundary) and draw the result to another figure.

K = [ [ 1, 0.1, 0 ], [ 0.1, 1, 0 ], [ 0.004, 0.002, 1 ] ]

Example result of this task is shown in figure 1.

For entering the points, the Matlab function `ginput`

, or the python function `matplotlib.pyplot.ginput`

, is suitable. Cross product of two vectors can be computed using the function `cross`

or `numpy.cross`

.

Develop a simulation of perspective camera projection – wire-frame model. Let the 3D object be given. The object is composed from two planar diagrams, that are connected. Coordinates of vertices of both diagrams `X1`

and `X2`

are:

X1 = [ [-0.5, 0.5, 0.5, -0.5, -0.5, -0.3, -0.3, -0.2, -0.2, 0, 0.5 ], [-0.5, -0.5, 0.5, 0.5, -0.5, -0.7, -0.9, -0.9, -0.8, -1, -0.5 ], [ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 ] ] X2 = [ [-0.5, 0.5, 0.5, -0.5, -0.5, -0.3, -0.3, -0.2, -0.2, 0, 0.5 ], [-0.5, -0.5, 0.5, 0.5, -0.5, -0.7, -0.9, -0.9, -0.8, -1, -0.5 ], [ 4.5, 4.5, 4.5, 4.5, 4.5, 4.5, 4.5, 4.5, 4.5, 4.5, 4.5 ] ]

Wire-frame model contains edges, that connects vertices in `X1`

and `X2`

in given order, and additionally it contains edges connecting vertices between `X1`

and `X2`

, such that the vertex `X1(:,i)`

is connected to the vertex `X2(:,i)`

, ∀ `i`

.

The internal calibration matrix of the camera is:

K = [[ 1000, 0, 500 ], [ 0, 1000, 500 ], [ 0, 0, 1 ]]

See Task 0-2 notes for used convention of direction of a 3D rotation.

- Construct following camera matrices (keep the image
*u*-axis parallel to the scene*x*-axis):`P1`

: camera in the origin looking in the direction of*z*-axis.`P2`

: camera located at`[0,-1,0]`

looking in the direction of*z*-axis.`P3`

: camera located at`[0,0.5,0]`

looking in the direction of*z*-axis.`P4`

: camera located at`[0,-3,0.5]`

, with optical axis rotated by 0.5 rad around*x*-axis towards*y*-axis.`P5`

: camera located at`[0,-5,4.2]`

looking in the direction of*y*-axis.`P6`

: camera located at`[-1.5,-3,1.5]`

, with optical axis rotated by 0.5 rad around*y*-axis towards*x*-axis (i.e.,**-**0.5 rad) followed by a rotation by 0.8 rad around*x*-axis towards*y*-axis. See figure 4.

- Use the cameras
`P1`

to`P6`

for projection of given wire-frame model into an image. The edges inside`X1`

should be drawn red, the edges inside`X2`

should be drawn blue and the rest should be drawn in black.

The example projection by the camera `P1`

is shown in figure 3.

Let `u1`

, `v1`

be the image coordinates of projected vertices `X1`

and `u2`

, `v2`

be the coordinates of projected vertices `X2`

. The desired picture can be drawn, e.g., in python like this:

plt.plot( u1, v1, 'r-', linewidth=2 ) plt.plot( u2, v2, 'b-', linewidth=2 ) plt.plot( [u1, u2], [v1, v2 ], 'k-', linewidth=2 ) plt.gca().invert_yaxis() plt.axis( 'equal' ) # this kind of plots should be isotropic

or in Matlab like this:

plot( u1, v1, 'r-', 'linewidth', 2 ) hold on plot( u2, v2, 'b-', 'linewidth', 2 ) plot( [u1; u2], [v1; v2], 'k-', 'linewidth', 2 ); set( gca, 'ydir', 'reverse' ) axis equal % this kind of plots should be isotropic

See the probabilistic model of line and model fitting.

A 2D line `[-10, 3, 1200]`

generates set of `100`

points (inliers), corrupted by a gaussian noise. Additionally, there is a set of `200`

uniformly distributed random points not belonging to the line (outliers). There are three sets of points linefit_1.txt, linefit_2.txt, linefit_3.txt generated with the gaussian noise $\sigma=1$, $\sigma=2$, and $\sigma=3$, respectively. One set of points is shown in figure 5.

For such kind of data, non-robust estimation method, e.g. regression by least squares, cannot be used. A golden standard method for robust (i.e. in the presence of outliers) estimation in such a case is RANSAC. If more acurate result is needed, additional optimisation is applied to the result of RANSAC, i.e. least squares regression using the inliers only. One of the common modifications of RANSAC method employs support function derived as a maximum likelihood estimate (MLE), instead of a standard zero-one box function used in traditional RANSAC. This modification (MLESAC) has the same complexity and convergence, but tends to produce more accurate estimates directly. Comparison of a single run of these methods is in figure 6.

To show difference between results of the above methods, we run RANSAC and MLESAC on a single data set 100 times. Every run is then followed by least squares line regression using inliers. For visualisation purposes, every line is characterised by two parameters - its angle $\alpha$ and its orthogonal distance $d$ to origin, thus showing as a single point in $\alpha-d$ graph. The results are shown in figure 7 and figure 8. It is clearly visible, that MLESAC tends to produce more accurate estimates compared to RANSAC.

See Task 0-3 notes.

- Use these points for (non-robust) least squares estimation (regression) of the line. This is expected to fail.
- Use these points for robust estimation of the line. Use RANSAC to find initial estimate and to separate inliers and outliers, followed by least squares regression using the inliers.
- Repeat the previous step with MLESAC.
- For the estimation, do not use the parameters of the original generator (numbers of inliers, outliers, $\sigma$). Set a RANSAC threshold by hand.
- Draw the whole situation:
- Original line.
- Points.
- Line found by non-robust regression.
- Line found by RANSAC
- Line found by RANSAC followed by least squares regression
- Line found by MLESAC followed by least squares regression

Reading a point coordinates from a text file can be achieved as `x = load( 'linefit_1.txt' )';`

(matlab) or `x=numpy.loadtxt( 'linefit_1.txt' ).T`

(python); note the transpose to obtain column vectors. Random sample of `2`

from `n`

points can be generated as `i = randperm( n, 2 );`

or `rng = numpy.random.default_rng()`

called once followed by `i = rng.choice( n, 2, replace=False )`

.

Consider a pair of images of a scene with two dominant planes, e.g. a pair of images from a triple below. Each plane generate one homography between the image pair. i.e. there are two homographies **Ha** and **Hb**.

When a set of tentative correspondences between the images is known these homographies can be estimated. Additionally, since these two homographies are generated by two scene planes, there is a line in both images, that is a projection of the intersecting line of the planes. The image points **u1**, **u2** laying on that line in both images are related by both homographies, i.e., **u2** ∼ **Ha** **u1** ∼ **Hb** **u1**, which can be written as inv(**Hb**) **Ha** **u1** = λ **u1**. Thus these two homographies must be estimated such that these constraint is enforced (two arbitrary regular 3×3 matrices generally do not ensure the existence of such a common line. Note that this constraint means that there exist a common line that is transformed same way by the two complementary **line homographies** inv(**Ha**') and inv(**Hb**') (this line always exists for any two regular 3×3 matrices) but additionally that this line is transformed by both homographies from the first image to the same line in the second one **point-wise**. Note that this **H** maps points from the first image to the same image.

The special transformation composed from the such two homographies (given by two planes observed by a pair of cameras ) **H** = inv(**Ha**) **Hb** is called **planar homology**, and has only 5 degrees of freedom (3 correspondences needed), compared to 8 DoF of a general homography.

There are many methods how to estimate multiple geometric models (homographies in this case) from given data. Two simplest possibilities are:

- Sequential RANSAC
- Estimate the first homography using standard RANSAC scheme for a single homography. Typically the most dominant one is found.
- Remove inliers of the first found homography from the data.
- Estimate the second homography using the remaining data, such that the common homology constraint is enforced (3 correspondences needed)

- Multi RANSAC with conditional sampling
- Estimate both homographies simultaneously in RANSAC scheme by sampling 7-tuple of correspondences.
- In every sample:
- Draw 4 random correspondences and estimate
**Ha** - Remove inliers of
**Ha**from the data set - Draw 3 random correspondences and estimate
**Hb**with homology constraint - Verify support of both homographies, accept the best one

Common homology estimation from three: when **Ha** is estimated, the sampled three points **u_2** (corresponding to three points **u_1**) are mapped to the first image by inv(**Ha**) as **u_21**. Then these correspondences **u_1** and **u_21** are used for estimating homology **H** from the three correspondences. Finally, the second homography is computed as **Hb** = **Ha** **H**.

Note that having two homographies an inlier must be decided to which it belongs to. This can be done based on minimal error or according to on which side of common line the point is laying.

See Task 0-4 notes.

- Select at least one pair of images from the set above. Note that the pair 1-2 is the easiest one and the pair 1-3 is the hardest one considering the stability of solution.
- Use interest points detected in the images: books_u1.txt, books_u2.txt, books_u3.txt for the image 1, 2, 3, respectively.
- Use set of tentative correspondences between a selected pair of images: books_m12.txt, books_m13.txt, books_m23.txt
- Estimate the two dominant homographies by a method of your choice, with homology constraint.
- Compute the common line.
- Show the results - show in both images outliers and inliers of either homography (in different colours) and the common line. Show the correspondences as needle map. See example in figure 9.

courses/tdv/labs/0_geometry.txt · Last modified: 2021/09/16 10:25 (external edit)