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Step 4: Homography estimation

Recall from the lectures that we can estimate a homography between two images from point correspondences using the Direct Linear Transform:

Homography estimation with DLT

We can often achieve a better result if we perform the estimation on normalized point correspondences, using the Normalized DLT:

Homography estimation with Normalized DLT

4. Understand how we estimate the homography

Lets take a look at how we have implemented all this math! Go to the class HomographyEstimator in lab_mosaic.py:

  • Look at HomographyEstimator._dlt_estimator():
    • Try to identify steps 1-4 in the DLT above.
  • Look at HomographyEstimator._normalized_dlt_estimator():
    • Try to identify steps 1-3 in the normalized DLT above.
    • What does the normalizing similarity do?
  • Look at HomographyEstimator._ransac_estimator():
    • How many point correspondences do we sample each iteration of the RANSAC loop? Why?
    • Which homography estimation algorithm do we use in the RANSAC loop? DLT or normalized DLT? Why not the other one?
  • Look at HomographyEstimator.estimate():

5. Compute the reprojection error

To make the homography estimator work, we need to finish HomographyEstimator._compute_reprojection_error() in order to compute the reprojection error in the RANSAC inlier test.

In this context, reprojection error is a measure of how well a homography fits with a correspondence ui ↔ ui`:

\varepsilon_i = \lVert H(\mathbf{u}_i)-\mathbf{u}_i' \rVert + \lVert \mathbf{u}_i - H^{-1}(\mathbf{u}'_i) \rVert

Here, H maps pixels from one image to the other according to

H:\mathbf{u} \mapsto \tilde{\mathbf{u}} \mapsto \mathbf{H}\tilde{\mathbf{u}} = \tilde{\mathbf{u}}' \mapsto \mathbf{u}'

and H-1 is its inverse:

H^{-1}:\mathbf{u}' \mapsto \tilde{\mathbf{u}}' \mapsto \mathbf{H}^{-1}\tilde{\mathbf{u}}' = \tilde{\mathbf{u}} \mapsto \mathbf{u}

In HomographyEstimator._compute_reprojection_error() you need to compute the reprojection error for a point correspondence.

Hint: You can use the functions homogeneous() and hnormalized() in common_lab_utils.py

When you are happy with your implementation, run the program. Choose a reference and perform matching by pressing . Use debugging tools or printouts to the console to check that your implementation computes reasonable results.

Now, lets use the computed homography to combine the current frame with the reference in an image mosaic! Please continue to the next step.