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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! First, study the class declaration and documentation in homography_estimator.h. Then go to homography_estimator.cpp:

  • Look at HomographyEstimator::dltEstimator:
    • Try to identify steps 1-4 in the DLT above.
  • Look at HomographyEstimator::normalizedDltEstimator:
    • Try to identify steps 1-3 in the normalized DLT above.
    • What does the normalizing similarity do?
  • Look at HomographyEstimator::ransacEstimator:
    • 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?
    • This method returns a PointSelection. What is that? (Hint: Check homography_estimator.h).
  • Look at HomographyEstimator::estimate:

5. Compute the reprojection error

To make the homography estimator work, we need to finish HomographyEstimator::computeReprojectionError 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::computeReprojectionError you need to compute the reprojection error for a point correspondence.

Hint: Use Eigen::Matrixbase::homogeneous(), Eigen::Matrixbase::hnormalized() and Eigen::Matrixbase::norm().

When you are happy with your implementation, compile and 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.

You definitely want to compile and run the program in release mode (see lab-corners) when you are finished debugging.

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