Multiple View Geometry in Computer Vision, Richard Hartley and Andrew Zisserman, Solutions

An index to the worked solutions

A 3 minute read, posted on 3 Jan 2020
Last modified on 20 May 2020

Tags computer vision, projective geometry, problem solution

After searching in vain for solutions to the exercises in this book, I decided to start documenting my solutions with the hope that it might provide encouragement to others like me on the path of self-study.

On the note of self-study, I would like to provide some feedback to those who have just begun or are contemplating using this book to learn computer vision. Firstly, if you don’t know this already (I didn’t), multiple view geometry is just one, albeit major, facet of computer vision, not the whole of it. You might want to first explore computer vision breadth-wise before you decide to commit to this one particular area.

Secondly, I found this book to be more of a compendium of research papers on multiple view geometry rather than an introductory textbook for beginners in the field. So if you’re starting from scratch like me, I strongly recommend becoming conversant with projective geometry, probability & statistics, linear algebra, calculus, optimization and some image processing, before attempting the material, to get the most out of it.

Finally, if you put in the work, you will find the material to be rewarding. Through this book, I’ve been able to learn things that I didn’t even know were possible. It has literally broadened my horizons (pun intended).

I strongly recommend that you try to attempt the exercises yourself before looking at the solutions. Even if you initially feel that you’re making no progress, the mind is a strange and wonderful instrument and you might just get the answer after a restful nap.

My answers are organized into chapters just like in the book. If you’re satisfied with your attempt and feel that you can’t proceed without help or you just want to confirm your answer then take a look! Click on the chapter’s name to go to my solutions for that chapter or use the numbered link to go directly to the chosen exercise.

Chapter 2: Projective Geometry and Transformations of 2D

If you are finding it hard to grasp the ideas in this chapter, I suggest going through an introductory text on projective geometry. One book I highly recommend is “Introduction to Projective Geometry” by C.R. Wylie Jr. I also recommend that you play with the interative 3D graphs that are part of the solution set for the book on this blog. Just hit the “View in GeoGebra” link and modify the lines and conics to get a feel for perspective projection. The point $C$ is the center of projection, the image plane is $z = 0$ and the object plane is $y = 0$.

Here are quicklinks to the exercise solutions in this chapter.

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Chapter 3: Projective Geometry and Transformations of 3D

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Chapter 4: Estimation - 2D Projective Transformations

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Chapter 5: Algorithm Evaluation and Error Analysis

This chapter discusses determining the lower bounds of the bias and variance of an estimator, and measuring the uncertainty in the estimated parameters. The lower bound for the bias gives us a good benchmark to measure the performance of any new estimation algorithm, and the uncertainty measure helps us figure out how reliable the results are for any given input.

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Chapter 6: Camera Models

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Chapter 7: Computation of the Camera Matrix P

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Chapter 8: More Single View Geometry

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