Digital Video Processing (EE392J)Department of Electrical EngineeringStanford UniversityProblem Set No. 5Issued: Wednesday, February 21, 2007Due: Wednesday, February 28, 2007Brief Oral Project Descriptions: Wednesday, February 28, 2007 (see next page)Office hours: Wed (2/21) and Mon (2/26) after class, and Friday (2/23) 2:00-3:00.Please feel free to stop by office hours to talk about possible projects.Problem l. Constraint-based Signal Recovery: POCS when each constraint is a subspaceThis problem considers constraint-based signal recovery in the special case where each constraint corre-sponds to a subspace. Assume that the desired solution satisfies a linear equation of the form a1x1+ a2x2+... + aNxN= 0. This equation can be equivalently expressed as aTx = 0 where a = [a1, a2, ..., aN]Tandx = [x1, x2, ..., xN]T∈ RN. Therefore, the solution must lie in the (N − 1)-dimensional subspace definedby aTx = 0.(a) Determine an expression that describes how to compute the orthogonal projection of an arbitrary pointx onto the subspace. The orthogonal projection of an arbitrary point x onto a subspace is the point in thesubspace that is closest (in Euclidean distance) to the point x. (Hint: Think projection theorem or leastsquares.)(b) Assume that the desired solution satisfies two linear equations of the type shown above. Also assume thatthe corresponding subspaces intersect. Is there a unique solution? If not, what is the form of the solution,e.g., what is its dimension?(c) Assume you perform alternating orthogonal projections to determine an element in the solution set. Howdoes the convergence rate depend on the “angle” between the subspaces? (Hint: Consider the analogouscase of two 1-D subspaces in a 2-D space, i.e., when you have two linear equations in two unknowns. Whathappens when the subspaces are orthogonal? What happens when the angle between them is very small?)(See Back)Problem 2. Brief Oral Project DescriptionThe goal of this brief presentation is twofold: (1) to describe to your classmates what you are working onfor your final project, and (2) to provide an opportunity for additional feedback which may help improveyour final project.• Presentation length: Your presentation should only be about 7-8 minutes in length, followed by a fewminutes of discussion and Q&A. Therefore, you should have between 4 and 8 slides total, includingtitle slide.• Presentation format: Use whatever format is most convenient for you. If you have a laptop, feel freeto use it. If you don’t have a laptop and you would like to use powerpoint or PDF, just email me thecontent and you can use my laptop for the presentation. Also feel free to use transparencies if youprefer, as we have an overhead projector in class.• Suggestions for presentation content:– Name and high-level description of project– Why is this project interesting/fun for you?– What specific problems must be overcome?– How are you trying to overcome these problems? What prior art (e.g., algorithms from the classtextbook or from published papers) are you using or building upon?– What is your project roadmap for the remainder of the quarter, i.e., what is your plan for whatyou are trying to do and by when?– Possibly showing preliminary results: If you already have some preliminary results that youwould like to show that would be great. If not, that’s fine too. Alternatively, if you have justacquired the images or video(s) which you will use in your final project then you can show themto the class.I’m available to meet and discuss any possible projects, as well as to help you identify potential problemswithin the project and approaches to overcome these problems.Office hours: Office hours are listed on the first page. Please feel free to stop by office hours to talk aboutpossible