Midterm
CSC Second 242 10 May 2002
Write your NAME legibly on the bluebook. Work all problems. Best strategy is not to spend more than the indicated time on any question (minutes = points). Open book, open notes. 1. Grammar: 30 mins. We all remember: Wenn Fliegen iegen hinter Fliegen, Fliegen iegen Fliegen nach. And of course Die Mnner, die vor dem Schokoladen Laden Laden laden, laden Ladenmdchen zum Tanze a a ein. (These Germanic noun capitalizations make parsing these sentences easier, nicht? ) A. Make up a context-free phrase structure grammar with a lexicon made up only of transitive verbs, adjectives, and nouns that will parse (or generate) sentences of the form: cats ght dogs. troy cats ght dogs. cats ght albany dogs. troy cats ght albany dogs. cats love dogs albany residents hate. syracuse cats hate albany dogs albany residents love. rochester cats rochester dogs hate love syracuse cats albany dogs hate. .... But which do not include sentences of the form: cats ght. big troy cats ght dogs. the rochester cats ght large syracuse dogs. ... (Note that we dont need no stinkin capitalization.) B. Now consider your grammar with the following lexicon. Noun: bualo (as in bison). Adjective: bualo (as in from Bualo, NY). Verb: bualo (as in bae, frustrate). How many parses does your grammar give for the following sentence? bualo bualo bualo bualo bualo. C. What is the longest sentence your grammar can generate (or parse) consisting entirely of repetitions of the word bualo? Considered as a model of actual English usage, is the embedding property of these phrase structure grammars a problem? If not why not, and if so what do you suggest we do about it? D. Consider the sentences I dropped the lightbulb on the table and it broke. I dropped the anvil on the table and it broke.
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What mechanisms do NLU systems provide to resolve the reference of it in sentences like the above? Answer: A. I think something like this should work: S = ModN V ModN ModN = SubN | SubN NClause NClause = SubN ModN V | SubN V SubN = N | A N N = cats | dogs V = fight | love | hate A = troy | syracuse | albany | buffalo B. I get three, with the main verb being the 2nd, 3rd, or 4th bualo. C. Innite. Denite problem because the famous linguistic touchstone, the native informant, will not agree an innitely long sentence (or even a deeply embedded one) is grammatical. What to do, I dunno...one would have to check out some linguistics books. Clearly some sort of quick hack would be to put a limit on the number of times you invoke certain embedding rules like the 2nd and 3rd mutually recursive ones above. D. Lexical categories and information could encode some of this common sense knowledge, but also logical rules about object properties and the associated inferences may have to be used. 2. Vision and Linear Systems: 30 mins A. What does the convolution theorem for Fourier transforms tell us and why is it useful? B. What does the sampling theorem for Fourier transforms tell us and why is it useful? C. (see D.) What are three sorts of variation in an image that we might want to ignore for the purposes of object recognition? What algorithms or methods could implement such invariance calculations? D. (see C.) On the other hand, how could each of the irrelevant variations in your answer to C be used to tell you something about the object? Be technical and specic. E. On page 729 your text states that a Lambertian surface appears equally bright to an observer in any direction. That is, the number of photons per second impinging on a retinal receptor is constant for an observer looking at the surface from any angle. (Just how bright is given by the equation in the book, but thats irrelevant to this question.). Now I have another text that says that a Lambertian surface is one for which the photon ux, or amount of light (that is the number of photons per second emitted from the surface in any direction) is proportional to the cosine of the angle between the surface normal and the emittance direction. So one text says that the brightness is constant with angle A and the other says that the ux varies with A. In the gures, the arrow measuring brightness follows the semicircle and stays constant size, the arrow measuring ux follows around the circle and goes from 0 to 1 and back to 0 as the angle goes from 90 degrees to zero and back to 90. Since these are textbooks, they have to be correct, so how do you reconcile these statements?
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Surface Surface Normal
Normal
To Viewer
A
A
To Viewer
Length of Arrow is Brightness in its Direction
Length of Arrow is Flux in its Direction
Answer: A. convolution in time (spatial) domain is multiplication in (spatial) frequency domain, which yields elegant ways to think about the operation of linear operators. since Also, the fast FT is fast, the conv. th. allows convolution, usually an expensive (n2 in one dimension) operation, to be done quickly (n log n) by swapping domains. B. If you sample faster than the Nyquist frequency (twice the highest freq in the signal), you can reconstuct a continuous (but band-limited) function with a nite number of discontinuous samples. Basic to digital image science. C. Shading, texture variation, perspective distortion (size variation with distance), color appearance with dierent colored lights, etc... To beat shading you could lter or edge-nd to remove slowly-varying components. Or you could reason backward from assumptions of light source location and object surface reectance uniformity. Texture variation could maybe be beat by something like ane invariants, that is extracting geometric properties of the texels and their arrangements that are invariant to size, rotation, and skew. Color constancy is something humans are pretty good at, and ways to get it vary from trying to guess the color of the light source(s) to assuming all light sources and reectances are made up of a small number (3) of basis functions. Also the dierences of of colors between the various patches in the scene can be less variant than the colors themselves... this is basis of Lands color theory. Size-invariant measures can beat down the shrinks with distance phenomenon, as would an inverse camera model. D. Shape from shading, shape from texture: one approach is the equivalent of a reectance function mapping properties (for shading, just brightness, for textures, maybe some other properties) to a small subset of surface normals. Then some regularization and boundary condition assumptions can allow for iterative solution. Using location or size in an image to infer distance (above or smaller means farther) is quite common. E. The foreshortening of the surface when you see it from more oblique angles means that each receptor in your eye gets photons from a larger chunk of surface. In fact the size of the surface goes as 1/cosine, while the ux is going as the cosine of that angle. These eects cancel, leading to the uniform brightness you experience.
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3. Perceptrons: 40 mins A. You have a perceptron with a threshold permanently xed at zero (0) and two inputs, one for the x-coordinate of a data point and one for the y coordinate. It looks like this with T xed at 0. x
A Sum B Threshold T Output (0 or 1)
y
Inputs Weights
Can it discriminate between the lled and hollow data points in the data shown below for the left and right cases? Why or why not in each case?
(0,0)
(0,0)
left
right
B. You have a similar perceptron to the one in part A, only it can have a non-zero threshold. In fact you know that the weight on its x input is A, the weight on its y input is B, and its threshold is T . Specify the decision surface (in...