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TodayMid-term exam6.866 projects6.866 projects, continuedGradients and edgesSmoothing and DifferentiationRemaining issuesNoticeedgesSomething useful with edgesAnother useful, bandpass-filter-based, non-linear operation: Contrast normalizationBayesian methodsSimple, prototypical vision problemBayesian approachLikelihood function, P(obs|parms)Prior probabilityPosterior probabilityAb = 1 problemLoss functionsBayesian decision theoryLocal mass loss function may be useful model for perceptual tasksRegularization vs Bayesian interpretationsBayesian interpretation of regularization approachWhy the difference mattersExample imageMultiple shape explanationsGeneric shape interpretations render to the image over a range of light directionsLoss functionShape probabilitiesComparison of shape explanations12Today• Edges•Bayes• Motion analysis3Mid-term examProblem set 3– Open book, open web.– Work by yourself. But you can ask us questions for clarification.– Due Tuesday, Oct. 22 (in 5 days).46.866 projects• Proposals to us by Oct. 29 or earlier.• We will ok them by Oct. 31• 3 possible project types:– Original implementation of an existing algorithm– Rigorous evaluation of existing implementation.– Synthesis or comparison of several research papers.56.866 projects, continued• Some possible projects– Study conditions on shape and reflectance maps such that shape is interpretable from rendered image.– Pose and solve a problem: make an algorithm that detects broken glass, or that finds trash. Implement and evaluate it.– Evaluate accuracy of photometric stereo shape reconstructions.– Compare several motion estimation algorithms. Discuss how they’re different, the benefits of each, etc. Put them in a common framework.67Gradients and edges• Points of sharp change in an image are interesting:– change in reflectance– change in object– change in illumination– noise• Sometimes called edge points• General strategy– determine image gradient– now mark points where gradient magnitude is particularly large wrt neighbours (ideally, curves of such points).Forsyth, 20028There are three major issues:1) The gradient magnitude at different scales is different; which shouldwe choose?2) The gradient magnitude is large along thick trail; howdo we identify the significant points?3) How do we link the relevant points up into curves?Forsyth, 20029Smoothing and Differentiation• Issue: noise– smooth before differentiation– two convolutions to smooth, then differentiate?– actually, no - we can use a derivative of Gaussian filter• because differentiation is convolution, and convolution is associativeForsyth, 2002101 pixel3 pixels7 pixelsThe scale of the smoothing filter affects derivative estimates, and alsothe semantics of the edges recovered.Forsyth, 200211We wish to mark points along the curve where the magnitude is biggest.We can do this by looking for a maximum along a slice normal to the curve(non-maximum suppression). These points should form a curve. There arethen two algorithmic issues: at which point is the maximum, and where is thenext one?Forsyth, 200212Forsyth, 2002Non-maximumsuppressionAt q, we have a maximum if the value is larger than those at both p and at r. Interpolate to get these values.13Forsyth, 2002Predictingthe nextedge pointAssume the marked point is an edge point. Then we construct the tangent to the edge curve (which is normal to the gradient at that point) and use this to predict the next points (here either r or s).14Remaining issues• Check that maximum value of gradient value is sufficiently large– drop-outs? use hysteresis• use a high threshold to start edge curves and a low threshold to continue them.Gradient magnitudet1t2Labeled as edgePixel number in linked list along gradient maxima15Notice• Something nasty is happening at corners• Scale affects contrast• Edges aren’t bounding contours16Forsyth, 200217fine scalehigh thresholdForsyth, 200218coarse scale,high thresholdForsyth, 200219coarsescalelowthresholdForsyth, 200220edges•Issues: – On the one hand, what a useful thing: a marker for where something interesting is happening in the image.– On the other hand, isn’t it way to early to be thresholding, based on local, low-level pixel information alone?21Something useful with edgesDan Huttenlocherhttp://www.cs.cornell.edu/~dph/hausdorff/hausdorff1.html22Another useful, bandpass-filter-based, non-linear operation: Contrast normalization• Maintains more of the signal, but still does some gain control.• Algorithm: bp = bandpassed image.absval = abs(bp);avgAmplitude = upBlur(blurDn(absval, 2), 2);contrastNorm = bp ./ (avgAmplitude + const);amplitudelocal contrastContrast normalized output23Bandpass filtered (deriv of gaussian)Original image24Absolute value Blurred absolute valueBandpass filtered25Bandpass filtered and contrast normalizedBandpass filtered26Bandpass filtered and contrast normalizedBandpass filtered27Bayesian methods28Simple, prototypical vision problem• Observe some product of two numbers, say 1.0.• What were those two numbers?• Ie, 1 = ab. Find a and b.• Cf, simple prototypical graphics problem: here are two numbers; what’s their product?291 2 3 44321hyperbola of feasible solutionsab1 = a b30Bayesian approach• Want to calculate P(a, b | y = 1).• Use P(a, b | y = 1) = k P(y=1|a, b) P(a, b).Likelihood functionPrior probabilityPosterior probability31Likelihood function, P(obs|parms)• The forward model, or rendering model, taking into account observation noise.• Example: assume Gaussian observation noise. Then for this problem:222)1(21),|1(σπσabebayP−−==32Prior probability• In this case, we’ll assume P(a,b)=P(a)P(b), and P(a) = P(b) = const., 0<a<4.33Posterior probability• Posterior = k likelihood prior222)1()1|,(σabkeybaP−−==for 0 < a,b<4,0 elsewhere34Ab = 1 problemD. H. Brainard and W. T. Freeman, Bayesian Color Constancy, Journal of the Optical Society of America, A, 14(7), pp. 1393-1411, July, 199735Loss functions36D. H. Brainard and W. T. Freeman, Bayesian Color Constancy, Journal of the Optical Society of America, A, 14(7), pp. 1393-1411, July, 19973738Bayesian decision theoryD. H. Brainard and W. T. Freeman, Bayesian Color Constancy, Journal of the Optical Society of America, A, 14(7), pp. 1393-1411, July, 199739D. H. Brainard and W. T. Freeman, Bayesian Color Constancy, Journal of the Optical Society of America, A, 14(7), pp.


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MIT 6 801 - Mid-Term Exam

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