Computational VisionU. Minn. Psy 5036Daniel KerstenLecture 12: Coding Efficiency, Spatial statisticsInitialize‡Read in Statistical Add-in packages:In[34]:=Off@General::spell1D;SetOptions@ArrayPlot, ColorFunction Ø "GrayTones", DataReversed Ø True,Frame Ø False, AspectRatio Ø Automatic, Mesh Ø False,PixelConstrained Ø True, ImageSize Ø SmallD;SetOptions@ListPlot, ImageSize Ø SmallD;In[37]:=nbinfo = NotebookInformation@EvaluationNotebook@DD;dir =H"FileName" ê. nbinfo ê. FrontEnd`FileName@d_List, nam_, ___D ßToFileName@dDL;‡HistogramIn[38]:=myhistogram@image_D := Module@8histx<,histx = BinCounts@Flatten@imageD, 80, 255, 1<D;Return@N@histx êPlus üü histxDD;D;‡EntropyIn[39]:=entropy@probdist_D := Plus üü HIf@Ò == 0, 0, -Ò Log@2, ÒDD & êü probdistL‡Image dataIn[149]:=granite = ImageDataB F;In[150]:=N@Mean@Flatten@graniteDDDN@StandardDeviation@Flatten@graniteDDDwidth = Dimensions@graniteD@@1DDOut[150]=0.507939Out[151]=0.0773643Out[152]=64OutlineLast time‡First-order intensity statistics. Explaining point non-linearities in terms of efficient coding of natural imagesToday‡Form and function: overview of visual pathway‡2nd order spatial statistics and efficient codingWe've learned about localized spatial frequency filters in early vision. We now ask: Why?Retina to V1: Review of form & function2 12.SpatialCodingEfficiency.nbRetina to V1: Review of form & function(There are a number of web-based overviews, for example: http://www.sumanasinc.com/webcontent/anisamples/neurobiol-ogy/visualpathways.html).Overview of pathways from eye-to-cortexRoughly ten million retinal measurements are sent to the brain each second, where they are processed by some billion cortical neurons.The primate retina has about 10^7 cones that send visual signals to the optic nerve via about 10^6 ganglion cells. The optic nerves from the two eyes meet at the optic chiasm where about half of the fibers cross over and the other half remain on the same side of the underside of the brain. Before synapsing in the lateral geniculate nucleus, about 20% of these fibers that make up the optic tract branch off to the superior colliculus--a structure involved with eye movements. Other fibers project to various other nuclei, but the majority of the optic tract fibers synapse on cells in the lateral genicu-late nucleus. Cells in the lateral geniculate nucleus send their axons in a bundle called the optic radiation to layer IV (one of six layers) of primary visual cortex. A schematic representation of these pathways was shown in notes for an earlier lecture.RetinaEarlier we noted that retinal ganglion cells have a characteristic center-surround organization with excitatory centers and inhibitory surrounds (or inhibitory centers and excitatory surrounds). We modeled the spatial output of the retina as a linear filter that convolves the input image with a kernel determined by the center-surround receptive field weights--a so-called single channel model, because the kernel is assumed to be the same shape and size at different locations. The spatial frequency bandpass characteristics of the retina are determined by just one kernel.The left figure shows contrast thresholds for various light levels (from van Nes, & Bouman, M. A. (1967). Spatio modulation transfer in the human eye. J Opt Soc Am, 57(3), 401-406). The right figure is a replot of the left figure from: Atick, J. J., & Redlich, A. N. (1992). What does the retina know about natural scenes? Neural Computation, 4(2), 196-210. The solid lines show fits by Atick & Redlich based on an efficient coding model.12.SpatialCodingEfficiency.nb 3The left figure shows contrast thresholds for various light levels (from van Nes, & Bouman, M. A. (1967). Spatio modulation transfer in the human eye. J Opt Soc Am, 57(3), 401-406). The right figure is a replot of the left figure from: Atick, J. J., & Redlich, A. N. (1992). What does the retina know about natural scenes? Neural Computation, 4(2), 196-210. The solid lines show fits by Atick & Redlich based on an efficient coding model.The retina's temporal processing can also be thought of as differentiation, but in time rather than space, and can be mod-eled as a band-pass temporal frequency filter (see Enroth-Cugell and Robson, 1966). Analogous to the spatial frequency selectivity, retinal ganglion cells pass the contrast of medium temporal frequencies more effectively than either low or high frequencies. For a retinal ganglion cell, contrast sensitivity as a function of temporal frequency is an inverted U, qualitatively similar to the spatial CSF. Humans are insensitive to temporal frequencies higher than the temporal cut-off (for humans about 50-80 Hz, depending on the mean light level). That is why TV frames (60 Hz interlaced) or computer displays (now usually >70 Hz) are not seen to be flickering. An extreme consequence of the low temporal frequency attenuation, is that an image that is held stationary on the retina dissappears. A VLSI retina having similar spatial and temporal filtering properties was first built at Caltech by Mead and colleagues in the late 1980s (Mead, 1989). At the retina, one begins to see evidence for multiple visual pathways for spatio-temporal information. In cats, ganglion X-cells have smaller receptive fields and poorer temporal resolution than Y-cells, suggesting that the X channel carries information important for fine spatial detail, and the Y-cell channel conveys coarse-scale spatial information quickly. There is a similar distinction in primates, the, so-called magno-cellular (homologous to Y-cells) and parvo-cellular (homologous to X-cells) cells and pathways.Human temporal contrast sensitivity functions.4 12.SpatialCodingEfficiency.nbFunctions of the Chiasm and LGNThe optic chiasm routes neuronal information so that information from corresponding points on the left and right eyes can come together at cortex for binocular vision, and in particular stereo vision. Typically animals with frontal vision have nearly complete cross-over, and animals with lateral eyes (e.g. fish) have little or no cross-over. The nervous system has gone to considerable length to bring information from the two eyes together early on. This suggests that certain kinds of cortical computations cannot easily be done "remotely", but require close connectivity between neurons, and the resulting topographic maps. The neurons of lateral geniculate nucleus do more band-pass filtering, and