Computational VisionU. Minn. Psy 5036Daniel KerstenLecture 12: Coding Efficiency, Spatial statisticsInitialize‡Read in Statistical Add-in packages:In[11]:=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;In[13]:=nbinfo = NotebookInformation@EvaluationNotebook@DD;dir =H"FileName" ê. nbinfo ê. FrontEnd`FileName@d_List, nam_, ___D ßToFileName@dDL;‡HistogramIn[39]:=myhistogram@image_D := Module@8histx<,Needs@"Statistics`DataManipulation`"D;histx = BinCounts@Flatten@imageD, 80, 255, 1<D;Return@N@histx ê Plus üü histxDD;D;‡EntropyIn[15]:=entropy@probdist_D := Plus üü HIf@Ò == 0, 0, -Ò Log@2, ÒDD & êü probdistLImport image fileImport with a dialog box:granite = Import@Experimental`FileBrowse@FalseDD@@1, 1DD;Or put granite64x64.jpg in the same directory as this notebook, and use this:Or put granite64x64.jpg in the same directory as this notebook, and use this:In[16]:=granite = Import@"granite64x64.jpg", Path Ø dirD@@1, 1DD;In[17]:=N@Mean@Flatten@graniteDDDN@StandardDeviation@Flatten@graniteDDDwidth = Dimensions@graniteD@@1DDOut[17]=129.524Out[18]=19.7279Out[19]=64OutlineLast time‡Understanding intensity statistics, and point non-linearities in terms of efficient coding of natural imagesToday‡Form and function: overview of visual pathway‡Spatial statistics and efficient codingWe've learned about localized spatial frequency filters in early vision. We now ask: Why?2 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/neurobiology/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.12.SpatialCodingEfficiency.nb 3RetinaEarlier 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 retina's temporal processing can also be thought of as differentiation, but in time rather than space, and can be modeled as a band-pass temporal frequency filter (see Enroth-Cugell and Robson, 1966). Analogous to the spatial fre-quency 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 (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.nb12.SpatialCodingEfficiency.nb 5Functions 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 the cells are characterized by fairly symmetri-cal center-surround organization like the ganglion cells. They show even less response to uniform illumination than ganglion cells. Despite the fact that neurons from the two eyes exist within the same nucleus, no binocular neurons are found in LGN. We have to wait until cortex to see binocular neurons. The X- and Y-cell division of labor continues with the so-called parvocellular (with corresponding retina input from P cells in monkeys, or X cells in cats), and the magnocel-lular (Y cells or M cells) pathways. Again the experimental measurements are consistent with the idea the the M pathway