{"id":18301,"date":"2011-09-06T10:45:33","date_gmt":"2011-09-06T10:45:33","guid":{"rendered":"http:\/\/www.labri.fr\/perso\/barla\/blog\/?p=18301"},"modified":"2012-07-09T16:03:16","modified_gmt":"2012-07-09T16:03:16","slug":"vision-images-signals-and-neural-networks","status":"publish","type":"post","link":"https:\/\/www.labri.fr\/perso\/barla\/blog\/?p=18301","title":{"rendered":"Vision: images, signals and neural networks"},"content":{"rendered":"<p id=\"top\" \/><em>Jeanny H\u00e9rault<\/em><\/p>\n<h4>The visual system of primates<\/h4>\n<ul>\n<li>The occipital lobe is the main entry point and the main region which is specific of visual information. The superior colliculus is a structure involved in the control of eye movements. About 10-20% of the optic nerve&#8217;s fibers is routed to the superior colliculus (SC), making it aware of where something of possible interest is. As a reflex the superior colliculus sends an order to orient the eyes toward the object. This constitutes a first loop Eye-SC-Eye. The remaining 80-90% of the fibers reach the lateral geniculate nucleus (LGN) and from there is routed to the primary visual cortex V 1. The superior colliculus receives also a feedback from the primary visual area, which tells where, in the visual field, a texture, or a color is different from its surroundings, initiating a movement of the eye toward this location. This constitutes a second loop: Eye-LGN-Vl-SC-Eye. Further, the lateral intraparietal area (UP) sends a more elaborated information to SC, for example in order to verify if an object out of the visual field is still here. This is the third loop: Eye-LGN-Vl-LIP-SC-Eye. All these three loops of controls to SC work unconsciously. The only conscious control is sent by the frontal eye field (PEF) to SC to make a voluntary saccade in order to explore a given position or to do a pursuit movement in order to follow a moving object. p.47\u00a0 <em>This is probably an important stream of connections for probing the environment.<\/em><\/li>\n<li>There are three kinds of large eye movements: Saccades: they rotate both eyes and bring the image of interest onto the fovea, for example when reading this book. Due to the very low inertia of the eye, saccades may be very fast, up to 600 degrees per second . Vergence: it is an eye movement caused by looking at different objects at different distances. We talk about Convergence or Divergence, depending on the fact that we look from far to near or from near to far. Pursuit: if an object that we are looking at moves slowly, a pursuit movement is generated in order to keep its image still on the retina. p.51There are also three kinds of miniature eye movements: Drift movements: they are slow, with amplitudes of around 2-5 minutes and velocities around 1 minute of arc per second. Drift movements of each eye are not correlated. Microsaccades: they have the same function as the large saccades. They are primarily corrective, they occur at the end of each drift movement. Tremor: the amplitude of tremor is very small, about 5-10 seconds of arc, not far from intercone distance. p.51<\/li>\n<li>At the center of the fovea there is a smaller area: the foveola where the rods are absent. That is the reason why you cannot see a small star at night when looking directly at it, but you will see it when looking next to it. S cones are missing also in this area. p.54 <em>Some kind of nigt vision blind spot.<\/em><\/li>\n<li>Spatial coupling: Neighboring photoreceptors are coupled by electrical synapses called &#8220;Gap Junctions&#8221; (Attwell et aI., 1984): by this mean, a partial sharing of their internal potential, the photoreceptors&#8217; output image is a slightly smoothed version of the retinal image. p.55<\/li>\n<li>Adaptation: The range of light conversion by photoreceptors is limited to 1.5 decade. Because the range of light arriving on it is of the order of 6 decades, a photoreceptor adapts to the mean ambient Lighting, coding for 1.5 decade around this mean Lighting. p.55<\/li>\n<li>In other words, a photoreceptor (a bipolar cell) receives a negative feedback (feed-forward) signal from the network of horizontal cells, which extends over a wide spatial area. This action is in fact the origin of the processing of visual signals by the OPL: from an input signal (cones), we subtract a smoothed version of itself, this produces the shape of the receptor field of bipolar cells. [&#8230;] In fact, the processing which is operated here is a little more complicated because the cells have a temporal response.\u00a0 p.57-58<\/li>\n<li>The circuit for the S-cones is particular (no OFF bipolar cell, specific horizontal cell type). This has some implication for color processing. p.59<\/li>\n<li>Ganglion cells carry wide band signals, they are high-pass in spatial frequency, low-pass in temporal frequency and their response exhibits inseparability of time and space variables (Beaudot et at., 1993). They transmit color signals with chromatic opposition: Red\/Green or Blue\/Yellow. [&#8230;] Their response concerns the low spatial frequencies of the retinal image, with a temporal aspect markedly transient, probably due to the interaction with amacrine cells. p.62-63<\/li>\n<li>Summary: Output Signals of the Retina: It seems that first of all the Y ON-OFF cells tell the brain where and when something is happening in the visual field. Secondly, the ON and OFF Y cells send a gross signal giving the outlines of the visual scene. Then the signal of the X cells, providing the missing details arrives. This scheme might be of interest in a kind of &#8220;coarse-to-fine&#8221; process. After a first stage of processing, the retina divides the information into several streams that reach the brain at different time slots. p.64<\/li>\n<li>In fact, the retinal cells constitute only 10% of the LGN inputs. The remaining 90% are issued from VI and brain stem (non-visual inputs). p.66 <em>LGN seems to be like a hub.<\/em><\/li>\n<li>Among the possible functions of the LGN, one is thought to be the enhancement of information about contrast. [&#8230;]Another fact that may be of importance is that there exist two kinds of cells, which are temporally different: lagged and non-lagged cells. [&#8230;]From a theoretical point of view, these cells are supposed to help the temporal decorrelation of the visual information. p.67 <em>If LGN is like a hub, it could also act as a manager that attributes more spatial and temporal contrast based on higher levels of visual processing.<\/em><\/li>\n<li>As the retina and the LGN appear as signal-conditioning units, area VI can be seen as a signal analyzer. It maps the visual field according to a particular topology and decomposes each part of the retinal image into many local features. p.68 <em>Makes a catalog of potential features that are later combined\/manipulated for a specific task.<\/em><\/li>\n<li>The cortical structure of the whole brain is organized in functional columns orthogonal to the cortical layers. p.70<\/li>\n<li>Each ocular dominance column contains 15-20 orientation columns covering the full range of orientations. In these columns, a subgroup of cells is sensitive to the direction of displacement of the stimulus. p.71<\/li>\n<li>Spatial frequency columns: Columns that are sensitive to orientation are segregated into spatial frequency, again in a columnar organization. [&#8230;] The frequency selective columns are organized as concentric coronas, the iso-orientation columns being radially dispatched. p.72<\/li>\n<li>Blobs of cytochrome oxidase are regions located in the center of the pinwheels, where cells coding for color are found. They code almost exclusively for red\/green and blue\/yellow chromatic oppositions and not for luminance. p.72<\/li>\n<li>Simple cells respond to stimuli in a linear manner and they are well represented by Gabor filters. p.72<\/li>\n<li>Complex cells respond to stimuli in a non-linear manner. Their response to an adapted stimulus is the same whatever its position within the receptive field of the cell is. For simple cells, the response follows the position of the stimulus. Their response is well simulated by the sum of the squared responses of in-phase and in-quadrature simple cells. p.72<\/li>\n<li>It is well known that area V 1 receives modulatory feedback from other areas. The result is that the receptive fields of cells may change dynamically according to the context. p.73<\/li>\n<\/ul>\n<p><a href=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p47.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-thumbnail wp-image-18621\" title=\"p47\" src=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p47-150x150.jpg\" alt=\"\" width=\"150\" height=\"150\" \/><\/a><\/p>\n<h4>Basic model of the retina<\/h4>\n<ul>\n<li>The circuit of the electrical model of the membrane can be reduced to a unique voltage generator Eg in series with an internal resistor rg feeding in parallel the membrane resistance. rm and the membrane capacitance Cm. p.83\u00a0 <em>Represents exchange of different ions across the membrane of photoreceptors!<\/em><\/li>\n<li>The two connected cells share the same part of the membrane so that the potential in one cell is related to the potential in the other one, through a simple resistor R. p.84<\/li>\n<li>Both input and output signals are said to be spatiotemporal. They depend on the discrete spatial variable k and on the continuous variable t. p.85<\/li>\n<li>In other words, the form of the spatial response depends on the temporal content of the input signal, and vice versa. The form of the temporal response depends on the spatial content of the input signal. As a consequence of this inseparability, the behavior of this circuit will be of special interest: The spatial filtering is not static: it will evolve with time after a stimulation has been applied. When stimulated by an input signal where time and space are coupled, for example a moving stimulus x(k, t) = x(k &#8211; vt) of velocity v, the response will be a motion signal of the variable k &#8211; vt with the same velocity. p.87\u00a0 <em>It&#8217;s tracking-designed!<\/em><\/li>\n<li>Models of the Outer Plexiform Layer: According to the biological data, there are two possible structures of connections. In the feed-forward model, the horizontal cell inhibits the ON bipolar cell and excites the OFF bipolar cell. In this model, the signal is the difference between the ON and OFF cell. In the feed-back model the horizontal cells return an inhibiting signal back to the photoreceptors. p.88<\/li>\n<li>In both cases, we observe that, under an overall low-pass behavior in the region of high frequencies (imposed by the photoreceptors), there is a high-pass function in the region of low spatial and temporal frequencies (due to the inhibitory action of the horizontal cells). The global shape of this transfer function in the spatial frequencies, as in the spatiotemporal I frequencies is that of a volcano. p.90<\/li>\n<li>Why are the photo receptors coupled in a low-pass filter? The model to be considered is a set of ideally identical receptors receiving the input image and applying to it additive and multiplicative noises, representing the mismatch between individual thresholds and gains. Because this noise changes independently from cell to cell, it represents a random signal in the range of very high spatial frequencies. p.91<\/li>\n<li>Why does the retina behave as a high-pass filter? The main function of spatial high-pass filtering is precisely to enhance contours. Another fact to consider is that objects&#8217; surfaces may be submitted to a large range of illumination, widely changing their luminance distribution and appearance, whereas the contours remain more unchanged. [&#8230;] Besides, the statistics of natural images tells that their mean frequency spectrum is shaped in 1\/ f, either for the spatial frequencies in static images or for the spatiotemporal frequencies in moving sequences. The role of the retinal high-pass filter is to compensate this 1\/ f spectrum. In signal processing, this technique is called &#8220;spectral whitening&#8221; p.92<\/li>\n<li>Consequence of Time-Space Inseparability: It is interesting to observe that just after the light is turned on, the spatial behavior of the OPL is low-pass. It progressively becomes high-pass as time progresses. When light is turned off, a negative rebound appears only for the very low spatial frequencies. This behavior explains why the names &#8220;ON&#8221; and &#8220;OFF&#8221; are given to the two kinds of bipolar cells. p.93<\/li>\n<li>The Feed-Forward &#8211; Feed-Back Model: The Figure shows a comparative study of the three models of horizontal cells connectivity (feed-forward, feed-back and compound). The three curves of spatial transfer functions are scaled to have the same maximum. We can see how the compound model allows the widest range of transition from the region of low frequencies to the region of frequency of the maximum transfer. This is of utmost importance if we consider that one of the roles of the retina is to compensate for the 1\/ f spectrum of natural images. p.96<\/li>\n<li>The Midget and Diffuse Bipolar Model: The diffuse bipolar cells are known to connect several photoreceptors, the number of connections depending on the eccentricity of their position. In the simplest way, they can be modeled by a purely spatial low-pass filter with a space constant linked to the eccentricity, in cascade with a purely temporal filter linked to the cell&#8217;s membrane time constant. p.96 <em>Might be used to initiate tracking.<\/em><\/li>\n<li>Midget and Parasol Ganglion Cells: The midget ganglion cells are connected to only one bipolar cell and thereby present exactly the same transfer function as the midget bipolar cells. [&#8230;] The parasol ganglion cells connect several bipolar cells (depending on retinal eccentricity). Due to this connection scheme, they will be represented by a purely spatial low-pass filter in series with a purely low-pass temporal filter (membrane time constant). p.97 <em>What is their purpose if their trnasfer function is the same as those of bipolar cells? <\/em><\/li>\n<li>The Amacrine Cells: According to the properties of the magnocellular pathway, they seem to play an important role in relation with the parasol ganglion cells. They are likely to enhance the transient aspect of these cells, providing them with a response anticipating the one of the parvocellular pathway. p.98 <em>They may be thought to boost\u00a0 reaction times by anticipating processing in the parvocellular pathway<\/em>.<\/li>\n<li>ON, OFF and Bi-Stratified Ganglion Cells: The midget and parasol ganglion cells are connected to the bipolar and amacrine cells in the sub-layer of the IPL corresponding to their ON or OFF characteristic. These cells are known to present a rather linear response (the response to the sum of two signals is the sum of the responses to each signal presented individually) they are called &#8220;X type&#8221;. There is another type of ganglion cells called &#8220;bi-stratified&#8221; because they make connections in both sub-layers. These cells exhibit a non-linear response and, in particular, they respond as well to the onset of light as to the offset. In cats, they are called ON-OFF cells or Y cells. [&#8230;] The role of the Y ganglion cells is of major importance in signaling relatively large objects or objects which suddenly appear. They project to the VI area, but also to the Superior Colliculus and to archaic visual structures. They may be involved in automatic tasks as gaze control or obstacle avoidance.\u00a0\u00a0 p.98 <em>Is there any kind of gain control appart from linear\/absolute effects?<\/em><\/li>\n<li>p.99<\/li>\n<li>Due to the temporal high-pass filter added by the amacrine cells of the IPL, there is a hierarchy between the times of arrival of these signals to the cortical area: the magnocellular system reaches the cortex in advance with respect to the parvocellular one. The first wave is the one of Y cells signal. It seems to warn the brain when (and where) something is happening in the visual field. The second wave is the X-magnocellular signal that gives a rough idea of the scene components (i.e. a smoothed image where only the blobs of activity are present). And the third wave is the one of the parvocellular signal that brings information about the details of the picture and also about color. In short, the three phases are: warning information, coarse information, then detailed information. p.101 <em>Amacrine cells seem to act as delays.<\/em><\/li>\n<\/ul>\n<p><em><a href=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p82.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-thumbnail wp-image-18631\" title=\"p82\" src=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p82-150x150.jpg\" alt=\"\" width=\"150\" height=\"150\" \/><\/a><a href=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p87.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-thumbnail wp-image-18641\" title=\"p87\" src=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p87-150x150.jpg\" alt=\"\" width=\"150\" height=\"150\" \/><\/a><a href=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p97.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-thumbnail wp-image-18651\" title=\"p97\" src=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p97-150x150.jpg\" alt=\"\" width=\"150\" height=\"150\" \/><\/a><a href=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p99.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-thumbnail wp-image-18661\" title=\"p99\" src=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p99-150x150.jpg\" alt=\"\" width=\"150\" height=\"150\" \/><\/a><br \/>\n<\/em><\/p>\n<h4>Neuromorphics circuits and motion estimation<\/h4>\n<ul>\n<li>Frequency Domain Approaches: Let us start with the model of an image submitted to a translation movement. [&#8230;] This means that all the energy of the translating image is contained in a plane of the spatio-temporal frequencies domain. [&#8230;] With two spatial dimensions, the information about direction and intensity of the velocity vector is contained in the orientation of the frequency spectrum. p.114-115<\/li>\n<li>More General Cases. The Image May Vary in Time: The spectrum of the moving image lies no longer on a plane of equation ft + vx fx + vy fy = 0 in the spatio-temporal frequency the co-ordinates. [&#8230;] The spectrum of the moving image seems (only seems) to rotate in a direction ft + v fx = 0 compatible with motion, BUT its axis is NOT collinear with the motion direction as would be expected. [&#8230;] If the original image would have been of narrower temporal spectrum extent, its moving spectrum would have been more coaxial with the motion direction. p.118 <em>It&#8217;s the case of a changing environment with a moving observer (in quite slow motion).<\/em> <em>The lower the temporal frequencies of the signal, the closer the  spectrum is rotated along a plane which properly describes motion.<\/em><\/li>\n<li>If the test function, instead of being a Dirac (infinite length and zero width) oriented at the velocity Vtest. is a low-pass spatia-temporal filter, that is, with a finite length and a non-zero width, the same result is obtained. We will call such a filter a &#8220;velocity-tuned filter&#8221; (VTF). p.120 <em>Is the role of VTFs to probe time-varying images in ego-motion? <\/em><\/li>\n<li>Motion Transparency: Multiple motions happen in various cases, for example a moving shadow cast on a stationary surface, a moving object partially occluded by foliage in the wind, or a moving object seen by reflection when looking through a glass wind, or a moving object seen by reflection when looking through a glass window.These cases of motions are called motion transparencies. The process can be additive (glass window) or multiplicative (cast shadow). p.120 <em>With multiple VTFs, one could carry motion transparency to higher cortical areas for further processing (via population coding?).<\/em><\/li>\n<li>Estimation of Local Energy: two properties of the VTF&#8217;s response may be used to estimate velocity: the spatial shape of the response and the output energy of the filter. The spatial shape is not relevant because it varies according to the image&#8217;s shape. The output energy is much more interesting because it is largely independent on the image&#8217;s shape. p.129<\/li>\n<li>Combination of VTF: Now, for each velocity-tuned filter, we know the local energy. How is it possible to find the velocity vector. A solution could be to take a lurge number of selective VTFs, each of them tuned to a particular velocity. [&#8230;] We will prefer a solution widely used in nature: a limited number of loosely tuned filters. This solution is also used in radio-engineering to discriminate frequency modulation: take the difference of the signals from two filters tuned to two neighboring frequencies. [&#8230;] Because we want to obtain a result independent of the input signal, we normalize this difference by the sum of the output energies. [&#8230;] A better formula provides full linearity by introducing a third filter tuned to optimal velocity zero. p.131 <em>Does the HVS rely on such 2\/3 tuned combinations, or rather on a population response coding, or something else?<\/em><\/li>\n<li>The Aperture Problem. First Solution &#8211; Spatial Integration: the solution seems now easy to derive: if there is no aperture problem, the integration window should be small to get the best accuracy, and if the aperture problem is present, the integration window should be enlarged until the problem disappears. [&#8230;] However, this is far from being sufficient. Because the integration window is large, new problems may happen in the presence of multiple objects moving in different directions. p.133<\/li>\n<\/ul>\n<h4>Color processing in the retina<\/h4>\n<ul>\n<li>How to Build a Retina. Considering the statistical distribution of wave-length spectra for natural scenes, a peA on these data shows that any spectrum can be represented by a weighted sum of three spectral, preserving at least 90% of variance. p.144<\/li>\n<li>Choice of the Spatial Distribution of Spectral Sensitivities. In fact, if we look at the spatial details available according to wave-lengths in natural scenes, we see that the longer wave-lengths (in the range of green to red) carry much more details than the shortest wave-lengths (in the range of violet to green). [&#8230;] Consequently, in order to preserve spatial accuracy, the spatial density of long wave-length receptors should be much higher than that of short wave-length receptors. In other words, there is no need to densely sample an image in the range of blue light (recall Shannon&#8217;s theorem on sampling). p.145<\/li>\n<li>What Happens in the Cones Layer. Under the hypothesis of cones coupled only within the same type, the cone circuit is equivalent to three independent and non-overlapping low-pass filters, the output of which are multiplexed the same way. p.150 <em>Any foreseeable implication on<\/em> <em>the perception of paradoxical colors?<\/em><\/li>\n<li>According to this model, it appears that there is absolutely no need of color processing in the retina: as the luminance is spatio-temporally high-pass filtered, the chromatic information goes through the retinal circuitry almost without any perturbation. If this is true for the biological retina (and it seems to be), it has an important consequence on the economy of neural matter in the optic nerve part, corresponding to the fovea: instead of transmitting three signals (luminance plus two chromatic oppositions) for each receptor, there is only one Midget cell conveying both luminance and chrorninance in a multiplexed scheme with its neighbors. This represents a 66% savings of neural matter. p.152 <em>It may explain the importance of luminance (as opposed to color) gradients in image processing&#8230;<\/em><\/li>\n<li>Emergence of Chromatic Oppositions. These signals represent what is called &#8220;chromatic oppositions&#8221; in neurophysiology and in psychophysiology. [&#8230;] Observe that for the S photoreceptor, the chromatic opposition signal is not directly weighted by Ps, which provides it with the same relative importance than the others, even if the S cones are in small number! [&#8230;] We have, facing the L cones a mainly &#8220;L &#8211; M&#8221; (Red-Green) component, facing the M cones a mainly &#8220;M &#8211; L&#8221; (Green-Red) component and facing the Scones a mainly &#8220;S &#8211; (L + M)&#8221; (Blue-Yellow) component. p.158-159 <em>Therefore the small quantity of S cones has more an impact on the kind of opponency than on the ability to perceive blue colors.<\/em><\/li>\n<\/ul>\n<h4>Non-linear, irregular and non-stationary processes<\/h4>\n<ul>\n<li>The random sampling, though adding noise in high frequencies, is essential in order to reduce the aliasing. This is particularly important for the retinal coding of color, where the sampling is very sparse, especially for the blue cones. p.185<\/li>\n<li>The Log-polar model of retina-cortical projection presents an interesting . property when considering computational efficiency. The simple estimation of cortical horizontal and vertical mean velocities, allows the estimation of four ego-motion parameters, respectively: \u00b7 In the region corresponding to the foveal area, these make it possible to estimate translation motions of the retinal image. In the region corresponding to the retinal periphery, these estimate looming and rotation motions. p.187 <em>It could also be used to guide attention to image features during saccades&#8230;<\/em><\/li>\n<li>Adaptive Non-Linearity in IPL. The simultaneous record of a ganglion cell exhibits an adaptation behavior with a different time constant, according to the sign of the contrast change. p.196 <em>ON &amp; OFF cells are not only here because of biochemical limitations, but they may also be used to deal differently with bright and dark stimuli!<\/em><\/li>\n<\/ul>\n<h4>Cortical processing of images<\/h4>\n<ul>\n<li>In the primary visual area V1, the retinal signals are filtered by simple (in-phase and in- quadrature) and complex (energy dependent) cells. Each cell responds to One central spatial frequency and one orientation. They are organized into micro-columns of similarfrequency and orientation. There are approximately 6-7 central frequencies (on a log-scale) and 15-16 orientations (between 0\u00b0 and 180\u00b0). Micro-columns are organized in a pin-wheel fashion within a macro-column. To put it briefly, after retinal preprocessing, the task of V1 complex cells can be viewed as a log-polar sampling of the energy spectra of local patches by band-pass oriented filters over the whole visual field. p.216<\/li>\n<li>Log-Polar Representation of Frequency. It is known since De Valois (1991) that the frequency bandwidth of cortical filters increases with their central frequency, whereas their orientation bandwidth remains almost constant, This suggests that if the frequency bandwidth\/ central frequency ratio is constant, the progression of central frequencies is linear on a logarithmic scale. [&#8230;] This log-polar representation is particularly well suited to treat image zooms and rotations. If an image is zoomed (x -&gt; ax), its frequency spectrum is also zoomed (f ~ f\/a) and the effect in log-polar domain is a horizontal translation of -log(a). Similarly, if the image is rotated, its frequency spectrum is rotated by the same angle, and in the log-polar domain, it corresponds to a vertical translation. p.223-224<\/li>\n<li>Eq 7.9\u00a0 to 7.11 p.225 [viz]<\/li>\n<li>The Log-Normal Model. That is, zooms and rotations are merely transformed into pure translations on the filters. Furthermore, these filters being made steerable if their number is odd, they allow either invariance with, or pursuit of zoom and rotation. This property would not be reached without the 1\/ f factor. p.226<\/li>\n<li>It should be noticed that the Log &#8211; Normal function has no simple analytic form (Leipnik, 1991), so the shape of the impulse response can be derived only by computer simulation. p.228<\/li>\n<\/ul>\n<p><a href=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p223.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-thumbnail wp-image-18671\" title=\"p223\" src=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/wp-content\/uploads\/2011\/09\/p223-150x150.jpg\" alt=\"\" width=\"150\" height=\"150\" \/><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Jeanny H\u00e9rault The visual system of primates The occipital lobe is the main entry point and the main region which is specific of visual information. The superior colliculus is a structure involved in the control of eye movements. About 10-20% of the optic nerve&#8217;s fibers is routed to the superior colliculus (SC), making it aware &#8230; <a title=\"Vision: images, signals and neural networks\" class=\"read-more\" href=\"https:\/\/www.labri.fr\/perso\/barla\/blog\/?p=18301\" aria-label=\"Read more about Vision: images, signals and neural networks\">Read more<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[621],"tags":[],"class_list":["post-18301","post","type-post","status-publish","format-standard","hentry","category-books"],"_links":{"self":[{"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/posts\/18301","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=18301"}],"version-history":[{"count":11,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/posts\/18301\/revisions"}],"predecessor-version":[{"id":22721,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=\/wp\/v2\/posts\/18301\/revisions\/22721"}],"wp:attachment":[{"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=18301"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=18301"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.labri.fr\/perso\/barla\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=18301"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}