Mapping and Geospatial Data Analysis Using MATLAB

hello and welcome my name is Alan Wong and I work here at the math works and Technical Marketing I work with image and video processing products including mapping tool box so today I'm going to be discussing mapping and geospatial data analysis using MATLAB now there's quite a few things I like to cover so let's go ahead and get started with the agenda now the first thing we'll take a look at is an overview of mapping toolbox I like to go over the key features and functionality that's built into the product just give you an idea of the types of problems can help us solve but the majority of our time today will actually be spent inside MATLAB in the following three demos an oil spill simulation a weather avoidance demo and a train analysis demo now each of these will highlight different parts of the toolbox but will give us an idea of how we can use MATLAB and mapping toolbox to solve a geospatial problem after that we'll come back into PowerPoint and take a look at some related toolboxes and if there's any questions I'll be glad to answer them at the end in a question and answer session and so with that let's go ahead and get started the first thing I want to take a look at is an idea of how someone works with geospatial data and here we have the technical computing workflow which divides the process into three main stages at the beginning on the left hand side we need to access our geospatial data in some way and they can be stored in a file in a database or perhaps on the internet but we need to grab it in some way and bring it over to the second stage where we can process it and so in this stage we you might do some data analysis or modeling or algorithm development but basically process our data in some way to produce some output or extract some information and we pass those results over to the last stage where we share it in some way either by creating a report maybe creating a map visualization deploying an application or even just writing our process data out to a new file format so you can share it with other GIS software ideally we like to improve our productivity by automating the entire process and this is how MATLAB and mapping tool box can help us out when working with geospatial data and we'll see this in action as we get to those demos later on now the next thing I want to look at is where we see this workflow happening and so on this slide I've listed out a number of applications and industries that make use of geospatial data so whether you're working with aerial imagery and aerospace and defense sea temperature data or natural resource information or even weather data in the financial industry we see that there's this access processing and sharing is a really common way to work with geospatial data so all of these industries but they don't have much to do with each other the types of tasks and problems that they were solving are really quite in common and so this is really where mapping toolbox can come and help us out so here I've listed out the key features of the product and how they relate to those three stages of the technical computing workflow from importing exporting Geographic data in that access stage to creating 2d and 3d maps like on the right hand side and geospatial analysis and analyzing terrain data we'll go into each of these in more depth over the next few slides but I just want to give you a high-level view of what kind of capabilities the mapping toolbox can provide to us so let's go ahead and dive into that first bullet import and exporting Geographic data now there really are a lot of file formats that the product supports and if you want to get a comprehensive list you could I recommend going to our website at WWF works comm you can navigate over to the product page for mapping toolbox and there you'll find full documentation for all the functions inside the product and that includes a list of all the file formats that we support now I've listed here some notable ones that we support such as reshaped files KML files geo TIFF and digital elevation models we'll take a look at the actual functionality in practicing in the demos that are on how we actually read data into MATLAB but I want to focus a little bit more on that last bullet importing data from web map service or WMS servers now this is new functionality to mapping toolbox that came out in 2000 our 2009 b and for those who aren't familiar with WMS it's a protocol that was developed by the open geospatial consortium for serving rendered data render maps over the internet so there's a lot of organizations out there such as NASA USGS and NOAA that provide these WMS servers and provide data layers to the public so on the right hand side we've got an example data layer of the Larsen ice shelf and it's collapsed over I believe a few months period so this is an animated data layer and I've really seen a lot of a wide range of data layers available through WMS from air pollution to sea temperature as I mentioned before here we've got the Larsen I shelf we've got natural natural resources there's really a huge range of data available on the internet and to help you find the data set that's applicable to your to your problem the mapping toolbox provides a pre-qualified database of WMS servers as well as search functionality to help you narrow down quickly what you're looking for and so I think this is a great opportunity just let's go ahead and jump into MATLAB and show you how you can use this database and the search capabilities to find data through WMS so what I'm going to do here is use the WMS find function which is the function to search through this quality qualified database if so you can do is use a number of key words server URLs geographical locations and whatnot to basically find some data layers that you can use so for Mona if we pretend we're in Canada and we want to look for maybe in our kin which stands for Natural Resources Canada we can specify our search field to be server URLs so any server URL that has NR can inside the name let's return those servers and I returned them here into this variable a in which we can look at the size and see there's nearly 4,000 data layers out there for us to work with so let's go ahead and take a look at the top three and we see that there's four the first top three there's variety of data layers on mineral deposits the first one has to do with gold deposits so let's go ahead and take a look at that one now all the data here is already stored in this database that comes with mapping tool box if we want to update it you can use the WMS update function and let's go ahead and store this into a new variable B and what this will do is communicate with that server and download the most up-to-date information as you can see here we've downloaded an abstract that basically tells us a little bit more about this on data layer so it's something with the bedrock hosted gold deposits let's go ahead and just download this data and see what what it looks like so here you can use the WMS read function pass in the server name and matlab's go out and gone ahead and download our data into the workspace so what I can do is just do a quick imshow on this variable C and display this image remember these are rendered Maps coming down from these servers so we have a bunch of these kind of goldish is showing where these gold deposits are now let's go ahead and put this on to a map you might have noticed that this second variable over here are it's actually the referencing matrix it tells us how this image is oriented on a map so what I can do is let me first load in a coastline and we'll create a Mercator map axis and I'll just go ahead and plot those coastline down onto our map so now we have a little bit of reference of where these gold deposits are going to be so what I can do is type hold on and use the Geo show function to display our gold deposits using this referencing matrix and if i zoom in we can see it's already aligned our gold deposits for us to show where they where they're located so that's a quick way to find data using these WMS fine functions there's also a number of search routines to help you narrow down your search but basically it's very quick to download data into MATLAB and display it in a map so let's go ahead and now that we've seen in a quick example of accessing web map service service let's take a look at the next key feature inside mapping toolbox which is creating 2d and 3d maps so we already saw this a little bit in that quick demo but there's a lot of map displays that mapping toolbox supports so you can display point data line data polygons imagery on the right hand side here we have the Blue Marble image which was downloaded in via WMS in this case it's been rendered in many different map projections and this is one of the key things that mapping toolbox provides the ability to manage map distortions with over-65 map projections as part of creating maps you need to analyte 8 or customize your map so here I'll also mentioned a little bit about the markers colors skew rulers and North arrows that you can use and we'll see this in more depth as we go into our first time at the oil spill demo before we're creating well where we will create some maps the next thing I like to take a look at is once you imported your data into MATLAB we need to go to that second stage where we process it in some way and so here I've listed out a number of geospatial analysis functionality such as calculating distances coordinate transformations navigational calculations unfortunately don't have too much time to go into depth with each of these so I find a good way to talk about them is just to look at the images on the right where different geospatial analysis functionality is being highlighted in the top image we're basically calculating a buffer zone so if you want to calculate may be a threat zone maybe five kilometers away from a border you can use one of the functions in bakwin toolbox to do that and here we were doing that for the Great Lakes border and calculating area probably about 100 miles radius around that on the lower right we're plotting some small circles and figure out which polygons are inside of this small circle and so if you need to calculate distances like how far do I need to go to achieve its objective those are the types of things that you can solve inside mapping tool box on the lower left we're doing some 3d coordinate transforms to orient this radar dome onto the surface of the planet to show its effective range so there's quite a few things that you can do and we'll get into more deputies in the actual demos as we move into MATLAB now there's also a lot of functionality to analyze terrain data so this is an image that's that we're going to be seeing a little bit in the terrain analysis demo but basically if you need to calculate things like gradients viewsheds line-of-sight visibility and slopes that type of functionality is available to you in mapping toolbox now another nice thing that you can do is create a virtual reality world from your elevation data so if you need to do things like fly throughs and analyze your data in that way that functionality is available to you as well and lastly there's basically a lot of utility functions that that are that underlies all this analysis capabilities inside mapping toolbox so the basic functions to work with vector and raster data merging lines segmenting them combining and changing the spatial resolutions all those type of utilities are available to you through the toolbox as well and as we'll see in a little bit map trimming is one of the the key functions that we'll look at in the oil spill demo and with that we've basically seen the key features for mapping toolbox from access to processing to you know creating map visualizations so I think this is a great opportunity to just jump into MATLAB and take a look at this some epic functionality in in action so in this first demo what we'd like to do is simulate an oil spill so what we have here is a picture of a Bay and some current fields on it it's what we want to do is pick a location maybe there's a ship that runs the ground over here and there's unfortunately an oil spill and based on the title movements how is this oil slickin to move around inside the bay so that's what this sells the script is going to show us so I've already made this script and divide it up into a number of cells that I'll run from a high level just to show us what's going on inside the simulation so in this first cell we're basically you have that access stage of the computing workflow and we want to import some data into MATLAB so here I use the shape tree function to read in some shape files and then particular this one has some Coast Guard stations in it now I've also nested a number of Chez Paree functions into this custom script over here called import my data and so I'm basically automated the process of importing many shape files and we have that data over here in the upper left in the workspace so here we have some potami tree data some depth information of the water our shipping lanes our shoreline data let's go ahead and just take this information and put it onto a map and visualize our datasets and that's what I do here in this cell I use the USA map function with a particular latitude and longitude extent it'll automatically pick a map projection for me and I use the Geo show function to basically overlay these datasets on top of each other so here's our shoreline of railroads coastguard and so forth so I'll go ahead and let this processing create our map visualization for us and so here we've got basically our vector datasets our shoreline and whatnot all displayed on to this map now as I mentioned before mapping toolbox gives us ability to annotate and customize our map so in this next sole I'd like to customize it by inserting a scale ruler north arrow and some text now what does will do is label our dredge channel so this is where the ship shipping lane are down here as well and also skill ruler north arrow so we can tell from this that our bays about five miles in width now one other thing I'd like to do is put in a little inset map over here to show where this Bay is in relation to the rest of the Texas coastline and so I do that in this cell using similar functionality that we've already used to create this map only I create a secondary map axes as you can see here and basically plot the data into this particular location and I overlay a little red square to show what our primary axis is looking at so this is a pretty quick way to just take our data and create a map visualization of it now the next thing I like to do since this is all vector data I just want to quickly show how you can combine vector data with raster data and so this is what our best imagery data is going to be showing what I've done with this create map function over here is taken some key functions from these visualization cells up here to create a basic map for me to work with and so here I've displayed a latitude longitude and a title as well as just the shoreline data so what I want to do is now take a look at what the depth looks like in this bay over here so that's what the bit symmetry data will help me with if I type of temperature over here we basically have three thousand data points and if I drill down into the first element we see it's basically just point data with a particular latitude and longitude and a depth so what I can do is extract this information from this structure into these variables and I'm just create a new figure and plot out these data points and we see that we've got an a regularly sampled set of data points what we want to do is now interpolate this into a regularly regularly spaced grid and that's what we do over here with the geo look to grid function geo locations to grid it'll go through and interpolate and create a nice regular grid for us to display into a map in this next cell here rather than display the entire the temperature data for the entire Bay and ocean I'm just going to trim it down to this particular latitude and longitude extent and here's where we see some map trimming functionality rather than to split thing we just trim it down and overlay it on to the original image so here I've displayed a color map as well as our data over here so you can see we can now combine both vector and raster datasets together overlay them properly to do it communicate some more information so perhaps as our boat was coming in maybe a ran aground at a certain point over here and that's why the oil spill occurred so this could be information that we might need to insert buoys in particular locations now let's go ahead and move on to the actual oil spill simulation so as I mentioned before what's going to determine how this oil slick is going to move around is primarily title' title forces so what I need to do is create a current field a current model now there's a number of organizations and academic organizations out there that perform these kind of models unfortunately didn't have access to them so I'm going to create them on my own and that's the really nice thing about MATLAB that I can create these models and just test them out with with our simulations so here what I'm doing is again defining a particular location I want to model my my currents and then create a regular grid of data points using the mesh grid function now these regular data points are some might line up on top of land masses so when we figure out what our current fields are we need to make sure that those data points on land is set to zero and that's what I do inside this cell I figure out which points are inside on these shore line polygons and I just set them to a value 1 so I'm creating a binary mask where we have ones inside land masses in zeros elsewhere so here I'm going to display it and so we have here's our ones and our zeros elsewhere the reason why this is important is because I can invert this image so that the land masses have value 0 and the water values have value ones so if I overlay that mesh grid essentially I'll have a number of data points with the current vector for each one and for any one that lines on top of land I can just zero it out so now the next step is how do we find how do we determine our current field well what I did was actually use some functionality in image processing toolbox called ROI poly now what that allows me to do is I can go through and interactively select out a particular location so perhaps at the mouth of the bay I might want this kind of a field like this and what it does is it for this particular location I can now create a vector field such as an outward or inward moving circular vector field and so by entering that process over all these different channels in this Bay I can then create a current field and here I'm just going to display particular time steps in our current model so this is a time varying current vector field and as you can see here's basically as tides moving up we now have a vector field pointing out from this from this mouth and now tides can go back in and you can see it running down these channels up and down so this is a very rudimentary way to basically create a current field and now we have this time bearing field that we can simulate an oil stick oil slick moving around inside so that's what we'll do in this next few cells simulate how that oil spill moves around here I define where that slick is going to originate from and here I'll just create a figure again using our create map function and display a triangle where are our oil particles initially are located so at this point we now have basically a current field and what I can do is use Euler's method to move these particles in particular directions and so in this cell here I basically minimize this for loop but essentially what's happening is for each particle I interpolate from my current field which direction it should move in and move it a tiny step based off of this this time step and if I ate that many many times I can basically create a simulation a movie of how these particles are moving around I just need to capture each figure and that's what I've gone ahead and done I'll go ahead and display this model the simulation using the ion play function which will display this avi file that I've created so if I play it we can have our we can see our oil particles in our current field overlaid on top and we see the title forces are initially pushing it outwards and as part of my scripts I actually highlight any polygon that gets touched by by I'll in red is a here tides move back in we can see that the oil particles are starting to disperse even more and touch even more of these little islands around bay again tidal forces are moving it back out and once these particles reach these islands because the current forces are zeroed out inside the masses they are left here along the edge so we can record all this information let me own and close this basically I was recording where these particles are moving around so that I can compare later on and cross-correlate you know if these islands have oil on it what's what else is being affected and that's what happens here in the following cells I create a new map and overlay some rookeries onto it and these are areas where birds will nest and they're very environmentally sensitive so I've used these great polygons to display where these these birds are nesting what I can do is cross correlate where those oil particles ended up and see which of these have been affected and that's what happens inside the cell using the in polygon function to determine which of these polygons are now affected and we enough hat and I've highlighted them here in green now as similar similarly we might want to figure out within a particular location this oil slick origin maybe we want to know everything inside a 5 nautical mile radius so what we can use is this circle function over here to put these types of objects onto our map and help us better manage maybe cleanup for these types of oil spills so this was a quick introduction to this oil spill demo but basically we're able to create some app visualizations create some numerical modeling and simulate how this oil spill moves around now in this next demo the weather avoidance demo what we're basically going to do is take a look at a couple new different capabilities inside mapping tool box first the ability to access WMS servers here download some backdrop this blue marble image from JPL as well as NEXRAD weather data that you can see overlaid on top of this image what we'll do is plan a fly plat from say San Francisco to Boston and if that flight path goes through any of these storm systems let's try and reroute it so we'll see some navigational function as well as some visualization capabilities inside mapping toolbox in this demo so here we'll go ahead and switch over into the weather avoidance script again this is divided up into a number of cells that I'll run from a high level in this first cell I create a visualization basically create a map axes and the reason why I want to do this I use the USA map function to display a map projection for the continental United States I want to do this to capture the latitude and longitude extent because I want to use these parameters as input to WMS read function later on so in this cell over here I use the WMS find function just like we used before to search for Jet Propulsion Laboratory or JPL servers I refine my search using this method here to look for only the blue marble image and then I can use the ws read function along with these extents that had recorded before to have the server customized image for me and display it so here I'll go ahead and communicate with the server and have it download it into MATLAB and display it on to that map and you can see here I've used the jewish-owned function to display this state borders on to our map just so we have a little reference to what we're looking at now in this next step what I'll do is actually go ahead and dock this image over here and I'll interactively select using the input M function a start and stop so let's start from San Francisco and fly to Boston and now recorded those start and stop destinations and we can now plot some navigation paths onto our figure in this first cell I plot the great circle which is the shortest distance from point A to point B if we were to extend this completely around the world and back it would cut the world completely through this Center into two equal halves and so this is the shortest distance that you can fly on a on the planet but this is not the most convenient way to fly because your heading will constantly be changing instead of you might use rum lines to fly from a certain point a to certain point B with a rum line you fly along a particular heading that's constant and it will take you deviate from the great circle but it's a lot easier to navigate and you can see that it kind of comes down a little bit over here so if you're ever flying from the west coast of the United States over to Asia you will follow these this path that goes up and down from the Bering Strait and this is one of those rum line paths that we're flying along now you can see over here that I've calculated the distances for the Great Circle and rum line you can see that the brown line is a little bit longer so this could become more significant as your flight path increases in length now what I like to do is download our next ride weather data again we use the same functionality of WMS find our refines and WMS read and we play it display it onto a new map and you can see that there is a very very large storm system on top of the Midwest right now so what I'll do is combine these two datasets the backdrop and the next red weather data into a sim into a 1 figure so we've got some really nice visualizations we can go ahead and select a new and start and stop just in case you know the path from San Francisco to Boston didn't work out but it looks like it'll be just fine here so let's fly from San Francisco to Boston and we see that our flight path will take us right through the center of the storm so we might want to reroute around it now to do this we need to figure out where the storms are when to do some segmentation so here I use the image processing toolbox to segment our storm on from the background and so we have white where there is considered a big storm and what we can do is use some morphological operators to basically dilate this and maintain just this big storm system as well as these kind of isolated ones out to the side with that we can then go ahead and calculate the sizes of all those storm systems you can see I'm plotting them out over here we have a very large storm system over here I've also plotted the center of them these are all found using image processing tool box functionality called region props what region props does is I can take that binary image and calculate things like the convex hull and centroid that's what we see here display tonight in our figure the centroid and this convex hull I then use mapping tool box define a little buffer zone around this this border maybe like a no-fly zone so we see that the file path takes us through these two separate storm systems what I want to do then is figure out all the flight paths that I can take to wrap myself around it and here I've created this recursive function that basically goes from a certain point and goes through each of these storm systems figuring out how to get us around it and I define these midway points based off of our initial heading and just turning a 90 degree angle out to the side figure out where that intersection is and creating that is our new Waypoint so this is not the most sophisticated complex way to rewrite around us around these storms as you can see we're kind of hitting these outer edges but with these new paths you can use mapping tool box functionality to calculate how long each of these paths are and the reroute us appropriately so here in gray we have the original path and this new proposed redline flight path that takes us a little bit around these little storms tries to reorient us around this major system over here and so as you can see our new course adds about 100 nautical miles on to our on to our flights and so it does a pretty good job at figuring out how to get around this forum system but in future iterations and if you want to make this more sophisticated what one could do is grab driving as layers for the previous maybe 30 minutes and see how these storm systems are progressing and then adjust these fly paths as necessary you may we may not need to fly out this far if the storm is already going to move away so those are the types of things that you can do with mapping toolbox in this in this field so let me go ahead and clear our data clothes are off figures and then we'll take a look at our our last demo in which case is this terrain analysis demo and we already saw this figure in a previous slide but essentially what we'd like to do here is work with some digital elevation model download some aerial photography from WMS and then drape it on to this figure okay so we can make these nice beautiful math visualizations and after that we'd like to maybe a pick a point somewhere on this mountain range and figure out from there from that vantage point what can we see in the valley alright so here we are back inside MATLAB again I have a script that's divided into a number of cells that I'll run from a high level in our first cell here we're at the access stage of the technical computing workflow and I read a digital elevation model of San Francisco using the USGS 24km function again I capture the latitude and longitude extent just like we did for the weather avoidance demo so that I can use those extents to customize the WMS query but first let's go ahead and visualize this data that we've imported using the US a map and geo show function I use the Dempsey map function over here to color ID apply an automatic coloring so that higher elevation data is colored in Browns and lower elevations colored in greens so it's a nice way to view our our elevation data now alternatively alternatively we could change the viewing angle by modifying the camera position and what we're looking at so this kind of gives me a side profile of this valley over here so in the next step what we'll do is use that latitude and longitude data that we captured and communicate that to the WMS server and have them re-- sample the data and pass us a rendered map so here or communicating with the Microsoft error server which has a lot of ortho photos and topographic maps there we're having them read sample and basically trim and cut the data so that it aligns perfectly with our data and so you'll notice here that we're specifying the latitude and longitude extent as well as the image height and image width and this is all based on the digital elevation model that we imported so that when we get this data we can easily align it and drape the imagery on top so let's go ahead and let this finish downloading into MATLAB okay so here's our bird's eye view of South San Francisco and this image has already been ortho rectified so that you know lens distortion and camera tilt take into account and you can measure true distances here so what we can do with this image that we've downloaded is go ahead and drape it on to our digital elevation model the same functionality that we did before well create a new figure use the US a map in geo show function to plot our elevation model but in this case we'll use the C data parameter and set the ortho image to our color data so because of the alignment that's already been performed by the WMS servers it aligns perfectly and you can see that the mountainous range from the image aligns up correctly with the digital elevation model similarly we can create a new figure in overlay the topographic information on top and at this point let's go ahead and manually select out a point and calculate that view shed so what I'll do is all take a look at our first figure manually select out a point let's say we select out to a point in this little middle valley so what we would expect to see is areas within here maybe out on the water a bit but we shouldn't see anything off to the side so let's go ahead and select that point will calculate our view shed using the view shed function again where we're just passing our latitude and longitude as well as the height at that point as well as the digital elevation model and here we can go ahead and create a figure and display what we were looking at so we see here that we're able to see up and down the mountainous range let's change our viewing angle and we're able to see up the slope but of course not on the other side and a little bit out into the ocean over here now alternatively we could pick out the highest peak in in the data set and this is something that MATLAB is really great for by looking at this image which is basically a matrix or of the digital elevation model we can easily find the top top value using the fine M function I can do that and and by putting our point over here and displaying again what areas were able to see we see that the visibility is much higher at this vantage point so if this was maybe a cell tower that we were placing ideally we'd like to put it up here rather than down inside this valley where there would be limited number of subscribers that could view use the cell network so at this point let's go ahead and jump back into PowerPoint and wrap this up by looking at the key benefits for mapping toolbox as we've seen from the technical computing workflow the toolbox will give this a lot of features and functionality for each of those specific portions and stages of a workflow in particular it gives us the ability to access a variety of file formats from from say as reshape files as we saw today also WMS servers we're able to visualize many types of maps about 2d and 3d and of course analyze choose spatial data so we with these three primary components we look at those three stages inside the technical computing workflow now some similar toolboxes that might come in useful the first one I like to discuss is the image processing toolbox and we saw this a little bit in the weather avoidance demo when we were segmenting those weather systems now unfortunately don't have time to go into too much depth with this toolbox but because a lot of geographic data or raster datasets are basically images image processing toolbox is extremely useful when you need to analyze parts of your of your data maybe segment out certain portions in this case you on the right hand side we're looking at some multispectral imagery and analyzing it another toolbox that might be useful is the parallel computing toolbox and often you know geospatial data is is really really huge there's a lot of information involved and what you might need to do is distribute a job onto many many computers onto a cluster of computers and divide that job up so that can get done a lot faster and that's what's being done here on the right hand side this is one of the demos that before so you won't have too much time we're going to but it's a LAN color cover classification demo and basically what we want to do is we have data at say maybe areas that are forest forests grasslands urban areas at a very high resolution and what we want to do is if we zoom out we need to represent this now more coarse grain with a particular color and so this is a very paralyzing process and as you can see here where we are calculating for particular states dividing it up onto different computers so that the job can be done a lot faster on a local machine running through all the data sets takes about 40 seconds whereas if we divide the job up we can reduce the amount of time needed and so this part for if you're interested this will probably coming to play a lot more power for is basically a parallel for loop so as we loop through each of these states you can split that up onto a number of computers now just to give you a little idea of what people have done with mapping tool box here's an example of a company that's been using the toolbox to visualize missile tests and the fallout from these missile tests now we used other products such as simulating in the aerospace block set to simulate how these the debris is going to move around in space but to visualize it the mapping toolbox was used to create these maps and so we can see these folks here we're using the toolbox for visualization capabilities now if you're interested in finding out more about the toolbox there's a variety of places that you can go and take a look as I mentioned before our website WWE are calm has a variety of information there for you to learn a lot more in particular there's a number of demos such as the ones I've listed here the first one is a WMS demo accessing meteorological data for Katrina and animating it into a movie there's also all the documentation available for you there and if you're interested if you're ready using the tool and you want to see what else is out there whether code you can go to the matlab central file exchange and this is a place where other users will code up some some algorithms and post it there for other people to use so if you're looking for a particular functionality that might be a good place to look to see if you can find it and of course please go to our website and check out if there's any upcoming seminars or webinars that you might be interested in and you can register there to attend and with that we've basically come to the end of our webinar so if you have any questions I'll be glad to answer them at this time you can type them into the lower right I believe and I'll go ahead and pause for a moment to collect up the questions but I'll make sure to go through them all so again thank you so much for your time I hope you found this interesting and informative thank you very much you

Channel: MATLAB

Views: 23,754

Rating: 4.9215684 out of 5

Keywords: MATLAB, Simulink, MathWorks

Id: J8RD22MTejw

Length: 37min 50sec (2270 seconds)

Published: Sat Apr 29 2017