Topic 6 Visualisations in Matlab√ If you use the plot command with a single matrix, MATLAB plots the values versus the index number √ Usually this type of data is plotted on a

CITS2401 Computer Analysis & Visualisation

SCHOOL OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING

FACULTY OF ENGINEERING, COMPUTING AND MATHEMATICS

Topic 6 Visualisations in

Matlab

Material from MATLAB for Engineers, Moore, Chapters 5,14 Additional material by Peter Kovesi and Wei Liu

The University of Western Australia

Two Dimensional Plots √  The x-y plot is the most commonly used plot by engineers

√  The independent variable is usually called x

√  The dependent variable is usually called y


Consider this x-y data

time, sec Distance, Ft 0 0 2 0.33 4 4.13 6 6.29 8 6.85 10 11.19 12 13.19 14 13.96 16 16.33 18 18.17

Time is the independent variable and distance is the dependent variable



In the default mode the figure window is free floating, and appears on top of the MATLAB desktop.

It is often convenient to “dock” the figure, using the docking arrow



Engineers always add …

√  Title: use command title(‘Your title here’)!

√  X axis label, complete with units: xlabel(‘x-Label’)

√  Y axis label, complete with units: ylabel(‘y-Label’)

√  Often it is useful to add a grid: grid on

These commands apply to the most recently plotted figure.

You can use sprintf to use formatted output. E.g.:!

title(sprintf(’F(x) between %d and %d', x(1),x(20)))!



To add an apostrophe to a title (or other annotation) you must enter the single quote twice – otherwise MATLAB interprets the single apostrophe as the end of the string.


Creating multiple plots

√  MATLAB overwrites the figure window every time you request a new plot

√  To open a new figure window use the figure function – for example figure(2)

√  hold on •  Freezes the current plot, so that an additional plot can be

overlaid

√  When you use this approach the additional line is drawn in blue – the default drawing color


The first plot is drawn in blue

To unfreeze the plot use the hold off command


Each set of ordered pairs will produce a new line

You can also create multiple lines on a single graph with one command


The legend function

√  The legend function is used to provide legends for each data set that have been displayed simultaneously on one set of axes via the plot(X1, Y1, X2, Y2, X3, Y3, ...) function.

√  For example:

>> legend('Label 1', 'Label 2', ..., 'Label N');

% Adds a legend for each data set in the plot.

√  Type help legend at the Matlab command prompt to learn more about the many options available with this function.


Creating and switching between figure windows

√  The figure function is used to control different figure (or plot) windows.

>> figure; % Creates a new figure window.

% This becomes the "current" figure.

>> figure(N); % Makes figure N the current figure window.

% This function creates a new figure window

% called N if N does not already exist.

>> delete(N); % Deletes figure N.


Variations √  If you use the plot command with a single matrix, MATLAB

plots the values versus the index number √  Usually this type of data is plotted on a bar graph √  When plotted on an x-y grid, it is often called a line graph


If you want to create multiple plots, all with the same x value you can… √  Use alternating sets of ordered pairs

•  plot(x, y1, x, y2, x, y3, x, y4)

√  Or group the y values into a matrix •  z=[y1, y2, y3, y4]

•  plot(x,z)


The peaks(100) function creates a 100x100 array of values. Since this is a plot of a single variable, we get 100 different line plots

The peaks(100) function creates a 100x100 array of values, and will plot itself .


Plots of Complex Arrays √  If the input to the plot command is a single array of complex

numbers, MATLAB plots the real component on the x-axis and the imaginary component on the y-axis


Line, Color and Mark Style √  You can change the appearance of your plots by

selecting user defined •  line styles

•  color

•  mark styles

√  Try using help plot

for a list of available styles


Available choices Table 5. 2 Line, Mark and Color Options

Line Type Indicator Point Type Indicator Color Indicator

solid - point . blue b

dotted : circle o green g

dash-dot -. x-mark x red r

dashed -- plus + cyan c

star * magenta m

square s yellow y

diamond d black k

triangle down v

triangle up ^

triangle left <

triangle right >

pentagram p

hexagram h


Specify your choices in a string √  For example

√  plot(x,y,':ok') •  strings are identified with a tick mark

•  if you don’t specify style, a default is used

–  line style – none

–  mark style – none

–  color - blue


dotted line

circles

black


Axis scaling √  MATLAB automatically scales each plot to completely fill the

graph √  If you want to specify a different axis – use the axis command

axis([xmin,xmax,ymin,ymax])

√  Lets change the axes on the graph we just looked at



Subplots

√  The subplot command allows you to subdivide the graphing window into a grid of m rows and n columns

√  subplot(m,n,p)

rows columns location


Other Types of 2-D Plots

√  Polar Plots √  Logarithmic Plots √  Bar Graphs √  Pie Charts √  Histograms √  X-Y graphs with 2 y axes √  Function Plots


Polar Plots

√  Some functions are easier to specify using polar coordinates than by using rectangular coordinates

√  For example the equation of a circle is •  y=sin(x) in polar coordinates


Logarithmic Plots √  A logarithmic scale (base 10) is convenient when

•  a variable ranges over many orders of magnitude, because the wide range of values can be graphed, without compressing the smaller values.

•  data varies exponentially.

√  plot – uses a linear scale on both axes √  semilogy – uses a log10 scale on the y axis √  semilogx – uses a log10 scale on the x axis √  loglog – use a log10 scale on both axes


√  plot – uses a linear scale on both axes √  semilogy – uses a log10 scale on the y axis √  semilogx – uses a log10 scale on the x axis √  loglog – use a log10 scale on both axes


x-y plot – linear on both axes

semilogx – log scale on the x axis

semilogy – log scale on the y axis

loglog – log scale on both axes



Bar Graphs and Pie Charts

√  MATLAB includes a whole family of bar graphs and pie charts •  bar(x) – vertical bar graph •  barh(x) – horizontal bar graph •  bar3(x) – 3-D vertical bar graph •  bar3h(x) – 3-D horizontal bar graph •  pie(x) – pie chart •  pie3(x) – 3-D pie chart




Histograms √  A histogram is a plot

showing the distribution of a set of values


X-Y Graphs with Two Y Axes √  Sometimes it is useful to overlay two x-y plots onto the same

figure. However, if the order of magnitude of the y values are quite different, it may be difficult to see how the data behave.


plotyy: plots two functions with separate axis.


5.3 Function Plots

√  Function plots allow you to use a function as input to a plot command, instead of a set of ordered pairs of x-y values

√  fplot('sin(x)',[-2*pi,2*pi])

function input as a string

range of the independent variable – in this case x


Three Dimensional Plotting

√  Line plots

√  Surface plots

√  Contour plots

√  Cylinder plots


Three Dimensional Line Plots

√  These plots require a set of order triples ( x-y-z values) as input


Just for fun

√  try the comet3 function, which draws the graph in an animation sequence

√  comet3(x,y,z) √  If your animation draws too slowly, add more data points √  For 2-D line graphs use the comet function


Surface Plots √  Represent x-y-z data as a surface

•  mesh - meshplot

•  surf – surface plot


Using mesh with 3 variables √  If we know the

values of x and y that correspond to our z values, we can plot against those values instead of the index numbers



Shading

√  There are several shading options •  shading interp

•  shading flat

•  faceted flat is the default

√  You can also adjust the color scheme with the color map function


Default shading (faceted)

Shading interp

Shading flat


Colormaps autumn bone hot spring colorcube hsv summer cool pink winter copper prism jet (default) flag white


Contour Plots √  Contour plots use the

same input syntax as mesh and surf plots

√  They create graphs that look like the familiar contour maps used by hikers


To demonstrate these functions we used a more interesting example

•  A more complicated surface can be created by calculating the values of z, instead of just defining them

•  We’ll need to use the meshgrid function to create 2-D input arrays – which will then be used to create a 2-D result

•  [X,Y] = meshgrid(x,y) will return 2 n x m matrices where x is an n-vector and y is an m-vector.!



Example

√  For example, imagine generating two plots one above the other:

>> subplot(2, 1, 1), plot(x1, y1), title('Plot 1');

% The first call to subplot breaks the figure window into

% 2x1 cells, making the top one the active cell.

>> subplot(2, 1, 2), semilogy(x2, y2), title('Plot 2');

% A second call to subplot is needed to plot to the

% second cell.


Another example >> xs = 0 : 0.1 : 10;

>> subplot(2, 1, 1), plot(xs, sin(xs)), xlabel('x'), ylabel('sin(x)');

>> subplot(2, 1, 2), plot(xs, cos(xs)), xlabel('x'), ylabel('cos(x)');


3D line plot example

√  For example, the following commands:

>> t = 0 : 0.1 : 10;

>> x = exp(-0.2*t) .* cos(2*t);

>> y = exp(-0.2*t) .* sin(2*t);

>> plot3(x, y, t), xlabel('x'),

ylabel('y'), zlabel('time');

produces the plot:


Surface plotting in 3D

√  The plot3 function is suitable for displaying two variables that are a function of one independent variable:

x = f1(t) y = f2(t)

√  When you have a single variable that is a function of two independent variables, say:

z = f(x, y)

√  then a surface display is more appropriate.


mesh, surf, and contour functions √  Matlab's mesh, surf, and contour functions are used for displaying this

kind of data:

>> mesh(X, Y, Z) % Creates a mesh or wire-frame plot.

>> surf(X, Y, Z) % Creates a surface plot.

>> contour(X, Y, Z) % Creates a contour plot.


The meshgrid function √  The meshgrid function is extremely useful for generating the X and Y

2D arrays for a 3D plot. >> Xvals = -2 : 1 : 2; % Generate axis vectors.

>> Yvals = -2 : 1 : 2;

>> [X, Y] = meshgrid(Xvals, Yvals) % Make a mesh grid.

X =

-2 -1 0 1 2

-2 -1 0 1 2

-2 -1 0 1 2

-2 -1 0 1 2

-2 -1 0 1 2


Meshgrid Example Continues

Y = -2 -2 -2 -2 -2

-1 -1 -1 -1 -1

0 0 0 0 0

1 1 1 1 1

2 2 2 2 2

% Now calculate the function values on this mesh.

>> Z = X .* exp(-(X.^2 + Y.^2)); % A 2D Gaussian .* X.

>> mesh(X, Y, Z); % Display the plot as a mesh.

>> surf(X, Y, Z); % Display the plot as a surface.

>> contour(X, Y, Z); % Display the plot as a contour.


Mesh, Surface and Contour Plot

æ  By decreasing the increment in the axis vector generation commands, we can produce plots like:


Surface plotting with a single argument √  You can also invoke these surface displaying functions with a

single argument.

√  This single argument is treated as the 2D array of Z values for each point.

√  The X and Y arrays default to a range of [1 .. the number of rows/columns of Z].

√  For example:

>> surf(Z)


Parametric surface representation √  This specification of surfaces by three separate arrays - one for each of the x, y

and z coordinates of each data point is actually a parametric surface representation.

√  A parametric surface is parameterized by two independent variables, i and j, which vary continuously over a rectangle.

√  You can then specify functions x(i,j), y(i,j), and z(i,j) which determine the way in which x, y and z coordinates vary with i and j over the surface.

√  For example, the surface of the earth is parameterized in terms of longitude and latitude.

√  You can draw a map of the earth as a flat rectangle, but at each longitude and latitude we can determine the x, y and z values that correspond to the actual location of that point in space.


2D arrays == flat ?

√  In discrete terms, Matlab represents parametric surfaces in terms of 2D arrays of X, Y, and Z values that correspond to each (i, j) grid point on the surface.

√  Hence the arguments to the mesh, surf, and contour functions are 2D arrays of points.

√  In the plots shown so far, the X and Y 2D arrays are "flat", but there is no need for them to be like that...


A triangular prism æ  A very wide variety of shapes can be

represented, e.g. a triangular prism can be represented by: X =

1.0 -0.5 -0.5 1.0

1.0 -0.5 -0.5 1.0

Y = 0.0 0.866 -0.866 0.0 0.0 0.866 -0.866 0.0

Z =

0.0 0.0 0.0 0.0 1.0 1.0 1.0 1.0


Functions for generating points √  Matlab provides some other handy functions for generating the

points of parametric surfaces, eg. cylinder and sphere...

>> [X, Y, Z] = cylinder(R, N);

% Generates an N sided cylinder. R is an array of

% radius values to be used along the length of

% the cylinder.

>> [X, Y, Z] = cylinder([1 1], 3);

% Produces a 3 sided prism.

>> [X, Y, Z] = cylinder([1 0], 4);

% Produces a 4 sided pyramid.

>> [X, Y, Z] = sphere(N);

% Produces a parametrically defined sphere.


Parametric Surface Plots


Nice looking surfaces

√  For a really nice looking surface use:

>> surfl(X, Y, Z)

>> shading interp

√  The surfl function behaves just like surf but renders a surface according to the current lighting model (the default lighting model is fine for almost every purpose you might have).

√  The shading across the surface is no longer a function of its "height".

√  Instead the shading is a function of the relative angle between the incident light and the surface normal at each point.


Example >> [X Y Z] = cylinder([3 1 3 4 2 2 ],500); >> surfl(X, Y, Z) >> shading interp


Images

√  Another way of viewing a single variable that is a function of two independent variables is to display it as a 2D image.

√  The Z, or height, values of the function is encoded by an intensity value or colour.

√  The human visual system is highly proficient at interpreting this form of display if the data represents intensity.

√  However, the visual system can have difficulty if the data represents some other quantity such as height.

√  Images in standard formats such as gif, tiff, and jpeg are stored so that each pixel value lies in the range [0,255].


Why 0-255 √  The range [0, 255] is chosen so that each pixel can be represented by

8 bits (one byte).

•  0: Black

•  128: Grey

•  255: White

√  Colour images will typically consist of three 2D arrays in the range [0 , 255] for each of the red, green, and blue channels.

√  When you read an image into Matlab, Matlab stores the image as a matrix of numbers - each number lies in the range [0, 255].

√  To save memory, Matlab uses the type uint8 (an unsigned 8 bit integer which can represent the values [0, 255] only) to represent the numbers in the image matrix:

Documents

Topic 6 Visualisations in Matlab√ If you use the plot command with a single matrix, MATLAB plots the values versus the index number √ Usually this type of data is plotted on a