MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Advanced Graphics Chapter 14A

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Objectives After studying this chapter you should be able to: Understand how MATLAB handles the three different types of image files Assign a handle to plots and adjust properties using handle graphics Create an animation using either of the two MATLAB techniques Adjust lighting parameters, camera locations and transparency values Use visualization techniques for both scalar and vector information in three dimensions

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Section 14.1 Images MATLAB includes two basic functions for displaying images stored as a MATLAB matrix image Imagesc To understand how these functions work, first lets look at an old friend – the peaks function

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The peaks function is built into MATLAB for use in demonstrating graphics capabilities. The title and axis labels were added in the usual way.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Rotate the image in three dimensions with the Rotate 3-D tool

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A view of the surface plot of the peaks function looking down the z-axis

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. An easier way to create the previous image is to use a pseudo color plot. It’s the the same thing as the view looking straight down at a surface plot. pcolor(peaks) Pseudo color plots organize the data based on the right hand rule, and start at the 0,0 position on the graph z(1,1)z(1,2)z(1,3)…z(1,n) z(2,1)z(2,2)z(2,3)…z(2,n) z(3,1)z(3,2)z(3,3)…z(3,n) z(m,1)z(m,2)z(m,3)…z(m,n) …………… …………… …………… x - axis y - axis In this image we also removed the grid lines and adjusted the shading option to flat by using the command shading flat

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. This strategy makes sense because of the coordinate system we typically use in graphing It does not make sense for representing images such as photographs. When images are stored in matrices, we usually represent the data as starting in the upper left hand corner of the image, and work across and down

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. z(m,1)z(m,2)z(m,3)…z(m,n) …………… …………… z(1,1)z(1,2)z(1,3)…z(1,n) …………… …………… …………… x - axis y - axis

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. This is the peaks function rendered using the imagesc function Images are usually represented starting in the upper left hand corner, and working across and down, like we read a book. The pcolor plot and the imagesc plot are vertical mirror images.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Notice that the image is flipped in comparison to the pseudo color plot. Of course in many graphics applications it doesn’t matter how the data is represented, as long as we understand the convention used. However, a photograph would be upside down in a vertical mirror image – clearly not an acceptable representation.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Image Types Intensity (or grayscale) images Indexed images RGB (or true color) images

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Intensity Images Also called grayscale or scaled images Use the imagesc function In this approach the colors in the image are determined by a color map The values stored in the image matrix are scaled, and the values correlated to one of the maps This approach works well when the parameter being displayed does not correlate to an actual color

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. For example… In the peaks representations we’ve looked at, the color corresponds to a z axis value. Red corresponds to large values of z Blue corresponds to small values of z It’s an arbitrary choice based partially on aesthetics, but color maps can also be used to enhance features of interest in the image. We use this same color map when we create an image using imagesc

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. X-rays X-ray images traditionally were produced by exposing photographic film to X-ray radiation. Now, many X-rays are processed as digital images and stored in a data file – there is no film involved. We can manipulate that file however we want, because the intensity of X- ray radiation does not correspond to a particular color.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. MATLAB ships with a built in sample file called spine. When you load the file, a matrix is created that represents the data collected from a digital x-ray Use the imagesc function to create the picture shown in figure 1 The colors are determined by scaling the values stored in the matrix X, and using the default colormap (jet)

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. A representation that looks more like a traditional X-ray is returned if we use the bone color map. However, remember that there never was a real photograph taken on x-ray film. These colors just represent different intensity levels measured with the digital x-ray machine.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The axis image command forces the same scale to be used for the x and y axis

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The spine file also included a custom color map, which happens to corresponds to the bone color map. This array is called map. Custom color maps are not necessary to display intensity images

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Determining Image Data Type It is convenient to think of image data as a matrix It is not necessarily stored that way in the standard graphics formats. MATLAB includes a function, imfinfo, that will read standard graphics files and determine what type of data is contained in the file

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Consider this file… mimas.jpg was downloaded from the internet www.saturn.jpl.gov

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Create the image using the imagesc function, because we know that it is a “grayscale” image. The result is sometimes called a false color image – the default colormap is ‘jet’ To read a file from a standard graphics format, such as jpeg, use the imread function, and store the results with a variable name such as x

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Use the axis image command to adjust the dimensions, and the colormap(gray) function to change the image to black and white (grayscale).

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Indexed Image Where color is important, use an indexed image. The matrix, instead of being a list of intensity values, is a list of colors. The image is created like a paint-by- number painting. Each element contains a number that corresponds to a color.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Colormap The colors are listed in a separate matrix, called a color map. The color map is an nx3 matrix that defines n different colors, by identifying the red, green and blue components of each color. A custom color map can be created for each image, or a built- in color map could be used

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The image function is used for data stored in an indexed image mandrill is a built-in file that ships with MATLAB. It includes the image information, and a map of the appropriate colors Until we specify a colormap, the image is displayed using the default colormap – in this case jet

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The axis image command is used to adjust the image to the appropriate dimensions Custom colormap

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. image vs imagesc similar can give very different results

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. If we use the image function, the colormap does not correspond to the colors stored in the intensity image. This resulted in the washed out image and lack of contrast Because we know that the mimas image is stored as a grayscale image, the appropriate function is imagesc.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. It is important to recognize what kind of file you are displaying, so that you can make the optimum choice of how to represent the image.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Clip art stored using the GIF file format – this image comes standard with Microsoft Word Files stored using the gif format are often stored as indexed images imfinfo('drawing.gif') ans = 1x4 struct array with fields: Filename FileModDate FileSize Format etc. This doesn’t really tell us much!!

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The import wizard can be used to create an indexed image matrix and color map from a GIF file. Rename to a different name such as map

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Hint There are a number of sample images built into MATLAB and stored as indexed images. You can access these files by typing load The available images are: flujetdurer detailmandrill clownspine capeearth gatlin

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Each of these image files create a matrix of index values called X and a color map called map. For example, to see the image of the earth type load earth image(X) colormap(map) You’ll also need to adjust the aspect ratio of the display and remove the axis. axis image axis off

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. True color (RGB) images The third technique for storing image data is in a three dimensional file, mxnx3. Recall that a three dimensional matrix consists of rows, columns and pages. These image files consist of 3 pages, one for each color intensity; red, green or blue

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. blue rows columns pages green red True color images use a multidimensional array to represent the color of each element

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. imfinfo('airplanes.jpg') ans = Filename: 'airplanes.jpg' FileModDate: '12-Sep-2005 17:51:48' FileSize: 206397 Format: 'jpg' FormatVersion: '' Width: 1800 Height: 1200 BitDepth: 24 ColorType: 'truecolor' FormatSignature: '' NumberOfSamples: 3 CodingMethod: 'Huffman' CodingProcess: 'Sequential' Comment: {} indicates a page for each color intensity

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. True color image of airplanes – All of the color information is stored in a three dimensional matrix. (Picture used with permission of Dr. G. Jimmy Chen, Salt Lake Community College Department of Computer Science.)

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Notice in the workspace window that X is a 1200 x 1800 x 3 matrix – one page for each color. We don’t need to load a color map, because the color intensity information is included in the matrix

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Example 14.1 Mandelbrot and Julia Sets Benoit Mandelbrot Largely responsible for the current interest in fractal geometry. Built upon work by Gaston Julia

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Mandelbrot Images Advances in Julia’s work had to wait for the development of the computer, and computer graphics in particular. In the 1970’s Mandelbrot, then at IBM, revisited and expanded upon Julia’s work, and actually developed some of the first computer graphics programs to display the complicated and beautiful fractal patterns that today bear his name.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The Mandelbrot image is created by considering each point on the complex plane, x+yi.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. The series either seems to converge, or head off toward infinity The Mandelbrot set is composed of the points that converge. The beautiful pictures you have probably seen were created by counting how many iterations were necessary for the z value at a point to exceed some threshold value, often the square root of 5. We assume, though we can’t prove, that if that threshold is reached the series will continue to diverge and eventually approach infinity.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. M-file to calculate and plot a Mandelbrot image

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Mandelbrot Image – The figure was created by determining how many iterations were required for the calculated element values to exceed the square root of 5.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Another way to look at the set We know that all of the elements in the solid colored region of the image will be below the square root of 5. We could create an image based on those values instead of the number of iterations needed to exceed the threshold. We’ll need to multiply each value times a common multiple in order to achieve any color variation (otherwise the values are too close to each other.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. An image based on the Mandelbrot set, showing how the members of the set vary. The really interesting structures are at the boundary of the set.

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright and written permission should be obtained from the publisher prior to any prohibited reproduction, storage in a retrieval system, or transmission in any form or by any means, electronic, mechanical, photocopying, recording, or likewise. For information regarding permission(s), write to: Rights and Permissions Department, Pearson Education, Inc., Upper Saddle River, NJ 07458. Images created by “zooming” in on the Mandelbrot set, using the MATLAB program described in Example 14.1

MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright.

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MATLAB for Engineers 4E, by Holly Moore. © 2014 Pearson Education, Inc., Upper Saddle River, NJ. All rights reserved. This material is protected by Copyright.

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