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E7 University of California, Berkeley Sengupta, Spring 2020 only have one input (which is name), the code will load scores lab04.mat as the variable scores lab04. • You will also need to include the original scores lab04.mat file in your submis- sion. 2 A Mouse and a Cheeseboard function path = cheeseBoard(current, prev path) Input Type Description current 1x2 double Current position (row, column) of the mouse on the grid prev path mxn char array Previous path of the mouse, made up of a se- quence of ‘R’s and ‘D’s. m represents the num- ber of paths, and n represents the number of moves. Output Type Description path pxq char array or an empty array, [] Previous path, plus an additional step, unless the mouse reaches a position in which it is un- able to move both right and down. If this is the case, then path will be an empty array. Details A mouse confined to a 5x5 grid board is trying to reach a piece of cheese on the opposite corner. However, there are mouse traps on certain spots of the board that the mouse must avoid on its journey to obtain the cheese. You will write a function that finds all possible shortest paths the mouse can take to reach the cheese while avoiding all the traps. You only want to know the shortest paths, which means the mouse can only move right (‘R’) or down (‘D’). The mouse can only move one grid at a time. The mouse starts at the position (1, 1), i.e., the top left corner of the board, and is attempting to reach the cheese located at (5, 5), i.e., the bottom right of the board. Note that you may not know where the traps are beforehand. However, you will have a function myCheck.m whose input is a 1x2 double array which represents the position that the mouse is trying to get to, and its output is a logical value. If the output of myCheck.m is true, this means the mouse can go to that position. If the output of myCheck.m is false, then the mouse cannot go to that position. Tips • The top-left square of the board is (1, 1), and the bottom right of the board is (5, 5). This way we can treat each tile of the board as a matrix component. Note that the first index refers to the vertical direction going downwards and the second index refers to the horizontal direction going to the right. • An example of a valid path for the above picture would be ‘RRDDRRDD’. However, there exist several other shortest paths that your code must also find. 2 E7 University of California, Berkeley Sengupta, Spring 2020 • You can technically see where the traps are located beforehand by reading through the myCheck.m file. However, do not hard code around the trap locations in the file. We will test your code with di↵erent locations and number of traps. • Recursion could be helpful here. 3 Berkelize it! function mat = berkelize(mat, row, col, prev color, new color) Input Type Description mat MxNx3 unint8 The matrix representation of the image row 1x1 double The row of the selected pixel col 1x1 double The column of the selected pixel prev color 1x1x3 unint8 The color of last painted pixel new color 1x1 char 'B' for ’Berkeley Blue’ or 'G' for ’California Gold’ Output Type Description mat MxNx3 unint8 The matrix representation of modified image Details You have probably seen a Paint Bucket Tool in Photoshop or MS Paint. With this tool, you pick a pixel and designate it a color. The tool then fills all adjacent pixels whose colors are identical to the pixel you clicked to the designated color. In this problem, you will write a function that recursively implements this tool for an input image. A script file berkelizeDriver.m is provided to you on bCourses. It shows you how your function can be called to “Berkelize” a switch logo image. An example is shown below. If you only run line 4 (i.e. to comment out line 6 12), you should see the output image on the left; If you run the whole script, you should see the output image on the right: 3 E7 University of California, Berkeley Sengupta, Spring 2020 Figure 1: Original Image Figure 2: Berkelized! Tips • An image is represented by a 3D array with dimensions MxNx3. The first two dimensions M and N describe the size/dimensions of the image, and each of the “layers” in the third dimension represent the RGB values corresponding to that pixel location in the image. • Look up the O�cial Berkeley Color and the HEX representation of color to RGB values. This website can help you convert HEX to RGB values. • Your function should make only one color change each time it is called. However, your function should be able to change a previous color into either Berkeley Blue or California Gold, depending on what the previous color is. • Be careful when trying your function out on large images: it might cause the infamous stack overflow. 4 Floating point numbers function [result] = myDoubleFromBinary(binary) Input Type Description binary 1x64 logical Logical array of length 64 made of only zeros (false) and ones (true). Output Type Description myDouble 1x1 double The floating point number (in base 10) that is represented by binary using the IEEE-754 64-bit double precision binary representation. Details In lab02 you already programmed your own decimalToQuaternary function that could convert the data between di↵erent data types. For this problem, you will write a conversion function that converts binary data into floating point numbers, i.e., numbers with fractional or decimal components. There are multiple ways to represent floating point numbers in binary format (i.e. with only zeros and ones). The Institute of Electrical and Electronics Engineers (IEEE) defined a standard (called IEEE-754) for representing floating point numbers in binary format. IEEE-754 specifies 4 https://brand.berkeley.edu/colors/ https://www.rgbtohex.net/hextorgb/ https://en.wikipedia.org/wiki/Stack_overflow https://en.wikipedia.org/wiki/Stack_overflow https://www.binaryconvert.com/convert_double.html E7 University of California, Berkeley Sengupta, Spring 2020 di↵erent formats, depending on how many bits (e.g. 16 bits, 32 bits, 64 bits, 128 bits) are used to represent each floating point number. The formats that use 32 bits and 64 bits to represent each number are commonly known as “single precision” and “double precision”, respectively. In MATLAB, you can experiment by defining a variable a that contains the value 1 (a = 1;) in the command window and then using the function whos to inspect the variables currently defined in the workspace. You should see that the class of variable a is “double” and that it occupies 8 bytes = 64 bits of memory. Here is a figure showing the components of a double floating number in IEEE-754. In this problem, we will consider double precision representations, where each number is repre- sented using 64 bits (i.e. a sequence of 64 zeros (false) and/or ones (true)). You will write a function to convert this logical array to its double floating point equivalent. We index the bits from left to right: the left-most bit is the 1st bit and the right-most bit is the 64th bit. Instead of utilizing each bit as the coe�cient of a power of 2, floats allocate bits to three di↵erent parts: the sign indicator, s, which says whether a number is positive or negative; characteristic or exponent, e, which is the power of 2; and the fraction, f , which is the coe�cient of the exponent. To be more specific, the number represented by a sequence of 64 bits can be calculated using the following formula: n = (�1)s · 2e�1023 · (1 + f) where n represents the decimal format of the floating number. For example: 0 01111111111 01000000000000000000000000000000000000000000000000000. Its exponent decimal is s = 0 e = 1 · 20 + 1 · 21 + 1 · 22 + 1 · 23 + 1 · 24 + 1 · 25 + 1 · 26 + 1 · 27 + 1 · 28 + 1 · 29 = 1023 f = 1 · 1 22 (�1)0 · 21023�1023 · (1 + 0.25) = 1.25 However, there are special cases where the above formula does not necessarily apply. Remember to program these cases into your function. • if e = 0 and f 6= 0, n = (�1)s · 21�1023 · (1 + f) • if e = 0 and f = 0, n = 0 • if e = 2047 and f = 0, n = (�1)s1 • if e = 2047 and f 6= 0, n = NaN(Not a Number) 5 E7 University of California, Berkeley Sengupta, Spring 2020 5 Floating Point Precision vs Fixed Precision function [double stats, int64 stats] = comparePrecision(binary) Input Type Description binary 1x64 logical Logical array of length 64 made of only zeros (false) and ones (true), representing a single precision floating point number. Output Type Description double stats 1x3 double The value, precision, and maximum representable num- ber in double format, listed in that order. int64 stats 1x3 int64 The value, precision, and maximum representable num- ber in int64 format, listed in that order. Each numerical data type in Matlab has a finite amount of di↵erent numbers it can represent, associated with the number of bits it uses in your computer’s memory. Thus, rather than there being a continuous number line that we would expect in mathematical theory, for numbers stored in a computer, the number line is a set of discrete values that can be represented with gaps in between them where no numbers can be stored. The distance(absolute di↵erence) between these representable numbers is often referred to as machine precision and represents the accuracy with which we can store numbers and perform mathematical operations on a computer. For this problem, we will explore how the machine precision varies for two di↵erent data formats: double and int64, both of which take up 64 bits (and thus can represent the same quantity of distinct numerical values). In this problem, you are given the binary representation of a double precision number. You should take this data and do the following once for the double data format and once for the int64 data format. 1. Calculate the numerical value, getting the value of the double from the binary data and getting the value of the int64 from the double data. 2. Find the machine precision to which this number is represented in int64 or double data format. 3. Find the largest positive (non-infinite) numerical value that can be represented in int64 or double data format. Tips • Again, note that the output type should be double and int64. • Restriction: You should not use the Matlab built-in functions eps or realmax for this problem. 6 Memory Consumption There is a quiz in bcourse