1. Manipulating 2D arrays in Java
The Picture Lab
Manipulating 2D arrays in Java

2. RGB colors
Colors are typically represented by their components in red, green, and blue
Each component ranges from 0-255
Each component requires 8 bits
See the java.awt.Color class in java
Consider a 640x480 picture, calculate the required number of:
Pixels
Bytes
Bits
Let’s run the ColorChooser.java program to look more closely at how Colors are constructed in Java

3. ColorChooser.java How do you create:
Red Green Blue
White Gray Black
Cyan Magenta Yellow
How do you increase/decrease the intensity of a color
Light Yellow vs Dark Yellow

4. Storing pixels in an image
Run the file PictureExplorer.java
The image beach.jpg should load a 640x480 image
What is the row and column index for the top left corner?
What is the largest row index?
What is the largest column index?
How does the row index change as you move from left to right?
How does the column index change as you move from top to bottom?

5. Representing pixels in Java
Most images are manipulated as 2D arrays, an array of arrays
int[] oneDim = new int[10];
int[][] twoDim = new int[3][5];
More dimensions are possible:
int[][][] threeDim = new int[1][2][3];
Why an array of arrays?
Convenient way to represent a table of values

6. Creating 2D Arrays
// A 2D array with 4 rows,
// 3 columns, 12 elements
int[][] array = {{41,38,91},{63,10,21},{29,28,13},{70,52,39}};
int[][] array = { {41,38,91},
{63,10,21},
{29,28,13},
{70,52,39}};
array[0][1]
array[1][0]
array[1][1]
array[0][0]
array[0][2]
array[2][0]
array[2][2]
array[3][0]
array[1][2]
array[2][1]
array[3][1]
array[3][2] 63 28 13 21 29 91 38 41 10 70 52 39
Col Index 1 2 41 38 91 63 10 21 29 28 13 3 70 52 39
Row Index

7. 2D Arrays of pixels = image
Instead of ints, it would be better to think of a picture as a 2D array of pixels:
RGB(255,0,0)
RGB(0,255,0)
RGB(0,0,255)
RGB(255,255,255)
RGB(128,128,128)
RGB(0,0,0)
RGB(255,255,0)
RGB(255,0,255)
RGB(0,255,255)
Pixel[][] data = {{RGB(255,0,0), RGB(0,255,0), RGB(0,0,255)},
{RGB(255,255,255), RGB(128,128,128), RGB(0,0,0)},
{RGB(255,255,0), RGB(255,0,255), RGB(0,255,255)}};

8. Row Major Representation
So far, all 2D arrays shown have been in row major order.
All of the data for the first row is stored before the next row
int[][] array = {{41,38,91},{63,10,21},{29,28,13},{70,52,39}};
There is also column major order
You should assume all 2D arrays are row-major for our course
Inner arrays represent columns 41 38 91 63 10 21 29 28 13 70 52 39 41 38 91 63 10 21 29 28 13 70 52 39

9. Row vs. Column Major int[][] rowM = {{1,2,3}, {4,5,6} };
int[][] colM = { {1,4}, {2,5}, {3,6} };

10. Enhanced for loops (for-each)
When traversing an array of elements the typical programming pattern is this:
for(int n = 0; n < array.length; n++)
System.out.println(array[n]);
This is equivalent syntax to:
for(int item : array) //for each value in the array
System.out.println(item);
This loop is a required pattern for our course
Convenient but…
Can only loop over every element exactly once
Avoids missing an item or traversing past the end

11. Looping a 2D array Here is the typical programming pattern:
//the outer loop gets each internal array
//internal arrays = rows when row major order is used
for(int row = 0; row < array.length; row++) {
//inner loop traverses internal array (a single row).
//Each entry in the row is accessed by its column location
for(int column = 0; column < array[0].length; column++)
//array[row][column] is the current element }
Or with an enhanced for loop (for-each):
for(int[] row : array)//get each internal array from the 2D array
for( int entry : row )//get each entry in the internal array
//entry is the equivalent to array[row][column] above

12. Summary of Operations

13. Exercise The Matrix class
You will need to complete the skeleton code for the Matrix class
This is an exercise intended to help you become more familiar with 2D arrays before we move on to image processing
You must use enhanced for loops in your code
Not every loop but some
What follows are examples of how the methods in the Matrix class work and a bit about the algebra of Matrices.

14. public Matrix(double[][] values)
This will create a matrix with the given values
Assume row major order
For example, Matrix({{1,2},{3,4},{5,6}}) would initialize values_ to
[1,2]
[3,4]
[5,6]
DO NOT SIMPLY ASSIGN values_ = values;
This creates a “shallow copy” which is shared by more than just your object
That means your object is not the only thing that can change values_!
Make a deep copy by creating a new 2D array and copying the values

15. public int getRows() and getColumns()
These accessor methods should tell the caller how many rows and columns are contained in the Matrix
Do not store these during construction
Use the 2D array to retrieve the information

16. public String toString()
returns a String representation of the Matrix
For example, if values_ = {{1,2},{3,4},{5,6}};
toString would return:
“[1\t2]\n[3\t4]\n[5\t6]”
[1 2]
[3 4]
[5 6]
Other formats are possible as long as they make some similar attempt at a pleasing visualization of the array

17. public Matrix scalar(double value)
Scalar multiplication of a Matrix is when a number is multiplied with the Matrix:
Your method should not alter the existing object, but instead return a new Matrix that represents the scalar product

18. public Matrix add(Matrix m)
Addition of two Matrices works as follows:
Your method should not alter any of the existing Matrices involved in the addition but instead return a new Matrix that is the resulting sum
IMPORTANT!!!
This only works if the two matrices have the same number of rows and columns
If they don’t have the same number of rows and columns this operation is undefined
return null;

19. public Matrix subtract(Matrix m)
Subtraction of two Matrices works as follows:
Your method should not alter any of the existing Matrices involved in the subtraction but instead return a new Matrix that is the resulting difference
HINT: Think carefully, you’ve already done all of work needed for subtraction
IMPORTANT!!!
This only works if the two matrices have the same number of rows and columns
If they don’t have the same number of rows and columns this operation is undefined
return null;

20. public Matrix multiply(Matrix m)
When multiplying two Matrices it’s not as obvious how it works…
The row of the left matrix is multiplied on each column of the right matrix.
It works like this: 1 2 3 4 5 6 3 6 2 5 1 4
1x3+2x2+3x1
1x6+2x5+3x4
4x3+5x2+6x1
4x6+5x5+6x4 10 28 28 73

21. public Matrix multiply(Matrix m) cont’
This will (likely) be the trickiest problem for you to solve
Pre conditions (LM = left Matrix, RM = right Matrix)
LM.columns = RM.rows
LM.rows = RM.columns
Return null if any of these fail
The resulting Matrix will have:
rows = LM.rows
columns = RM.columns
The method should return a new Matrix and not alter the existing ones
One of the biggest challenges you may find is that a lot of tasks are occurring at the same time in order to accomplish this in just one method
Good programmers break down difficult problems into manageable sub problems

22. public Matrix multiply(Matrix m) cont’
Suggested helper methods (not required)
public double[] getRow(int r)
Gets a single row from the Matrix and returns it as an array
public double[] getColumn(int c)
Gets a single column from the Matrix and returns it as an array
public static double dotProduct(double[] row, double[] col)
Remember, row.length == col.length or you should have returned null
You can now simply loop through each spot doing a product on the value in the index and a sum of all products
this method is (and should be) static because it does not access any private member variables during execution (belongs to no particular instance of the class)

23. public Matrix multiply(Matrix m) cont’1 2 3 4 5 6 3 6 2 5 1 4
Left Matrix (left)
Right Matrix (right)
Now I can fill the example matrix like this (use a loop in your solution)
dotProduct(left.getRow(0), right.getColumn(0)) = 10
dotProduct(left.getRow(0), right.getColumn(1)) = 28
dotProduct(left.getRow(1), right.getColumn(0)) = 28
dotProduct(left.getRow(1), right.getColumn(1)) = 73

24. public Matrix transpose()
Transposing a Matrix is the process of exchanging all rows and columns:
array[i][j]  array[j][i]
It’s probably easiest to see in an example:
Create a method to return a new Matrix which is the transposed form of this 1 4 2 5 3 6 1 2 3 4 5 6

25. Solving systems of equations
One of the major applications of Matrices is solving a system of equations.
For example, you may have encountered a math question like this:
2x + 3y = 1
x – y = –2
We can think about this as three matrices:
A coefficient matrix
A variable matrix
A constant matrix 2 3 1 –1 x y 1 –2

26. Solving Systems of Equations
Or as a mathematical expression: So
x = –1 and y = 1
In other words, the inverse of the coefficient matrix multiplied by the constant matrix will give us the solution to the system
I have written the solve method for you
Write a main program that takes in the coefficients and constant terms for a system of equations and solves it using our Matrix class 2 3 1 –1 x y 1 –2 –1 x y 2 3 1 –1 1 –2 –1 1