Seam Carving (Nifty2015)

In this assignment, you will create a data type that resizes a W-by-H image using the seam-carving technique. Finding and removing a seam involves three parts and a tiny bit of notation:

1. Notation. In image processing, pixel (x, y) refers to the pixel in column x and row y, with pixel (0, 0) at the upper left corner and pixel (W − 1, H − 1) at the bottom right corner.

1. Warning: this is the opposite of the standard mathematical notation used in linear algebra where (i, j) refers to row i and column j and with Cartesian coordinates where (0, 0) is at the lower left corner.
2. We also assume that the color of each pixel is represented in RGB space, using three integers between 0 and 255. This is consistent with the java.awt.Color data type.
3. Energy calculation. The first step is to calculate the energy of a pixel, which is a measure of its importance—the higher the energy, the less likely that the pixel will be included as part of a seam (as we'll see in the next step). In this assignment, you will use the dual-gradient energy function, which is described below. Here is the dual gradient of the surfing image above:

1. The energy is high (white) for pixels in the image where there is a rapid color gradient (such as the boundary between the sea and sky and the boundary between the surfing Josh Hug on the left and the ocean behind him). The seam-carving technique avoids removing such high-energy pixels.
2. Seam identification. The next step is to find a vertical seam of minimum total energy. This is similar to the classic shortest path problem in an edge-weighted digraph, but there are three important differences:
   • The weights are on the vertices instead of the edges.
   • The goal is to find the shortest path from any of the W pixels in the top row to any of the W pixels in the bottom row.
   • The digraph is acyclic, where there is a downward edge from pixel (x, y) to pixels (x − 1, y + 1), (x, y + 1), and (x + 1, y + 1). assuming that the coordinates are in the prescribed ranges.
3. Seams cannot wrap around the image (e.g., a vertical seam cannot cross from the leftmost column of the image to the rightmost column).

Finding a horizontal seam is analogous.

1. Seam removal. The final step is remove from the image all of the pixels along the vertical or horizontal seam.

The SeamCarver API. Your task is to implement the following mutable data type:

public class SeamCarver {

   public SeamCarver(Image picture)                  // create a seam carver object based on 

                                                     //      the given picture

   public   Image picture()                          // current picture

   public     int width()                            // width of current picture

   public     int height()                           // height of current picture

   public  double energy(int x, int y)               // energy of pixel at column x and row y

   public   int[] findHorizontalSeam()               // sequence of indices for horizontal 

                                                     //      seam

   public   int[] findVerticalSeam()                 // sequence of indices for vertical seam

   public    void removeHorizontalSeam(int[] seam)   // remove horizontal seam from current 

                                                     //      picture

   public    void removeVerticalSeam(int[] seam)     // remove vertical seam from current 

                                                     //      picture

}

• Corner cases. Your code should throw an exception when called with invalid arguments, as specified below.
  • By convention, the indices x and y are integers between 0 and W − 1 and between 0 and H − 1 respectively, where W is the width of the current image and H is the height. Throw a java.lang.IndexOutOfBoundsException if either x or y is outside its prescribed range in energy().
  • Throw a java.lang.NullPointerException if either removeVerticalSeam() or removeHorizontalSeam() is called with a null argument.
  • Throw a java.lang.IllegalArgumentException if either removeVerticalSeam() or removeHorizontalSeam() is called with an array of the wrong length or if the array is not a valid seam (either an entry is outside the height/width bounds or two adjacent entries differ by more than 1).
  • Throw a java.lang.IllegalArgumentException if either removeVerticalSeam() or removeHorizontalSeam() is called when the width or height of the current picture is 1, respectively.
• Constructor. The data type may not mutate the Picture argument to the constructor.
• Computing the energy of a pixel. You will use the dual-gradient energy function: The energy of pixel (x, y) is Δ_x²(x, y) + Δ_y²(x, y), where the square of the x-gradient Δ_x²(x, y) = R_x(x, y)² + G_x(x, y)² + B_x(x, y)², and where the central differences R_x(x, y), G_x(x, y), and B_x(x, y) are the absolute value in differences of red, green, and blue components between pixel (x + 1, y) and pixel (x − 1, y). The square of the y-gradient Δ_y²(x, y) is defined in an analogous manner. To handle pixels on the borders of the image, calculate energy by defining the leftmost and rightmost columns as adjacent and the topmost and bottom most rows as adjacent. For example, to compute the energy of a pixel (0, y) in the leftmost column, we use its right neighbor (1, y) and its left neighbor (W − 1, y).
• Consider the 3-by-4 image with RGB values (each component is an integer between 0 and 255) as shown in the table below.

• Non-border pixel example. The energy of pixel (1, 2) is calculated from pixels (0, 2) and (2, 2) for the x-gradient
  • R_x(1, 2) = 255 − 255 = 0,
  • G_x(1, 2) = 205 − 203 = 2,
  • B_x(1, 2) = 255 − 51 = 204,

yielding Δ_x²(1, 2) = 2² + 204² = 41620;

and pixels (1, 1) and (1, 3) for the y-gradient

• R_y(1, 2) = 255 − 255 = 0,
• G_y(1, 2) = 255 − 153 = 102,
• B_y(1, 2) = 153 − 153 = 0,

yielding Δ_y²(1, 2) = 102² = 10404.

Thus, the energy of pixel (1, 2) is 41620 + 10404 = 52024. Similarly, the energy of pixel (1, 1) is 204² + 103² = 52225.

• Border pixel example. The energy of the border pixel (1, 0) is calculated by using pixels (0, 0) and (2, 0) for the x-gradient
  • R_x(1, 0) = 255 − 255 = 0,
  • G_x(1, 0) = 101 − 101 = 0,
  • B_x(1, 0) = 255 − 51 = 204,

yielding Δ_x²(1, 0) = 204² = 41616;

and pixels (1, 3) and (1, 1) for the y-gradient

• R_y(1, 0) = 255 − 255 = 0,
• G_y(1, 0) = 255 − 153 = 102,
• B_y(1, 0) = 153 − 153 = 0,

yielding Δ_y²(1, 2) = 102² = 10404.

Thus, the energy of pixel (1, 2) is 41616 + 10404 = 52020.

Finding a vertical seam. The findVerticalSeam() method returns an array of length H such that entry i is the column number of the pixel to be removed from row i of the image. For example, consider the 6-by-5 image below (supplied as 6x5.png).

The corresponding pixel energies are shown below, with a minimum energy vertical seam highlighted in pink. In this case, the method findVerticalSeam() returns the array { 3, 4, 3, 2, 2 }.

• Remember that seams cannot wrap around the image. When there are multiple vertical seams with minimal total energy, your method can return any such seam.
• Finding a horizontal seam. The behavior of findHorizontalSeam() is analogous to that of findVerticalSeam() except that it returns an array of length W such that entry j is the row number of the pixel to be removed from column j of the image.
• Performance requirements. The width(), height(), and energy() methods should take constant time in the worst case. All other methods should run in time at most proportional to W H in the worst case.

Analysis of running time. Estimate empirically the running times (in seconds) to remove one row and one column from a W-by-H image as a function of W, and H. Use tilde notation to simplify your answer. Answer the relevant questions in the readme.txt file.

Extra credit.

• Write a program SeamCarverGUI.java that resizes the image as the user resizes the window.
• Optimize SeamCarver.java so that you can run the algorithm at interactive speed for small images.

Submission. Submit SeamCarver.java, and any other files needed by your program (excluding those in stdlib.jar and algs4.jar).