Programming Assignment 7: Seam Carving

Seam-carving is a content-aware image resizing technique where the image is reduced in size by one pixel of height (or width) at a time. A vertical seam in an image is a path of pixels connected from the top to the bottom with one pixel in each row. (A horizontal seam is a path of pixels connected from the left to the right with one pixel in each column.) Below left is the original 505-by-287 pixel image; below right is the result after removing 150 vertical seams, resulting in a 30% narrower image. Unlike standard content-agnostic resizing techniques (e.g. cropping and scaling), the most interesting features (aspect ratio, set of objects present, etc.) of the image are preserved.

As you'll soon see, the underlying algorithm is quite simple and elegant. Despite this fact, this technique was not discovered until 2007 by Shai Avidan and Ariel Shamir. It is now a feature in Adobe Photoshop (thanks to a Princeton graduate student), as well as other popular computer graphics applications.

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:

0. 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. This is consistent with the Picture data type in stdlib.jar. 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.

   a 3-by-4 image

   (0, 0)	(1, 0)	(2, 0)
   (0, 1)	(1, 1)	(2, 1)
   (0, 2)	(1, 2)	(2, 2)
   (0, 3)	(1, 3)	(2, 3)

   We also assume that the color of a pixel is represented in RGB space, using three integers between 0 and 255. This is consistent with the java.awt.Color data type.

1. Energy calculation. The first step is to calculate the energy of each pixel, which is a measure of the importance of each pixel—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 implement the dual gradient energy function, which is described below. Here is the dual gradient of the surfing image above:
   

   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 except for the following:

   • The weights are on the vertices instead of the edges.

   • We want to find the shortest path from any of 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 range.

   Seams should NOT wrap around the image (e.g. you can't have a vertical seam that directly crosses over from the leftmost column of the image to the rightmost column).

   
3. Seam removal. The final step is remove from the image all of the pixels along the seam.

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

public class SeamCarver {
   public SeamCarver(Picture picture)
   public Picture 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 picture
   public    void removeVerticalSeam(int[] seam)     // remove vertical seam from picture

}

• Computing the energy of a pixel. We 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 (leftmost and rightmost column, top and bottom row), we calculate energy by defining the leftmost and rightmost columns as adjacent and the topmost and bottommost rows as adjacent. For example, to compute the energy of a pixel at position (x, 0) (i.e. the top row), we use its bottom neighbor at coordinate (x, 1), and its top neighbor at coordinate (x, H-1), where H is the height of the image in pixels. This is equivalent to treating the image like a torus.

  As a more thorough example, consider the 3-by-4 image with RGB values (each component is an integer between 0 and 255) as shown in the table below.

  (255, 101, 51)	(255, 101, 153)	(255, 101, 255)
  (255,153,51)	(255,153,153)	(255,153,255)
  (255,203,51)	(255,204,153)	(255,205,255)
  (255,255,51)	(255,255,153)	(255,255,255)

  Non-border pixel example

  The energy of pixel (1, 2) is calculated as follows:

  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.

  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) in calculated as follows:

  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.

  Since there is no pixel (x, y - 1) we calculate between pixel (x, y + 1) and pixel (x, height − 1).
  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.

  20808.0	52020.0	20808.0
  20808.0	52225.0	21220.0
  20809.0	52024.0	20809.0
  20808.0	52225.0	21220.0

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

  ( 78,209, 79)	( 63,118,247)	( 92,175, 95)	(243, 73,183)	(210,109,104)	(252,101,119)
  (224,191,182)	(108, 89, 82)	( 80,196,230)	(112,156,180)	(176,178,120)	(142,151,142)
  (117,189,149)	(171,231,153)	(149,164,168)	(107,119, 71)	(120,105,138)	(163,174,196)
  (163,222,132)	(187,117,183)	( 92,145, 69)	(158,143, 79)	(220, 75,222)	(189, 73,214)
  (211,120,173)	(188,218,244)	(214,103, 68)	(163,166,246)	( 79,125,246)	(211,201, 98)

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

  57685.0	50893.0	91370.0	25418.0	33055.0	37246.0
  15421.0	56334.0	22808.0	54796.0	11641.0	25496.0
  12344.0	19236.0	52030.0	17708.0	44735.0	20663.0
  17074.0	23678.0	30279.0	80663.0	37831.0	45595.0
  32337.0	30796.0	4909.0	73334.0	40613.0	36556.0

  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() as analogous to that of findVerticalSeam() except that it should return an array of W such that entry y is the row number of the pixel to be removed from column y 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. For faster performance, do not construct an explicit EdgeWeightedDigraph object.

• Exceptions. Your code should throw an exception when called with invalid arguments.

  • By convention, the indices x and y are integers between 0 and W − 1 and between 0 and H − 1 respectively. Throw a java.lang.IndexOutOfBoundsException if either x or y is outside its prescribed range.

  • Throw a java.lang.IllegalArgumentException if 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 removeVerticalSeam() or removeHorizontalSeam() is called when the width or height is 1, respectively.

Analysis of running time.  Analyze your approach to this problem giving estimates of its time and space requirements by answering the relevant questions in the readme.txt file.

• Estimate empirically the running time (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.

Challenge for the bored.

The challenges worth extra credit are quite difficult, and you certainly shouldn't attempt them just for the points. Make sure to note if you've successfully completed them in your readme.

1. Your energy() method implemented the dual gradient energy function. Try out other energy functions. No extra credit.

2. Add methods addVerticalSeam() and addHorizontalSeam() that expand the image by one column and and one row, respectively. One point of extra credit.

3. Optimize multiple calls to the find seam and remove seam methods by caching the previous call's information. One point of extra credit if you can get the algorithm running at interactive speeds for small images (say 200 x 200).

4. Implement an interactive object-removal feature: The user highlights an area of the image, and that portion of the image is forced to zero energy. Rows and columns are then successively removed until every pixel in that zero-energy region has been removed. One point of extra credit.

Submission.  Submit SeamCarver.java, and any other files needed by your program (excluding those in stdlib.jar and algs4.jar). Finally, submit a readme.txt file and answer the questions.