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Quadtree encoding (a simple compression algorithm)

Problem 287

Published on 10 April 2010 at 09:00 am [Server Time]

The quadtree encoding allows us to describe a 2N×2N black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:

  • the first bit deals with the complete 2N×2N region;
  • "0" denotes a split:
    the current 2n×2n region is divided into 4 sub-regions of dimension 2n-1×2n-1,
    the next bits contains the description of the top left, top right, bottom left and bottom right sub-regions - in that order;
  • "10" indicates that the current region contains only black pixels;
  • "11" indicates that the current region contains only white pixels.

Consider the following 4×4 image (colored marks denote places where a split can occur):

This image can be described by several sequences, for example : "001010101001011111011010101010", of length 30, or
"0100101111101110", of length 16, which is the minimal sequence for this image.

For a positive integer N, define DN as the 2N×2N image with the following coloring scheme:

  • the pixel with coordinates x = 0, y = 0 corresponds to the bottom left pixel,
  • if (x - 2N-1)2 + (y - 2N-1)2≤ 22N-2 then the pixel is black,
  • otherwise the pixel is white.

What is the length of the minimal sequence describing D24 ?


Answer:
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