Problem 287
Quadtree encoding (a simple compression algorithm)
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 ?
四分树编码（一个简单的压缩算法）
四分树编码使我们能够将一个2N×2N的黑白图片表示为一个比特串（0和1）。该比特串可以从左到右解读为：
- 第一个比特位描述完整的2N×2N区域；
- “0”表示对当前区域进行分割：
将当前的2n×2n区域分割为4个2n-1×2n-1区域，
接下来的比特位按照左上、右上、左下、右下的顺序分别描述这四个区域； - “10”表示当前区域只包含黑色像素；
- “11”表示当前区域只包含白色像素。
考虑如下的4×4图象（彩色标记意味着需要进行分割的位置）：
这个图像可以用多个不同的比特串表示，例如：
“001010101001011111011010101010”，长度为30，或者
“0100101111101110”，长度为16，后者是描述这幅图象的最短比特串。
对于一个正整数N，记DN是按照如下染色方案得到的2N×2N图象：
- 左下角的像素其坐标为x = 0，y = 0，
- 如果(x - 2N-1)2 + (y - 2N-1)2 ≤ 22N-2，那么该坐标的像素为黑色，
- 否则该坐标的像素为白色。
描述D24的最短比特串长度是多少？