Quadtree encoding (a simple compression algorithm)

The quadtree encoding allows us to describe a $2^N×2^N$ 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 $2^N×2^N$ region;
• "$0$" denotes a split:
  the current $2^n×2^n$ region is divided into $4$ sub-regions of dimension $2^{n-1}×2^{n-1}$,
  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 $D_N$ as the $2^N×2^N$ image with the following coloring scheme:

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

What is the length of the minimal sequence describing $D_{24}$ ?