Compression of an image - compression

Question

I have been calculating the uncompressed and compressed file sizes of an image. This for me has always resulted in the compressed image being smaller than the uncompressed image which I would expect. If an image contains a large number of different colours, then storing the palette takes up a significant amount of space, and more bits are also needed to store each code. However my question is, would it be possible the compression method could potentially result in a larger file than the uncompressed RGB image. What would the size (in pixels) of the smallest square RGB image, containing a total of k different colours, for which this compression method is still useful? So we want to find, for a given value of k, find the smallest integer number n for which an image of size n×n takes up less storage space after compression than the original RGB image.

Answer 1

Let's begin by making a small simplification -- the size of the encoded output depends on the number of pixels (the actual proportion of width vs. height doesn't really matter). Hence, let's generalize the problem to number of pixels N, from which we can always calculate n by taking a square root.
To further simplify the problem, we will also ignore the overhead of any image headers/metadata, such as width, height, size of the palette, etc. In practice, this would generally be some relatively small constant.
Problem Statement
Given that we have
N representing the number of pixels in an image
k representing the number of distinct colours in an image
24 bits per pixel RGB encoding
LRGB representing the length of a RGB image
LP representing the length of a palette image
our goal is to solve the following inequality
in terms of N.
Size of RGB Image
RGB image is just an array of N pixels, each pixel taking up a fixed number of bits given by the RGB encoding. Hence,
Size of Palette Image
Palette image consists of two parts: a palette, and the pixels.
A palette is an array of k colours, each colour taking up a fixed number of bits given by the RGB encoding. Therefore,
In this case, each pixel holds an index to a palette entry, rather than an actual RGB colour. The number of bits required to represent k values is
However, unless we can encode fractional bits (which I consider outside the scope of this question), we need to round this up. Therefore, the number of bits required to encode a palette index is
Since there are N such palette indices, the size of the pixel data is
and the total size of the palette image is
Solving the Inequality
And finally
In Python, we could express this in the following way:
import math
def limit_size(k):
return (k * 24.) / (24. - math.ceil(math.log(k, 2)))
def size_rgb(N):
return (N * 24.)
def size_pal(N, k):
return (N * math.ceil(math.log(k, 2))) + (k * 24.)

Answer 2

In general no, but your question is not precise.
If we compress normal files, they could be larger. E.g. if you compress a random generated sequence of bytes, there is not much to compress, and so you get the header of compression program, which tell which compression method is used, and some versioning. This will enlarge the file, and ev. some escaping. Good compression program will see that compression will not shrink the size, and so they should just not compress, and tell in the header that it is a flat file. Possibly this is done by region of program.
But your question is about images. Compression is done inside the file, and often not all file, but just the image bits. In this case program will see that there is no need to compress, and so they would keep the file uncompressed. But because the image headers are always present, this change only a flag, and so no increase of size.
But this could depends also on file format. You wrote about "palette", but this is not much used nowadays: compression is done finding similar pattern on file. But again: this depends on the image format. If you look in Wikipedia, for particular file format, you may see a table with headers parameters (e.g. bit depth or number of colours (palette), definitions of colours, and methods used to compress).
Then, for palette like image, the answer of Dan Mašek (https://stackoverflow.com/a/58683948/2758823) has some nice mathematical explanation, but one should not forget that compression is much heuristic and test of real examples: real images have patterns.