Is DCT (Discrete Cosine Transform) of Type-2 lossless or lossy?

Question

As I know, in general, the DCT is lossless. But I'm not exactly sure about Type-2 of DCT. Is it lossless or lossy?

Answer 1

Mostly yes, but it depends on the context. Let us elaborate.

DCT-II is one of the many forms of Discrete Cosine Transforms, and probably the most widely used one, as it is (somehow) present in JPEG or MP3 formats. "Lossy" often refer to the compression standard which uses it, because the main loss results from quantization (and generally not the transform by itself). By itself, the transform is invertible, even more orthogonal or orthonormal, so in theory you have no loss (it is bijective).

As @Fat32 points out, its coefficients sometimes involve cosines, hence irrational numbers (except at specific values) that are not easily represented in finite arithmetic (float or int), and can induce round-off errors.

But:

• DCT-II of size $1$ is $1$, DCT-II of size 2 (unnormalized) is: \begin{smallmatrix} 1 & 1 \\1 & -1\end{smallmatrix} so DCT-II can be lossless (in special cases though),
• For integer data (like images), it seems possible, with enough bit-depth computations (related to the initial integer range and the DCT size), to remain practically lossless, as you know that the transform/inverse transform result should be integer,
• Very accurate integer, or dyadic approximations exist, meant for integer hardware: for instance the binDCT.

My final answer is thus: DCT-II is mostly lossless, if you take care of it.