What does this formula mean?

Question

The formula in the image is depicting quantization

I would like to know if anyone can help me understand what is going on in the above formula? Am I supposed to first perform x = [c/s1] and then perform s1 * x?

Please help me understand this.

The paper in question.

Answer 1

In a nutshell, JPEG works by applying the discrete cosine transform to 8x8-pixel blocks of the image and then quantizing the resulting 8x8 matrix in order to compress it into fewer bits.

The quantization is what makes JPEG lossy. Different quantization matrices lead to different levels of compression (and decompressed image quality).

The formula in your question represents JPEG compression followed by decompression. The DCT yields an 8x8 matrix of float coefficients. Then:

• The DCT coefficient matrix is divided element-by-element by the quantization matrix.
• The results get rounded to integer.
• The integers are then multiplied back by the quantization coefficients.

The point that the paper is getting at is that when this process is done more than once (potentially with different quantization coefficients), this is detectable and can be used to find doctored JPEG images.

P.S. It seems to me that the author's notation is a bit sloppy. If I am not mistaken, JPEG rounds to the nearest integer whereas the paper uses the floor function, which rounds towards negative infinity. That aside, the main point of the paper stands.