JPEG SOF0 subsampling components - encoding

Question

with JPEG snoop for an image 4:2:2 hor (YYCbCr)
I see this in the SOF0:
Component[1]: ID=0x01, Samp Fac=0x21 (Subsamp 1 x 1), Quant Tbl Sel=0x00 (Lum: Y)
Component[2]: ID=0x02, Samp Fac=0x11 (Subsamp 2 x 1), Quant Tbl Sel=0x01 (Chrom: Cb)
Component[3]: ID=0x03, Samp Fac=0x11 (Subsamp 2 x 1), Quant Tbl Sel=0x01 (Chrom: Cr)
Now where are the values 0x21 and 0x11 coming from?
I know that sampling factors are stored like this: (1byte) (bit 0-3 vertical., 4-7 horizontal.)
but I don't see how 0x11 relates to 2x1 and 0x21 to 1x1.
I expected to see 0x11 for Y component and not 0x21.
(not sure how you get 0x21 as result).
Can somebody explain these values and how you calculate them for example 4:2:2 horizontal (16x8)?

Answer 1

JPEG does it bassackwarks. The values indicate RELATIVE SAMPLING RATES.
The highest sampling rate is for Y (2). The sampling rate for Cb and Cr is 1.
Use the highest sampling rate to normalize to pixels:
2Y = Cb = Cr.
Y = 1/2 Cb = 1/2 Cr.
For every Y pixel value in that direction you use 1/2 a Cb and Cr pixel value.
You could even have something like according to the JPEG standard.
4Y = 3Cb = 1Cr
Y = 3/4Cb = 1/4 Cr
or
3Y=2Cb=1Cr
Y=2/3Cb=1/3Cr
But most decoders could not handle that.
The labels like "4:4:4", "4:2:2", and "4:4:0" are just that: labels that are not in the JPEG standard. Quite frankly, I don't even know where those term even come from and they are not intuitive at all (there is never a zero sampling).
Let me add another way of looking at this problem. But first, you have to keep in mind that the JPEG standard itself is not implementable. Things necessary to encode images are undefined and the standard is sprawling with unnecessary stuff.
If a scan is interleaved (all three components), it is encoded in minimum coded units (MCUs). An MCU consists of 8x8 encoded blocks.
The sampling rate specifies the number of 8x8 blocks in an MCU.
You have 2x1 for Y + 1x1 for Cb and 1x1 for Cr. That means a total of 4 8x8 blocks are in an MCU. While I mentioned other theoretical values above, the maximum number of blocks in an MCU is 10. Thus 4x4 + 3x3 + 2x2 is not possible.
The JPEG standard does not say how those blocks are mapped to pixels in an image. We usually use the largest value and say that wave a 2x1 zone or 16x8 pixels.
But all kinds of weirdness is possible under the standard, such as:
Y = 2x1, Cb = 1x2 and Cr = 1x1
That would probably mean an MCU maps to a 16x16 block of pixels but your decoder would probably not support this. Alternatively, it might mean an MCA maps to a 16x8 block of pixels and the Cb component has more values in the 8 direction.
A final way of viewing this (the practicable way) is to use the Y component as a reference point. Assume that Y is always going to have 1 or 2 (and maybe a 4) as the sampling rate in the X and Y directions and define the rates on Cb and Cr are going to be 1 (and maybe 2).The Y component always defines the pixels in the image.
These would then be realistic possibilities:
Y Cb Cr
1x1, 1x1, 1x1
2x2, 1x1, 1x1
4x4, 1x1, 1x1
2x1, 1x1, 1x1
1x2, 1x1, 1x1