Effective JPEG compression for HF content?

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I have an image with grayscale background and 2 thin lines (1 pixel wide) drawn in color. I'm trying to use various JPEG compression table (luminance and chrominance) to get the best possible result while staying under a certain file size. The grayscale background compresses well and looks decent. The thin vertical and horizontal color lines get mutilated / smeared. The current JPEG algorithm uses 2x2 sub-sampling on the Cb and Cr channels and the chrominance compression table is fairly aggressive (high compression).

Is there any way to embed a "BMP" type data into JPEG image. Basically specify a color for specific pixels to be applied after the JPEG is de-compressed? Any other ways clean up how thin color lines get encoded / decoded in JPEG without increasing the overall file size by a lot.

P.S. I'm testing all this stuff in Matlab.

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