Change the black background to blue (same as the image color)

Answer 1

[Image Analyst]

Try this (untested)

% Extract the individual red, green, and blue color channels.
redChannel = rgbImage(:, :, 1);
greenChannel = rgbImage(:, :, 2);
blueChannel = rgbImage(:, :, 3);
% Get a mask for pure black.
blackMask = redChannel == 0 & greenChannel == 0 & blueChannel == 0;
% Make black blue
redChannel(blackMask) = 0;
greenChannel(blackMask) = 0;
blueChannel(blackMask) = 255;
% Recombine separate color channels into a single, true color RGB image.
rgbImage = cat(3, redChannel, greenChannel, blueChannel);
imshow(rgbImage);

[Image Analyst]

I'd probably take a histogram of each color channel. Ignore bin 1 because that will be zero. YOu need to find the gray level of the second most common gray level. That will be the purple. Then assign that to the black mask regions.