Why is the laplace transform not popular for image processing convolution?
The Laplacian is indeed used in image processing routinely but, possibly not as much as Fourier transforms. Reasons (other than just the difference in span of uses, narrow vs wider) may be: Fourier transforms have been highly optimized due to their wide application, and are possibly less complicated theoretically than the Laplacian. sometimes the Laplacian of the Gaussian is taken for "blob detection".[2][3]

From the book Digital signal processing fundamentals By Ashfaq A. Khan p105:


  Convolution is the primarily tool in image processing while Laplace Transform is used mainly in signal processing, such as speech and controls systems.


[1] Laplace filter in image processing (with edge detection and motion estimation applications)

[2] Laplacian in blob detection intuition (mathoverflow)

[3] blob detection