For signals (as opposed to images), you should have a look at "Understanding Digital Signal Processing" by Richard G. Lyons (ISBN-13: 978-0131089891). I enjoyed this very much and if you grasped the contents from it, you should be able to understand audio-specific books easily.
I second Lyons. In addition, "Digital Signal Processing: A Practical Guide" by Steven Smith is also quite approachable -- immersion in calculus not required.
"A Digital Signal Processing Primer" by Ken Steiglitz is a nice but rigorous intro to the subject. Written by an EE academic, it's more mathematically rigorous than Smith or Lyons.
Allen Downey's "Think DSP" is also worth a look, though its focus is more conceptual than practical, IMO.
