You need to define what's - for you - "best image quality". Is it the lack of noise at hight ISO, the vibrancy of colours, sharpness?... As you can probably gather, no single compact camera will have all these features. For example, the now-discontinued Fujifilm F31FD had undoubtedly the least amount of noise at hight ISO, which made it quite unique in the compact camera world. However, the new Fujifilm model that replaces it does not come close in that context - see Gordon's review on this website.
Personally, I tend to think that the Canon compact cameras have the best balance of image quality and features, and a great build. However, the Panasonics are also very good. Which points to you having to be more precise as to what you expect a "compact camera" to be, size-wise. Would you consider the Panasonic TZ3 or Canon G9 - reviewed on this site - to be compact cameras, or do you think of them as being smaller? If you accept the larger size, then you have more options, i.e. the TZ3 or the G9.
One common thing, though. At the moment, all compact cameras will have similar performance, the differences being more about their features. For example, the Canon G9 gives you more manual control than most compact cameras on the market (Ricoh's Caplio, anyone?...). However, this comes at a price. Another example would be the Olympus Mju-series that are shock-proof and water-resistant to a certain extent. Like I said: features. But as far as image quality - and in the context of image-size for which they are aimed at - I think they are quite similar.
Let's see what others have to say also.