Question

What I want to happen, in order: Alert box pops up saying processing (no buttons), the main processing function is called, when that completes the Alert box disappears, and then my navigation controller moves to the next view in the series.

What actually happens: The window freezes while the app processes, then the Alert box shows and disappears instantly while the navigation controller changes to the next view.

Best suggestion on how to fix this?

Code:
[_navigationController dismissModalViewControllerAnimated:NO]; //Camera dismisses
UIAlertView *alert;

alert = [[[UIAlertView alloc] initWithTitle:@"Analyzing\nPlease Wait..." message:nil delegate:nil cancelButtonTitle:nil otherButtonTitles: nil] autorelease];
[alert show];

[self processImage];

[alert dismissWithClickedButtonIndex:0 animated:YES];
[self gotoResults];

Answer 1

alert = [[[UIAlertView alloc] initWithTitle:@"Analyzing\nPlease Wait..." message:nil delegate:nil cancelButtonTitle:nil otherButtonTitles: nil] autorelease];
[alert show];

When calling [alert show], the UI event is pushed into main run loop. You can process other tasks in another function. For example, a function named foo processes other tasks. And after calling [alert show], we call [self performSelector:@selector(foo) withObject:nil afterDelay:0.1] to invoke foo. That will give run loop some time to handle the showing event of alert view.

- (void)foo {
  UIAlert *alert = ... ///< come from above 
  [self processImage];

  [alert dismissWithClickedButtonIndex:0 animated:YES];
  [self gotoResults];
}

Edit:
You can read about -performSelector:withObject:afterDelay: in this apple official document. You can assign 0.0 as parameter for afterDelay, and the document said:

delay

---

The minimum time before which the message is sent. Specifying a
delay of 0 does not necessarily cause the selector to be performed
immediately. The selector is still queued on the thread’s run loop and
performed as soon as possible.