How to apply function f for each time value t, to each pixel from image, resulting in t images

I have a 2D array of complex values and I want to apply a function to every entry of the array, for every timepoint t in a list. I know how to apply my function to one entry for every time t in a list, and I know how to apply my function for one t to every entry in an array. (I keep finding answers to those problems when searching with google and on SE) Somehow I cant get both to work at the same tim, and I cant find an explanation of how to do it either. Maybe someone has experience in doing this?

Example:

In:
function[source_, t_] := source(1+Sin[2Pi*t]);
lena = Import["ExampleData/lena.tif"];
size = ImageDimensions[lena];
data = ImageData[lena];
times = Table[i, {i, 600}]/10;
complexred = Fourier[data[[1;;size[[2]],1;;size[[1]],1]]];
newdata = function[complexred, times];

Out:
Thread::tdlen : Objects of unequal length in
{{...some complex values here...}{<<1>>},<<48>>,<<430>>} cannot be combined.


Thanks!

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f(x) means f*x. Compare ImageDimensions[lena] and Dimensions[data] – ssch Sep 21 '13 at 23:50
Ah those brackets were a typo, sorry. Matlab automatisms :). Thank you for pointing out the difference in coordinates for images and matrices! Thats useful to me. – Leo Sep 22 '13 at 8:40

Consider this:

fn[img_Image][t_] := ImageApply[Evaluate[# (1 + Sin[2 Pi*t])] &, img]

times = Table[i, {i, 0, 1, 0.1}];

fn[lena] /@ times


I used SubValues notation for flexibility (direct mapping onto times) but it is not required. You can also use the function like this:

Table[fn[lena][i], {i, 0, 1, 0.1}]


I pre-evaluated the body of the Function using Evaluate for best speed, but this is also not required. (Note that Evaluate must be applied to the entire body of a Function for it to have the desired effect; to evaluate a piece of the body use With.)

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Thank you for answering my question and explaining how an why you perform each step. Nice and illustrative. I take it the /@ operator can be applied to arrays of complex numbers as well, as thats what bill s does. I dont quite understand the #*body &, img part yet. What do the # and & do there? – Leo Sep 22 '13 at 8:58
@Leo You're welcome. Let me know if you have any other questions about what I wrote. Yes, /@ which is short for Map can be applied to any expression, with control over level using the full notation Map[func, expr, levelspec]. Bookmark this post for reference. You can also select an operator such as /@, # or & in the Front End and press F1 to bring up the help page for it. (continued) – Mr.Wizard Sep 22 '13 at 9:04
# is short for Slot, and & is short for Function; they are the basic building blocks of anonymous functions in Mathematica. See this and linked resources for an explanation of levelspec. – Mr.Wizard Sep 22 '13 at 9:05
Both links are very useful, thanks. – Leo Sep 22 '13 at 9:13
@Leo I changed my With construct to Evaluate, which in this particular case is less baroque. Be careful to understand how Evaluate works before attempting to apply it elsewhere. – Mr.Wizard Sep 22 '13 at 10:41

You say that your data has the form of complex values, so I will make up some complex-valued data and a range of time:

data = RandomReal[{-1, 1}, {10, 10}] + I RandomReal[{-1, 1}, {10, 10}];
time = Range[0, 1, 0.05];


Now consider a function

f[x_, t_] := Abs[x]^t


The goal is to "apply the function to every element of the data, for every timepoint." One way to do this is:

g = f[data, #] & /@ time


which gives an array of size

Dimensions[g]
{21, 10, 10}


This makes sense since there are 21 points in the time list and the data is a 10x10 array.

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Thanks you for answering my question! Although strictly your answer more directly answers my question than mr.wizards (by using complex numbers and explaining the resulting array), I did accept his answer because both your answers are actually the same method and his answer also showed me some extra tricks for using /@ effectively. But I like your explanation about the dimensions of the resulting array, since that was in my question as well. So thanks. – Leo Sep 22 '13 at 8:47