We evaluate an integral having to do with vector averages over all orientations in an n-dimensional space.
Problem definition
Let ˆv be a unit vector in n-dimensions and consider the orientation average of
where →a1,…,→ak are some given fixed vectors. For example, if all →ai are equal to ˆx, we want the orientation average of vkx.
Solution
We’ll evaluate our integral using parameter differentiation of the multivariate Gaussian integral. Let
The expression in the second line follows from completing the square in the exponent in the first — for review, see our post on the normal distribution, here. Now, we consider a particular derivative of I with respect to the α parameters. From the first line of (2), we have
The second factor above is almost our desired orientation average J — the only thing it’s missing is the normalization, which we can get by evaluating this integral without any →a‘s.
Next, we evaluate the parameter derivative considered above in a second way, using the second line of (2). This gives,
The sum here is over all possible, unique pairings of the indices. You can see this is correct by carrying out the differentiation one parameter at a time.
To complete the calculation, we equate (3) and (4). This gives
Again, to get the desired average, we need to divide the above by the normalization factor. This is given by the value of the integral (5) when k=0. This gives,
Example
Consider the case where k=2 and →a1=→a2=ˆx. In this case, we note that the average of ˆv2x is equal to the average along any other orientation. This means we have
We get this same result from our more general formula: Plugging in k=2 and →a1=→a2=ˆx into (6), we obtain
The two results agree.