Difference between manually-created gaussian filter and fspecial

I am wondering why there is difference between these two, and how could the max vaule in fspecial (0.4026) is larger than manually-created gaussian (0.3989) when x = 0?

manually-created gaussian filter code:

x = [-2 -1 0 1 2];
gauss_filter = 1/(sqrt(2*pi)*sigma)*exp(-x.^2/(2*sigma^2))

gauss_filter =

    0.0540    0.2420    0.3989    0.2420    0.0540

while the fspecial create gaussian filter:

gauss_filter = 1/(sqrt(2*pi)*sigma)*exp(-x.^2/(2*sigma^2))'

gauss_filter =

    0.0545    0.2442    0.4026    0.2442    0.0545

Answer 1

[DGM]

I believe the original question was pasted incorrectly, since both examples appear to be identical (except a transpose). Neither shows the use of fspecial().

However, the difference between the two results is simple. For the filters to not alter the image mean, their sum must be 1.

sigma = 1;
x = [-2 -1 0 1 2];

fk1 = 1/(sqrt(2*pi)*sigma)*exp(-x.^2/(2*sigma^2)) % as calculated

fk1 = 1×5

0.0540 0.2420 0.3989 0.2420 0.0540

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sum(fk1) % not sum-normalized

ans = 0.9909

fk1 = fk1/sum(fk1) % normalize it

fk1 = 1×5

0.0545 0.2442 0.4026 0.2442 0.0545

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sum(fk1) % sum-normalized

ans = 1