Constructing frequency filter from a 2d spatial filter
I found it! Placing the original center at the upper-left corner, by circularly-shifting $h$, makes $h$ an even function. From the properties of Fourier Transform, we know the DFT of a real and even function is real. So, $H$ becomes real and its phase angle is zero. Hence, the phase of the filtered image is as the phase of the original image.
By the way, shifting $h$ does not affect the frequency spectrum of $H$.