Spatial filtering in Fourier domain

   Spatial filtering in Fourier domain is performed by the 2-dimensional discrete Fourier transformation (2D DFT) of the image (Fig. 1A to B), then multiplication of the transform with a 2D transfer function (see below; the product is Fig. 1E) and finally inverse 2D DFT (Fig. 1G). An algorithmic implementation of DFT is the fast Fourier transformation (FFT), but the abbreviations DFT and FFT are often synonymously used. The engineering term for this filtering method is the finite impulse response (FIR) filtering. In 2 dimensions FIR means that when the input of the filtering is a single ‘1’ pixel in an image of zeros (this is the impulse) the result decays to zero in a certain distance from the original ‘1’ (finite response). This response is the point spread function (PSF) of the filter. What is mathematically called convolution in space domain (see kernel convolution filtering) is simplified into multiplication in Fourier domain. The PSF is functionally identical to the kernel in the kernel convolution filtering. The 2D transfer function is therefore the Fourier transform of the convolution kernel.

Fig.1. Spatial Filtering in Fourier domain. (A) Maximum intensity projected wide-field fluorescence micrograph of a mito-DsRed2 expressing neuron, a 512´512 image, scaled at 0.27 mm/pixel. Fluorescence intensities were rescaled in the original image to span between 0 and 1000. Images are printed with the indicated gray scaling at g=1.6. Only the region in the rectangle is shown in spatial domain images below. (B) The Fourier domain of (A), shown in a logarithmic scale for better visibility. (C) High pass filter function with a steep, rectangular cut on edge (wcut_on=0.2 cycle/mm). (E) Effect of high pass filtering in Fourier domain and (G) in space domain. Note that the filter function cuts the middle of the Fourier domain with circular symmetry around the origin, and that the steep cut on edge leads to the appearance of ring or halo patterns around bright objects in (G). (D) Low pass filter with a steep cut off (wcut_off=0.2 cycle/mm), complementer of (C), i.e. the sum of the two filter functions is one at each w. (F) Effect of low pass filtering in Fourier domain and (H) in space domain. Because the two filter functions (C and D) are complementers, the sum (I) of images (G) and (H) is identical to the unfiltered image (A). (J-L) High pass filtering with a gradual cut on (J, K; Butterworth filter with wcut_on=0.2 cycle/mm and order=1.5; see 2) mitigates ring formation (K). Using band pass filter with the same cut on provides smoothing or noise suppression (J, L; using a wcut_off=1.2 cycle/mm and order=3). Scale bar, 10 mm.

The filter function has a biologically meaningful interpretation; can be used for selective transmission or rejection of given sized objects of the image; such as mitochondria (Fig. 1G) or bulkier local background (Fig. 1H). To transmit ~d sized objects, the filter coefficients should be >0 (~1 or greater) for sine wave components with a wavelength l and l/2≈d. To reject objects filter coefficients should be 0 for the given ls.. Because l=1/w, w=1/2d; e.g. 300 nm thick mitochondria will be represented around w=1/(2*0.3mm)=1.67 cycle/mm.

The Fourier domain images above are scaled between ±cycles/(2´pixel size) i.e. at 0.27 mm/pixel resolution (Fig. 1A) between ±1.85 cycles/mm (Fig. 1B,E,F), because for better visualization the origin was placed (both for space and Fourier domain images) into the center of the image. For a comparison, the highest spatial frequency transmitted by a wide-field microscope with an NA=1.3 lens for green light is w_max≈4cycles/mm (see supplementary material of ²). In an exact wording of the Nyquist Sampling Theorem the optimal resolution to completely determine a signal (or image in this case) is at double sampling frequency as compared to the w_max of the optics, thus at 1/(2´2´w_max) ≈1/(2´2´4 cycles/mm) =0.0625mm/pixel in our case.

  To perform filtering in Fourier domain we define the following basic filters:

1)      A high pass filter function (Fig. 1C-G) with an w=1/(2d) cut on (so the filter function is 1 above this w and 0 below) transmits objects larger and rejects objects smaller than d diameter. However, technically we use band pass filters instead of high pass filters, because the highest spatial frequencies of high resolution microscopic images carry only shot noise, but not image information. Therefore, for noise suppression, it is reasonable to cut down filter functions above the spatial frequencies of the details of the interest (Fig. 1J-L).

2)      A low pass filter (Fig. 1D-H) with an w=1/2d cut off (so the filter function is 0 above the cut off w and 1 below) can decrease the contribution of d diameter objects to the image, however due to blur and light scatter fluorescence intensity changes originating from smaller than d-sized objects cannot be completely omitted ¹.

3)      Filtering with a band pass filter (so the filter function is ~1 between the cut on and cut off ws and 0 otherwise) provides information about the amplitude of the given frequency range (band), or in fluorescent images, yields the intensity of puncti and filaments with a given range of sizes (see below at Ratiometric Optimized Spatial Filtering Technique and in ²).

Spatial filtering in Fourier domain is linear. The order of performing linear transformations does not affect the result. A mathematical property of linearity is that if image a+b=c then this is true for their Fourier transforms and the equality remains true when all three images are spatially filtered with the same filter. Fig. 1A was filtered with a high pass filter (C) resulting (G) and a low pass filter (D) resulting (H). These high and low pass filters were designed to be complementers (their sum is 1 for each w), and filtering with a filter function which is 1 everywhere does not change the image. Therefore the sum of the result images (G+H) provides an identical image (I) to the original one (A, rectangle). Of note, in some of our applications below we use absolute value calculation after spatial filtering, and by this step the linearity is lost.

Fourier transformation presumes that the signal (so the image) is cyclic. This means that the image is considered continuous by leaving through right side entering in the left, or leaving through the top entering in the bottom. This has that unfortunate consequence, that if there is a bright object on one edge of the image Fig. 2Ai, it will produce an edge artifact in the Fourier transform (Fig. 2Aii, arrows) and after filtering (Fig. 2Aiii, arrows), on both involved sides of the image. This can be prevented by mirrored tiling Fig. 2B; by creating a twice larger image which is mirror symmetric to its median axes. Then the mirrored-tiled image is transformed, however at four times of computational cost.