Digital image processing low-pass, high-pass, band-stop and band-pass filters

1. Filter type

        Spatial and frequency domain filters are generally divided into four types of filters - low-pass, high-pass, band-stop, and band-pass filters.

        Low Pass Filter: Allows only low frequency details to pass, attenuating high frequency details. Example: smoothing filter.

        High Pass Filter: Allows only high frequency details to pass, attenuating low frequency details. Example: Sharpen Mask filter.

        Band Stop Filter: Attenuates signals within a certain frequency range. Frequencies below a certain threshold and above another are allowed to pass.

        Bandpass filter: Allows only signals within a specific frequency band to pass, attenuating frequencies below a threshold and above another threshold to pass.

δ(x, y) is the unit pulse kernel

2. Generate a zone plate

        The zone plate is used to test the characteristics of the filter. There are several versions of the zone plate, the one I'll use can be generated by: , where x, y belong to [-8.2, 8.2] in steps of 0.0275, resulting in a 597x597 image.

         python reference code

def zone(x, y):
    return 0.5 * (1 + math.cos(x * x + y * y))

SIZE = 597
image = np.zeros((SIZE, SIZE))

start = -8.2
end = 8.2
step = 0.0275

def dist_center(y, x):
    global SIZE
    center = SIZE / 2
    return math.sqrt( (x - center)**2 + (y - center)**2)

for y in range(0, SIZE):
    for x in range(0, SIZE):
        if dist_center(y, x) > 300:
            continue
        y_val = start + y * step
        x_val = start + x * step
        image[y, x] = zone(x_val, y_val)

         Zone Plate Pictures

title

3. Apply a low-pass filter on the zone plate

        python reference code

kernel_size = 15

lowpass_kernel_gaussian = gkern(kernel_size)
lowpass_kernel_gaussian = lowpass_kernel_gaussian / lowpass_kernel_gaussian.sum()

lowpass_kernel_box = np.ones((kernel_size, kernel_size))
lowpass_kernel_box = lowpass_kernel_box / (kernel_size * kernel_size)

lowpass_image_gaussian = cv2.filter2D(image, -1, lowpass_kernel_gaussian)
lowpass_image_box = cv2.filter2D(image, -1, lowpass_kernel_box)

        example image

4. Apply a high-pass filter on the zone plate

        In the spatial domain, a high pass filtered image (like an unsharp mask) can be obtained by subtracting the low pass filtered image from the image itself.

        python reference code

highpass_image_gaussian = image - lowpass_image_gaussian
highpass_image_gaussian = np.absolute(highpass_image_gaussian)

highpass_image_box = image - lowpass_image_box
highpass_image_box = np.absolute(highpass_image_box)

        example image 

5. Apply a bandstop filter on the zone plate

        Similarly, a band-stop filtered image can be obtained by adding the low-pass filtered and high-pass filtered images (at different thresholds).

         python reference code 

bandreject_image = lowpass_image_gaussian + highpass_image_box

        example image 

6. Apply a bandpass filter on the zone plate

         Whereas a bandpass filtered image can be obtained by subtracting the bandstop filtered image from the image itself.

        python reference code 

bandpass_image = image - bandreject_image
bandpass_image = np.absolute(bandpass_image)

        example image