In the field of Image Processing, Butterworth Lowpass Filter (BLPF) is used for image smoothing in the frequency domain. It removes high-frequency noise from a digital image and preserves low-frequency components. The transfer function of BLPF of order is defined as-
- is a positive constant. BLPF passes all the frequencies less than value without attenuation and cuts off all the frequencies greater than it.
- This is the transition point between H(u, v) = 1 and H(u, v) = 0, so this is termed as cutoff frequency. But instead of making a sharp cut-off (like, Ideal Lowpass Filter (ILPF)), it introduces a smooth transition from 1 to 0 to reduce ringing artifacts.
- is the Euclidean Distance from any point (u, v) to the origin of the frequency plane, i.e,
Step 1: Input – Read an image
Step 2: Saving the size of the input image in pixels
Step 3: Get the Fourier Transform of the input_image
Step 4: Assign the order and cut-off frequency
Step 5: Designing filter: Butterworth Low Pass Filter
Step 6: Convolution between the Fourier Transformed input image and the filtering mask
Step 7: Take Inverse Fourier Transform of the convoluted image
Step 8: Display the resultant image as output
Implementation in MATLAB:
Input Image –
Note: A Butterworth filter of order 1 has no ringing artifact. Generally ringing is imperceptible in filters of order 2. But it can become a significant factor in filters of a higher order. For a specific cut-off frequency, ringing increases with an increase in the filter order.
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