the image smoothing using the frequency domain low pass filter(a) Ideal (b) Butterworth (c) Gaussian

Image smoothing is a process used in image processing to reduce noise and detail in an image. This can be done using frequency domain low pass filters such as Ideal, Butterworth, and Gaussian. Here's how each of them works:

1. Ideal Low Pass Filter (ILPF):

   • Transform the image from spatial domain to frequency domain using Fourier Transform.
   • Design an Ideal Low Pass Filter (ILPF). This filter will allow low frequency components to pass through and block high frequency components.
   • Apply the ILPF to the image in the frequency domain. This will result in the smoothing of the image as the high frequency components (which represent the noise and details) are removed.
   • Transform the image back to the spatial domain using Inverse Fourier Transform.

2. Butterworth Low Pass Filter (BLPF):

   • Transform the image from spatial domain to frequency domain using Fourier Transform.
   • Design a Butterworth Low Pass Filter (BLPF). Unlike the ILPF, the BLPF does not have a sharp cutoff at the cutoff frequency. Instead, it gradually reduces the amplitude of the frequencies beyond the cutoff frequency.
   • Apply the BLPF to the image in the frequency domain. This will result in a smoother image as the high frequency components are gradually reduced.
   • Transform the image back to the spatial domain using Inverse Fourier Transform.

3. Gaussian Low Pass Filter (GLPF):

   • Transform the image from spatial domain to frequency domain using Fourier Transform.
   • Design a Gaussian Low Pass Filter (GLPF). The GLPF uses a Gaussian function to reduce the amplitude of the high frequency components. This results in a smoother transition between the passband and the stopband, reducing ringing artifacts.
   • Apply the GLPF to the image in the frequency domain. This will result in a smoother image with fewer artifacts.
   • Transform the image back to the spatial domain using Inverse Fourier Transform.

In all these methods, the key idea is to remove or reduce the high frequency components of the image, which represent the noise and details, to achieve image smoothing.