Filtering in the Frequency Domain II

(adapted in part from http://venus.javeriana.edu.co/course/course-index.html)
In many disciplines, signals and systems problems in N-dimensions are tackled either in the time or frequency domains. In the time-domain, these problems are solved using classical methods for solving differential/difference equations. In the frequency-domain, problems are solved by addressing algebraic equations. Where is the advantage? If integration and differentiation can be transformed into algebraic equations then we have facilitated the problem solving process.

The original image and its magnitude spectrum (dimension 256x256) is shown below

---

Observe that in the magnitude spectrum we can identify coherent noise, "stars" of different sizes. Our objective is to eliminate this noise pattern by making those "stars" equal zero. Displayed below is a pattern that we can use to multiply in the frequency domain and cancel the noise. Pass the mouse over the magnitude spectrum of the image to determine the coordinates of the spikes and create your frequency mask accordingly.

After multiplying in the frequency domain we get

Depicted below is the original image after elimination of the coherent noise

The absolute difference between the original image and the filtered image can be seen below (its maximum amplitude is approx. 6 in a scale of 0-255)





Wiener Filtering

In image restoration the goal is to recover an image that has been corrupted or degraded. The more information we have of the degradation process, the better off we are. This is know as a priori knowledge. There are several techniques in image restoration, some use frequency domain concepts, others attempt to model the degradation and apply the inverse process. The modeling approach requires determining a criterion of "goodness" that will yield an "optimal" solution.

Assuming no noise is present in the system, in the frequency domain the degradation process can be seen as 

    P(u,v) = H(u,v) Q(u,v)
where
    P(u,v) is the degraded image
    H(u,v) is the degradation transfer function
    Q(u,v) is the original image

And the inverse filtering process as 

    Q(u,v) = P(u,v) / H(u,v)

Experiment: deblurring an image using Wiener filtering

There is a technique known as Least Mean Square or Wiener Filtering. This technique assumes that if noise is present in the system, then it is considered to be additive white Gaussian noise (AWGN).

The degraded image for our experiment is shown below.

Blurred image

The log(mag+1) spectrum of the blurring function is

The result of the Wiener filtering operation is depicted below.

Deblurred image