Homework 4 Solved

Description

In problems 1-3

• Do the operations without zero padding directly in the frequency domain. The distortion ensuing from circular convolution is less important than the extra effort to interpolate spectra to the desired size.

• It is best if you shift your spectra by 180^o so that DC is in the center. Similarly, your filters should be shifted to where M and N are the image dimensions as stated in the problems.

1. Unsharp Masking and High-Boost Filtering: Consider the 503×720 X-ray image of the chest (chestXray). Enhance the image using the Gaussian high-pass filtering approach. Recall that unsharp masking of an image is obtained as

where

1. Use a Gaussian high-pass filter , where . You can choose within 5% to 10% of the long image dimension of the image. Plot the corresponding filter mask as a heat map.

2. Plot the result of filtering with a Gaussian high-pass filter. Let 0 be represented as gray value (e.g., 80 or 128) and plot the absolute values of the GHPF output

3. Plot the result of unsharp masking the image, i.e., k=1 and the result of high-boost filtering for 1.6.

4. Histogram equalize the results of step c) and plot the results.

2. Moiré Noise Removal: Consider the newspaper image with moiré pattern (car-moire-pattern).

1. Plot the magnitude spectrum in perspective.

2. Identify the prominent spectral peaks: find their center points of the peaks interactively on the magnitude spectrum and estimate their diameters to contain most of the energy, e.g., 3 dB points.

3. Design a notch filter (NP: notch-pass) to extract the moiré pattern. You can use a Butterworth notch filter . Choose n = 4 and D according to your estimate of the diameters above. Note that if there is a spectral peak at (u,v) = (k,l), then there must be also a notch filter placed at (-k,-l). Furthermore, there should be such a notch filter pair for every spectral peak involved. Plot the extracted moiré pattern.

4. Plot the image with the moiré pattern removed.

3. Deblurring: Images can be degraded due to various environmental conditions and/or instrumental or imaging imperfections. If we can model the degradation, then we have a chance to recover the original image under certain noise conditions. Consider the 688×688 book-cover image.

1. Consider the model of an atmospheric turbulence where k controls the severity of the degradation. Set k = 0.0025. Add white Gaussian noise N(0, 625) to this image.

2. Plot side by side the original and the blurred & noisy version of the book-cover image.

3. Apply directly the inverse filter and also apply it with a cutoff frequency at radius 70 using a Butterworth low-pass filter of order 10. Comment on why low-pass filtering, which is expected to degrade the image, actually improves greatly the result of deblurring.

4. The image is degraded by camera motion and by AWGN N(0, 625). The model for linear camera motion distortion is where a, b denote the rate of motion along x and y directions and T the diaphragm aperture time. Take a= b = 0.1 and T = 1 sec. Plot the original and degraded images side by side.

5. Restore the image by direct inverse filtering and by Wiener filter, e.g., eq. 5.8.6.

4. Importance of the Phase

1. Reconstruct Trump from phase-only spectrum and magnitude-only spectrum,

2. Reconstruct Erdogan from phase-only spectrum and magnitude-only spectrum,

3. Reconstruct Trump from Erdogan’s phase spectrum and Trump’s magnitude spectrum

4. Reconstruct Erdogan from Trump’s phase spectrum and Erdogan’s magnitude spectrum

5. A few comments