Part I

1. Color Image Processing
   The purpose here is to improve the appearance of a color image of your choice. There are quite a few color images here. You may also want to scan in a color photo of personal interest. However, make sure that there is plenty of room for quality improvement in the color image chosen in terms of contrast, color saturation, and color biase (e.g., some color photos are too blue or red due to improper exposure and/or development), so that the effect of your processing will be most significant. Submit the color image of your choice as image 0.

   Note: Two pairs (four) of RGB-HSI conversion functions are provided in this program. They are based on, respectively, the RGB cube color model and the HSV hexcone model. Go through the code to make sure you understand how the conversion is done before you use it. For example, you should know that the intensity in the function is represented by a value between 0 and 1 (instead of 0 and 255!). Try both color models for the following and submit the better one for each. Comment on the difference (or similarity) between the two, if any.

   (a) Increase the dynamic range of the intensity I of the image by linearly stretching its intensity histogram while keeping the hue and saturation unchanged. Submit the result as image 1.

   (b) Increase the saturation S of the image while keeping their hue and intensity unchanged. Submit the result as image 2.

   (c) Modify the hue H by changing the hue angle. Submit the image with increased hue (e.g. by 45 degrees) as image 3 and the one with decreased hue as image 4.

   (d) Modify all three parameters H, S, and I of the image to make it most beautiful! Submit the result as image 5.

   (e) Choose a gray level image and convert it into a psudo color image by assigning a color (in terms of the R, G, and B values) to each of the 256 gray levels through a color-lookup table, a 256 (for gray levels) by 3 (for R,G,B) array. You are free to choose the contents of the table for the color mapping. Submit the result as image 6.

Part II

1. Spatial Frequency Filtering with DFT
   • Use the 1D FFT code in your own code for 2D Fourier transform of an N by N image. As discussed in class, all you need to do is to run 2N 1D transforms: first you transform each of the N columns of the image array and put the transformed columns back in the same order to form a new array, then you transform each of the N rows of this new array and put them back in the same order. The resultant N by N 2D array is the spectrum of your image. (You could also do the row transforms first and then the column transforms. The order is not important.) Remember DFT is a complex transform. You could use the image as the real part of the complex array and always use zeros for the imaginary part. Or, better, you can use the method (discussed in class and also in the handout on Fourier transform) to transform 2 real vectors at the same time and separate their spectra after the transform. You can save half of the operation during the column (or row) transform, after which the method can no longer be applied as the data is no longer real. (But I have an algorithm which can take this advantage all the way in the forward and inverse transforms so that computation will be reduced to half. If you are serious to know, I can provide the paper.)

     In order to visualize the spectrum to see how the energy is distributed in the frequency domain, convert the spectrum in the complex array into a real image. To do this, you first find the magnitude of each frequency component (with both real and imaginary parts), then normalize these values so that they range from 0 to 255. As most of the energy is contained in the DC component and low frequency components, you will most likely see a bright spot at the center of the spectrum (if centralized). To see better how the rest of the total energy is distributed, you can scale their magnitude nonlinearly by, say, y=pow(x, 0.3). Then save the spectrum as an image. Submit the image and its spectrum as images 7 and 8.

   • Frequency domain Filtering can be achieved by multiplying the complex spectrum by a "zonal mask", such as an ideal filter mask or a Butterworth mask. Whether the filter is low-pass or high-pass can be determined by, for example, whether the spectrum is centralized or not. This way the same zonal mask can be used for both low-pass and high-pass filters. Filter the image using the following filters:

     (i) an ideal low-pass filter (ILPF).
     (ii) an ideal high-pass filter (IHPF).
     (iii) a Butterworth low-pass filter (BLPF).
     (iv) a Butterworth high-pass filters (BHPF).
     Try a few different orders for Butterworth filter to see the effect. Also try different cut-off frequencies to see the filtering effect. When considering the cut-off frequency, always keep in mind that most of the energy is concentrated a round the DC component. In other words, the radius of the filter (representing cut-off frequency) can be small for low-pass filtering (as most low frequency components are kept), and it should be large to cover enough energy for high-pass filtering (as the DC and some low frequency components are filtered out).

   • Inverse transform the filtered spectrum back to spatial domain. You can use the same 1DFT code except now you need to set the switch "inverse" to 1. Submit the four images filtered differently as images 9, 10, 11, and 12. Comment on the differences between the high-pass and low-pass filters, and between the ideal and Butterworth filters (you should be able to observe the ringing effect). Note that you may need to do some post-processing (such as linear stretching) for the output images to look right.

2. Spatial frequency Filtering with DCT
   

   Use the 1D fast DCT function provided to implement your own DCT 2D filtering. Similar to the 2D DFT, here you transform the columns and then the rows to get the 2D spectrum. Print the spectrum as an image (after nonlinear scaling and normalization) to have a visual display of the spectrum. Submit this frequency spectral 2D array as image 13.

   Then both high-pass and low-pass filter the image. Different from the Fourier spectrum, now the lowest frequency component, the DC component, is at the upper-left corner and the highest sequency component is at the lower-right corner of the 2D array of the spectrum. You should design your filters accordingly. To avoid the ringing effects, use any of the smooth filters discussed in class (Gaussian, Butterworth, or Hamming filter). Then inverse transform the filtered spectrum back to the spatial domain and submit the two filtered images as images 14 and 15.

3. Spatial Sequency Filtering with WHT
   

   Now do the same for sequency filtering. Use the provided 1D fast WHT function to do 2D WHT filtering. Save the sequency spectrum as an image (after rescaling and normalization) and submit it as image 16.

   Then both high-pass and low-pass filter the sequency spectrum of the image in the same way as 2D DCT filtering (you can use exactly the same code). Then inverse transform the filtered spectrum back to the spatial domain. Note that the inverse transform is identical to the forward transform (How nice!). Submit the two filtered as images 17 and 18.

4. Spatial Sequency Filtering with discrete Haar transform (DHT)
   

   Repeat the previous part based on the DHT. Submit the results as images 19, 20, and 21.