Download the file yacht.tif for the following section. Click here for help on the Matlab image command.

In order to process images within Matlab, we need to first understand their numerical representation. Download the image file yacht.tif . This is an 8-bit monochrome image. Read it into a matrix using

A = imread('yacht.tif');

Type whos to display your variables. Notice under the "Class" column that the AA matrix elements are of type uint8 (unsigned integer, 8 bits). This means that Matlab is using a single byte to represent each pixel. Matlab cannot perform numerical computation on numbers of type uint8, so we usually need to convert the matrix to a floating point representation. Create a double precision representation of the image using B = double(A); . Again, type whos and notice the difference in the number of bytes between AA and BB. In future sections, we will be performing computations on our images, so we need to remember to convert them to type double before processing them.

Display yacht.tif using the following sequence of commands:

image(B);

colormap(gray(256));

axis('image');

The image command works for both type uint8 and double images. The colormap command specifies the range of displayed gray levels, assigning black to 0 and white to 255. It is important to note that if any pixel values are outside the range 0 to 255 (after processing), they will be clipped to 0 or 255 respectively in the displayed image. It is also important to note that a floating point pixel value will be rounded down ("floored") to an integer before it is displayed. Therefore the maximum number of gray levels that will be displayed on the monitor is 255, even if the image values take on a continuous range.

Now we will practice some simple operations on the yacht.tif image. Make a horizontally flipped version of the image by reversing the order of each column. Similarly, create a vertically flipped image. Print your results.

Now, create a "negative" of the image by subtracting each pixel from 255 (here's an example of where conversion to double is necessary.) Print the result.

Finally, multiply each pixel of the original image by 1.51.5, and print the result.

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Figure 2: Histogram of an 8-bit image

The histogram of a digital image shows how its pixel intensities are distributed. The pixel intensities vary along the horizontal axis, and the number of pixels at each intensity is plotted vertically, usually as a bar graph. A typical histogram of an 8-bit image is shown in Figure 2.

Write a simple Matlab function Hist(A) which will plot the histogram of image matrix AA. You may use Matlab's hist function, however that function requires a vector as input. An example of using hist to plot a histogram of a matrix would be

x=reshape(A,1,M*N);

hist(x,0:255);

where AA is an image, and MM and NN are the number of rows and columns in AA. The reshape command is creating a row vector out of the image matrix, and the hist command plots a histogram with bins centered at [0:255][0:255].

Download the image file house.tif , and read it into Matlab. Test your Hist function on the image. Label the axes of the histogram and give it a title.

Figure 3: Pointwise transformation of image

A pointwise transformation is a function that maps pixels from one intensity to another. An example is shown in Figure 3. The horizontal axis shows all possible intensities of the original image, and the vertical axis shows the intensities of the transformed image. This particular transformation maps the "darker" pixels in the range [0,T1][0,T1] to a level of zero (black), and similarly maps the "lighter" pixels in [T2,255][T2,255] to white. Then the pixels in the range [T1,T2][T1,T2] are "stretched out" to use the full scale of [0,255][0,255]. This can have the effect of increasing the contrast in an image.

Pointwise transformations will obviously affect the pixel distribution, hence they will change the shape of the histogram. If a pixel transformation can be described by a one-to-one function, y=f(x)y=f(x), then it can be shown that the input and output histograms are approximately related by the following:

Since xx and yy need to be integers in Equation 3, the evaluation of x=f-1(y)x=f-1(y) needs to be rounded to the nearest integer.

The pixel transformation shown in Figure 3 is not a one-to-one function. However, Equation 3 still may be used to give insight into the effect of the transformation. Since the regions [0,T1][0,T1] and [T2,255][T2,255] map to the single points 0 and 255, we might expect "spikes" at the points 0 and 255 in the output histogram. The region [1,254][1,254] of the output histogram will be directly related to the input histogram through Equation 3.

First, notice from x=f-1(y)x=f-1(y) that the region [1,254][1,254] of the output is being mapped from the region [T1,T2][T1,T2] of the input. Then notice that f'(x)f'(x) will be a constant scaling factor throughout the entire region of interest. Therefore, the output histogram should approximately be a stretched and rescaled version of the input histogram, with possible spikes at the endpoints.

Write a Matlab function that will perform the pixel transformation shown in Figure 3. It should have the syntax

output = pointTrans(input, T1, T2) .

Hints

Download the image file narrow.tif and read it into Matlab. Display the image, and compute its histogram. The reason the image appears "washed out" is that it has a narrow histogram. Print out this picture and its histogram.

Now use your pointTrans function to spread out the histogram using T1=70T1=70 and T2=180T2=180. Display the new image and its histogram. (You can open another figure window using the figure command.) Do you notice a difference in the "quality" of the picture?

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Smoothing operations are used primarily for diminishing spurious effects that may be present in a digital image, possibly as a result of a poor sampling system or a noisy transmission channel. Lowpass filtering is a popular technique of image smoothing.

Some filters can be represented as a 2-D convolution of an image f(i,j)f(i,j) with the filter's impulse response h(i,j)h(i,j).

Some typical lowpass filter impulse responses are shown in Figure 5, where the center element corresponds to h(0,0)h(0,0). Notice that the terms of each filter sum to one. This prevents amplification of the DC component of the original image. The frequency response of each of these filters is shown in Figure 6.

Figure 5: Impulse responses of lowpass filters useful for image smoothing.

(a) (b) (c)

Figure 6: Frequency responses of the lowpass filters shown in Fig. 5.

(a) (b) (c)

An example of image smoothing is shown in Figure 7, where the degraded image is processed by the filter shown in Figure 5(c). It can be seen that lowpass filtering clearly reduces the additive noise, but at the same time it blurs the image. Hence, blurring is a major limitation of lowpass filtering.

Figure 7: (1) Original gray scale image. (2) Original image degraded by additive white Gaussian noise, N(0,0.01)N(0,0.01). (3) Result of processing the degraded image with a lowpass filter.

(a) (b) (c)

In addition to the above linear filtering techniques, images can be smoothed by nonlinear filtering, such as mathematical morphological processing. Median filtering is one of the simplest morphological techniques, and is useful in the reduction of impulsive noise. The main advantage of this type of filter is that it can reduce noise while preserving the detail of the original image. In a median filter, each input pixel is replaced by the median of the pixels contained in a surrounding window. This can be expressed by

where WW is a suitably chosen window. Figure 8 shows the performance of the median filter in reducing so-called "salt and pepper" noise.

Figure 8: (1) Original gray scale image. (2) Original image degraded by "salt and pepper" noise with 0.05 noise density. (3) Result of 3×33×3 median filtering.

(a) (b) (c)

Download the files race.tif, noise1.tif and noise2.tif for this exercise. Click here for help on the Matlab mesh command.

Among the many spatial lowpass filters, the Gaussian filter is of particular importance. This is because it results in very good spatial and spectral localization characteristics. The Gaussian filter has the form

where σ2σ2, known as the variance, determines the size of passband area. Usually the Gaussian filter is normalized by a scaling constant CC such that the sum of the filter coefficient magnitudes is one, allowing the average intensity of the image to be preserved.

Write a Matlab function that will create a normalized Gaussian filter that is centered around the origin (the center element of your matrix should be h(0,0)h(0,0)). Note that this filter is both separable and symmetric, meaning h(i,j)=h(i)h(j)h(i,j)=h(i)h(j) and h(i)=h(-i)h(i)=h(-i). Use the syntax

h=gaussFilter(N, var)

where N determines the size of filter, var is the variance, and h is the N×NN×N filter. Notice that for this filter to be symmetrically centered around zero, NN will need to be an odd number.

Use Matlab to compute the frequency response of a 7×77×7 Gaussian filter with σ2=1σ2=1. Use the command

H = fftshift(fft2(h,32,32));

to get a 32×3232×32 DFT. Plot the magnitude of the frequency response of the Gaussian filter, |HGauss(ω1,ω2)||HGauss(ω1,ω2)|, using the mesh command. Plot it over the region [-π,π]×[-π,π][-π,π]×[-π,π], and label the axes.

Filter the image contained in the file race.tif with a 7×77×7 Gaussian filter, with σ2=1σ2=1.

Now write a Matlab function to implement a 3×33×3 median filter (without using the medfilt2 command). Use the syntax

Y = medianFilter(X);

where X and Y are the input and output image matrices, respectively. For convenience, you do not have to alter the pixels on the border of XX.

Download the image files noise1.tif and noise2.tif . These images are versions of the previous race.tif image that have been degraded by additive white Gaussian noise and "salt and pepper" noise, respectively. Read them into Matlab, and display them using image. Filter each of the noisy images with both the 7×77×7 Gaussian filter (σ2=1σ2=1) and the 3×33×3 median filter. Display the results of the filtering, and place a title on each figure. (You can open several figure windows using the figure command.) Compare the filtered images with the original noisy images. Print out the four filtered pictures.

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Image sharpening techniques are used primarily to enhance an image by highlighting details. Since fine details of an image are the main contributors to its high frequency content, highpass filtering often increases the local contrast and sharpens the image. Some typical highpass filter impulse responses used for contrast enhancement are shown in Figure 9. The frequency response of each of these filters is shown in Figure 10.

Figure 9: Impulse responses of highpass filters useful for image sharpening.

(a) (b) (c)

Figure 10: Frequency responses of the highpass filters shown in Fig. 9.

(a) (b) (c)

An example of highpass filtering is illustrated in Figure 11. It should be noted from this example that the processed image has enhanced contrast, however it appears more noisy than the original image. Since noise will usually contribute to the high frequency content of an image, highpass filtering has the undesirable effect of accentuating the noise.

Figure 11: (1) Original gray scale image. (2) Highpass filtered image.

(a) (b)

Download the file blur.tif for the following section.

In this section, we will introduce a sharpening filter known as an unsharp mask. This type of filter subtracts out the “unsharp” (low frequency) components of the image, and consequently produces an image with a sharper appearance. Thus, the unsharp mask is closely related to highpass filtering. The process of unsharp masking an image f(i,j)f(i,j) can be expressed by

where h(i,j)h(i,j) is a lowpass filter, and αα and ββ are positive constants such that α-β=1α-β=1.

Analytically calculate the frequency response of the unsharp mask filter in terms of αα, ββ, and h(i,j)h(i,j) by finding an expression for

Using your gaussFilter function from the "Smoothing Exercise" section, create a 5×55×5 Gaussian filter with σ2=1σ2=1. Use Matlab to compute the frequency response of an unsharp mask filter (use your expression for Equation 11), using the Gaussian filter as h(i,j)h(i,j), α=5α=5 and β=4β=4. The size of the calculated frequency response should be 32×3232×32. Plot the magnitude of this response in the range [-π,π]×[-π,π][-π,π]×[-π,π] using mesh, and label the axes. You can change the viewing angle of the mesh plot with the view command. Print out this response.

Download the image file blur.tif and read it into Matlab. Apply the unsharp mask filter with the parameters specified above to this image, using Equation 10. Use image to view the original and processed images. What effect did the filtering have on the image? Label the processed image and print it out.

Now try applying the filter to blur.tif, using α=10α=10 and β=9β=9. Compare this result to the previous one. Label the processed image and print it out.

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