CINDY 186

Activity 18 | Noise Models and Basic Image Restoration

noreply@blogger.com (Anonymous) — Mon, 05 Oct 2009 12:55:00 +0000

Image restoration is a basic image processing application where an image that has been degraded or damaged is being recovered. To recover the image, the degradation must be known and thus can be used in restoring the image.

A common source of degradation is the presence of noise in the image. Noise can hide useful information in an image. It is acquired mostly during image acquisition and transmission. Noise is composed of random values that follows a distribution which is expressed by a Probability Density Function of PDF. Figure 1 shows the respective PDFs of the different noise types to be used in this activity.

Figure 1. PDFs of different noise types.

In this activity we implement basic image restoration of noisy images using different filters. We use a test pattern shown by Figure 3 along with its PDF and then add noise to the image which will be used as the image to be restored. In restoring the noisy images, four different types of filtering, shown by Figure 2. are investigated for each type of noise.

Figure 2. Types of filtering used.

Figure 3. Test image and its PDF.

It can be seen that each peak in the PDF of the original test image has spread into the shape of the distribution added noise of the such that the three gray levels are almost coinciding with each other. This is the degradation that is done when adding noise. The same goes for the other distributions. The PDFs of the restored images simply shows how the filters restore the image which is by filtering out the unwanted gray values by thus thinning the fattened distribution for each peak gray value.

Figure 4 shows the noisy image for the added Gaussian noise with mean = 0.5 and standard deviation = 0.8. The resulting restored images using the five filters were almost the same achieving the three distinct gray value peaks which is close enough to the original.

Figure 4. Image with gaussian noise and restored images.

Figure 5 shows the noisy image for the Rayleigh Distribution. From the PDFs of the restored images, it can be said that the restored images using the arithmetic and geometric mean filter were much better than the other filters.

Figure 5. Image with rayleigh noise and restored images.

For the image with Gamma Noise shown in Figure 6, all the filters produced almost the same restored images except for the +Q Contraharmonic filter where the spread of the gray values is still visible.

Figure 6. Image with gamma noise and restored images.

The test image added with Exponential Noise is shown in Figure 7 and the restored image using the +Q Contraharmonic is clearly not a good result which is obvious from the image itself. Looking at the PDF for this restored image shows that the peaks were just shifted and broadened thus resulting to the noisy image.

Figure 7. Image with exponential noise and restored images.

Uniform Noise added image shown in Figure 8 shows great degradation due to the shape of the distribution itself where the is no defined peak value. Here, the arithmetic mean filter seems to have produced the best results although the peak gray values are shifted resulting to a lighter gray value for the area that is supposed to be black.

Figure 8. Image with uniform noise and restored images.

For the Salt and Pepper Noise it can be seen that the arithmetic mean filter just blurs the salt and pepper noise thus smoothening it out. It can also be noted that the +Q contraharmonic filter filters out the pepper noise while the geometric, harmonic and -Q contraharmonic filters can filter out the salt noise.

Figure 9. Image with impulse or salt and pepper noise and restored images.

The arithmetic mean filter basically blurs the image to smooth out the noise. The noise models and image restoration filters are also applied to a grayscale image which was used in a previous activity. The following series of images show the image with the different added noise types and the corresponding restored images using the five filters.

I give myself a grade of 10 for this activity.

Activity 17 | Photometric Stereo

noreply@blogger.com (Anonymous) — Thu, 10 Sep 2009 01:58:00 +0000

Light when shone upon an object gives us an idea on the reflectance of that object that can be used in obtaining information about the object like color. The information stored in an objects reflectance due to a light source is used in many image processing methods which we have explored in the previous activities. Aside from the information that we have been using, we can also obtain the profile or the structure of the object by using images of the object with different light source positions and reconstruct the 3D structure of that object. This is what we call Photometric Stereo.

When light illuminates an object it produces shadows and color gradients all throughout the illuminated surface. And by getting images of the object at different light source locations, we obtain intensity profile of the whole illuminated surface. Different light sources also affect the intensity profile of the captured image of an object. Figure 1 shows the basic idea of Photometric Stereo where the angle in which light hits the surface of the object affects the captured intensity given by Equation 1 for a light source at infinity for that point P.

Figure 1. Photometric Stereo.

Equation 1. Intensity at point P(x,y).

In this activity, we use four images of an object taken with different light source positions and use it to reconstruct the 3D structure of the illuminated area, as shown by Figure 2.

Figure 2. Images of object at different light source positions.

To calculate for the surface normal of the object we need to get the far away point source locations given by V which was already provided, and the intensity I which is simply the images . The equations needed for calculating the surface normal is shown in Figure 2.

Figure 3. Equations needed for solving the surface normal.

After getting the surface normal, we can now compute for the surface elevation of the object at point (u,v) which is given by z = f(u,v) as shown in the equation below.

Figure 4. Surface elevation at point (u,v).

The resulting reconstructed 3D surface is shown below in Figure 5. We can see that the reconstructed surface is not smooth having jagged peaks and grooves. Also there is a concave dip at the areas where a transition in the intensity is located by comparing it to the four object images shown previously. All in all, the semi-spherical shape obtained matches the expected structure of the object from the images. I give myself a grade of 9 in this activity due to the roughness of the reconstructed surface.

Figure 5. Reconstructed 3D surface of object.

Activity 16 | Neural Networks

noreply@blogger.com (Anonymous) — Thu, 10 Sep 2009 01:54:00 +0000

We have already investigated and implemented two methods for object classification using object features which are Minimum Euclidean Distance and Linear Discriminant Analysis. Another method that can be used for object classification is Neural Networks. This method imitates the way the human brain process information like a network of neurons thus the name Neural Networks.

The human brain manages information by forming paths for that certain information and thus 'learns' the information unlike LDA and the Minimum Distance which relies heavily on the features of the object. Examples and iterations develops or trains the network thus speeds up recognition. The framework of a Neural Network is shown below.

Figure 1. Neural Network Framework

A Neural Network is composed of three layers which are the input layer, hidden layer, and the output layer. The number of inputs in the input layer is determined by the number of features while the hidden layer can be composed of any desired number of nodes. The input receives the data/signals/features which in turn is forwarded to the hidden layer which acts on the inputs and finally passes the results to the output layer.

In this activity we feed the obtained features of our objects to a neural network composed of 3 input nodes that represent the 3 features used namely, area, perimeter and RGB sum. There are also 3 hidden nodes, and an output node. The learning rate used is 0.5 and the number of iterations is 1000.

To train the Neural Network we first entered into the network the training set and then followed by the test set. Figure 2 below shows the results obtained from the Neural Network. To implement this method, the code used by the previous AP186 class, made by Mr. Jeric Tugaff, was utilized. The classification showed 100% accuracy. Changing the order of the objects is also an interesting direction that can be took which might challenge the accuracy of the method since the Neural Network is not trained enough for recognition. I give myself a 10 for this activity.

Figure 2. Resulting classification obtained using Neural Networks.

Activity 15 | Probabilistic Classification

noreply@blogger.com (Anonymous) — Thu, 10 Sep 2009 01:53:00 +0000

Previously we have performed object classification by using the Minimum Euclidean Distance of the obtained features of the train and test objects. Another method for classifying objects is the Linear Discriminant Analysis or LDA. This method simply uses the probability that an object belongs to a certain group which depends on the features of that specific object and the mean feature vector for that certain group. This also depends on the number of samples present in a group. The highest probability that an object belongs to a group automatically classifies that object into that group. This is called Bayes Rule. LDA uses this rule but starts by assuming the distribution of each group which a multivariate Normal Distribution and that all groups share the same covariance matrix. The probability that an object belongs to a group is given by,

where, is the mean of features for group i, is the feature data of object/row k in x which is the matrix containing the features data, C is the pooled within group covariance matrix, and p is the prior probability vector for object i. Simply, object k is assigned to group with highest f. The results for the data collected from last activity is shown below.

The yellow highlighted areas are the maximum f values. The test objects were also jumbled to show that the method works no matter what the order of test objects is. It can be seen that there is a 100% accuracy in the classification of the objects. I give myself a grade of 10 for this activity for successfully implementing LDA and getting a 100% accurate object classification.

References: Kardi Teknomo's Page - Discriminant Analysis Tutorial http://people.revoledu.com/kardi/tutorial/LDA/index.html

Activity 14 | Pattern Recognition

noreply@blogger.com (Anonymous) — Thu, 27 Aug 2009 01:08:00 +0000

Sorting or classifying objects using visual information depends on the features of the objects. The human brain can draw, process and manipulate these features from a previous encounter to identify objects. This ability is being implemented in computer vision although it is not on the same level as the human brain.

In this activity, we implemented pattern recognition using images of different groups of objects and obtained sets of features to describe the objects and to use for recognizing new sets of objects. The groups of objects I used in this activity are 10 coins, 10 fasteners, and 10 leaves as shown by Figures 1-3. The features that were obtained are the area, perimeter and the sum of the RGB values of the object images.

Figure 1. Object group 1: coins.

Figure 2. Object group 2: fasteners.

Figure 3. Object group 3: leaves.

The first five objects of each group is used as the training set or the first set of examples where we get the basis for pattern recognition. The last five objects of each group is used as the testing set or the prediction set which will be used for testing the basis features. The same features was obtained for the test set and will be used for predicting which group of objects it belongs.

The features were obtained using basic image processing techniques such as segmentation by thresholding then counting of the number of pixels of the segmented image to get the area. The perimeter was obtained by getting the length of the contour of the segmented image while the RGB sum is simply the sum of the RGB values of the object image. The basis for pattern recognition is the mean of the features of the five sample objects for each group.

An object is classified into a group by computing the Minimum Distance of its features with respect to the basis vector or the mean feature vector. The test object is then classified into the group with the smallest feature distance. The figures below show the test objects and the Euclidean Distance for each feature. The highlighted areas represent the group with the minimum distance and thus the group where the test object is classified.

Figure 4. Euclidean Distances for test objects composed of coins.

Figure 5. Euclidean Distances for test objects composed of fasteners.

Figure 6. Euclidean Distances for test objects composed of leaves.

All in all the accuracy of the pattern recognition is 100%. I give myself a grade of 10 for this activity for the successful implementation of the Minimum Euclidean Distance for object classification.

Activity 12 | Color Image Segmentation

noreply@blogger.com (Anonymous) — Thu, 06 Aug 2009 07:44:00 +0000

It is a common problem in image processing when the region of interest is more than a problem of separating it from a background, or an application of threshold especially when the image involves a great variety of colors. In this activity the region of interest is isolated from the image by image segmentation using color in such a way that different brightness levels are not a problem by converting the RGB color space of the image to the Normalized Chromaticity Coordinates or NCC which is shown in Figure 1.

Figure 1. Normalized Chromaticity Coordinates

Image segmentation can be done in two ways. First is the parametric segmentation and second is non-parametric segmentation. Parametric segmentation involves the use of image patches of the color of the ROI and its histogram. The ROI can be isolated by doing histogram backprojection to the image using the histogram of the patch of the ROI. Figure 2 shows the original image used for this activity while Figure 3 shows the patches used and the resulting image after applying histogram backprojection to the image using the histogram of the color patch.

Figure 2. Original image. Hibiscus.

Activity 11 | Color Camera Processing

noreply@blogger.com (Anonymous) — Thu, 06 Aug 2009 07:43:00 +0000

One of the major factors used in judging the quality of digital cameras is its ability to capture colors satisfactorily. Most cameras nowadays give a set of options on the lighting conditions for capturing digital images. These set of options is often called White Balance Setting that gives a set of white balancing constants needed for different conditions.

In this activity, the digital camera of a Sony Ericsson phone was used to take pictures of objects using the different White Balancing Options which are bulb, cloudy, daylight, and fluorescent. Figure 1 below shows the images captured.

Figure 1. Different white balancing settings for differently colored objects (left) and objects of the same hue (right) like objects that have the color green. The white balancing settings are (top to bottom) bulb, cloudy, daylight, and fluorescent.

From the images obtained, it is observed that the bulb and fluorescent settings produced bluish images although the bulb setting images are more bluish and the fluorescent setting images are dimmer, while the cloudy setting produced images with slightly yellowish images and the daylight setting produced brighter colors and less yellowish images than the cloudy setting.

Now we apply two different White Balancing algorithms to these images namely the White Patch Algorithm and the Gray World Algorithm. The White Patch Algorithm is done by dividing the RGB channels of the original image by the RGB value of the white patch in the original image. Figure 2 shows the white patches used for each image in Figure 1 respectively and Figure 3 shows the resulting images after the White Patch Algorithm.

Figure 2. White patches used for White Patch Algorithm for the respective images in Figure 1.

Figure 3. White Balanced images using White Patch Algorithm.

As observed from the first row of images in Figure 3 it seemed that the white balancing went wrong. By examining the RGB channels (see Figure 4) of the original image of one of the wrongly white balanced images we see that there are areas in the image where the R values are very small and this includes the white patch used for the White Patch Algorithm. Performing the balancing on the red channel will result to very high values for the areas having high red values and very low values for the areas with very low red values thus resulting to the first row of images in Figure 3. It can also be noted that these images were captured with the White Balance setting turned to 'Bulb' which may be the source of error for the resulting image.

Figure 4. Respective R, G, and B channels of the original image for the resulting image (first row) in Figure 3.

Now we examine the second white balancing algorithm which is the Gray World Algorithm. In this algorithm the balancing constant is based on all the respective RGB values of the image and not just a white patch in the image. Figure 5 shows the resulting white balanced images of the original images in Figure 1.

Figure 5. White Balanced images using White Patch Algorithm.

Summary of Images

From the obtained white balanced images, we can see that the White Patch Algorithm gives better results but has a disadvantage when the RGB values of the original image has extreme values that could be very low or very high for particular areas. The Gray World Algorithm did not give a good white balancing result because it depends on the most dominant color on the image although extreme RGB values is not a problem with it.

I give myself a 9 for this activity since I was able to implement correctly the two white balancing algorithms and was able to identify possible sources of error, although I was not able to adjust the unsatisfactory results obtained for the images with 'Bulb' white balance setting.

I thank Kaye Vergel for lending me her Sony Ericsson phone with digital camera, Gilbert Gubatan for the red umbrella, Cherry Palomero for the pink clear folder and yellow hand fan, Luis Buno for the blue logbook, Shamandura Cabato for modeling her green shirt, Winsome Rara for the green eyeglass case, Jica Monsanto for the green box and the white piece of paper, NIP for the green floor paint and green colored chalk, and lastly Miguel Sison for the comments and recommendations on the unsatisfatory results for the white balancing of the images with 'Bulb' white balance setting.

Activity 10 | Preprocessing Text

noreply@blogger.com (Anonymous) — Thu, 06 Aug 2009 07:42:00 +0000

Activity 9 | Binary Operations

noreply@blogger.com (Anonymous) — Thu, 06 Aug 2009 07:41:00 +0000

Activity 8 | Morphological Operations

noreply@blogger.com (Anonymous) — Tue, 21 Jul 2009 01:28:00 +0000

Morphological operations simply refers to operating on the structure of something and these operations can be thought of like operations on sets like 'union', 'intersection', 'complement' and many others. An example of operations is the XOR which is union minus intersection and [NOT(A)] AND B which is the intersection of 'complement of A' and 'B'.

In this activity morphological operations like 'dilate' and 'erode' are implemented on 5 images namely a square, triangle, circle, a hollow square and a cross. Four structuring elements are used in the dilation and erosion of these images and these are a 4x4 square, 2x4 and 4x2 rectangles, and a 5x5 cross with a width=1. Figure 1 shows the structuring elements while Figure 2 shows the original image (column 1) and the respective resulting images for the dilation of the original with the order the same as the order of the structuring elements in Figure 1 while Figure 3 shows the results for erosion.

Figure 1. Structuring elements.

Figure 2. Resulting images after dilation.

Figure 2. Resulting images after erosion.

Before erosion a dilation was applied to the images by simulating the predicted resulting images. The shape of the predicted images are the same as the simulations. Only the dimensions of the resulting images were a bit different than the simulated ones but were close estimations. An simple example is for the 50x50 square when dilated with the 4x4 square. The predictions made resulted to a 54x54 square but the simulations gave a 53x53 square. This error is due to the assumption that the center and the axes used for the operations did occupy pixels.

I give myself a grade of 9 for this activity for the successful predictions of the shape of the resulting dilated and eroded imaged and for the good estimation of the dimensions.

Activity 7 | Enhancement in the Frequency Domain

noreply@blogger.com (Anonymous) — Tue, 21 Jul 2009 01:26:00 +0000

The Fourier Transform properties had been familiarized already in the previous activities. With this knowledge we use it to enhance images not by changing the image itself but its Fourier Transform. Now we examine the FTs of different patterns such as two dots with increasing radius where the smallest dots are two separate pixels as shown in Figure 1.

The FT of the two pixels is a series of lines that looks like a sinusoid. We can remember from previous activities that the FT of a sinusoid are two small dots along the axis of propagation of the sinusoid. Taking the FT of the two separate pixels is like doing the inverse FT of the FT of a sinusoid. The FTs of the two small circles on the other hand is like the product of the FT of the two pixels and a circle like that from Activity 5, and it is observed that increasing the size of the spots decreases the size of the resulting FT.

Figure 1. Two dots with increasing radius and their respective FTs.

We know that a pixel is square in shape. If we enlarge these squares the FT will change from that of a sinusoid pattern to like that of the two small circles only instead of the product of the FT two dots and a circle, the patterns looks like the product of two dots and a scquare. Increasing the size of the squares also decreases the size of the FT pattern just like in Figure 1. The images for the squares and their FTs are shown in Figure 2.

Figure 2. Two squares with increasing size and their respective FTs.

Now Figure 3 shows the two gaussian of varying variance and their FT while Figure 4 shows the Gaussian dots and the inverted counterparts and the real and imaginary parts of their FTs. It may not be visible but there is a small faint set of vertical lines shaped in a circle centered at the bright spot.

Figure 3. Two Gaussians with increasing variance and their respective FTs.

Figure 4. Two Gaussians with increasing variance and their respective FTs' real (middle row) and imaginary (last row) parts.

Using the concepts above we apply them in enhancing images in the frequency domain. We perform enhancement on three images namely a fingerprint for ridge enhancement, lunar landing scanned pictures for line removal, and a digital painting for canvas weave modeling and removal.

RIDGE ENHANCEMENT - Fingerprint

Figure 5. To be enhanced fingerprint image and its FT.

Enhancing this fingerprint such that the ridges are defined is the main goal for this part of the activity. Looking at the FT of the fingerprint we can see areas that are the brightest and could possibly be the FT pattern for the ridges. Basing from the FT we then create a mask to obtain the ridges of the fingerprint alone.

Mask 1: Thresholding

LINE REMOVAL - Lunar landing scanned pictures

Figure 5. To be enhanced fingerprint image and its FT.

CANVAS WEAVE MODELING and REMOVAL - painting

Activity 6 | Properties of the 2D Fourier Transform

noreply@blogger.com (Anonymous) — Thu, 09 Jul 2009 23:44:00 +0000

Previously in Activity 5, we applied image processing techniques involving the Fourier Transform such as edge detection and template matching using convolution and correlation. Now we investigate further the properties of the 2D Fourier Transform. The properties to investigate are that the Fourier Transform expresses any signal or image into a superposition of sinusoids and that the rotation of the sinusoids result to the rotation of the Fourier Transform.^[1]

Before anything else let us familiarize the Fourier Transforms of some 2D patterns (left) namely a square, an annulus or donut, a square annulus, two slits, and two dots along the x-axis that are symmetric about the center which are shown below together with their corresponding FTs (right) shown in Figure 1.

Figure 1. 2D patterns and their corresponding Fourier Transforms.

Now we investigate the anamorphic property of the Fourier Transform through the investigation of the superposition and rotation of sinusoids and their FTS in relation to the FT of different 2D patterns.

We create a 100x100 sinusoid with frequency f = (1, 4, 16) and get the modulus of its Fourier Transform. The images in Figure 2 show the produced 2D sinusoids (upper row) and their corresponding FTs (lower row) with increasing frequency values from left to right. The Fourier Transform of a 2D sinusoid results to an image with a peak at the frequency (+f or -f) of the said sinusoid thus we see two points along the axis of the sinusoid and symmetric from the center which is f = 0. As the frequency increases, the distances between the two points increases as expected. Code is taken from the AP186 Activity 6 sheet.

nx = 100; ny = 100;
x = linspace(-1,1,nx);
y = linspace(-1,1,ny);
[X,Y] = ndgrid(x,y);
f = 1; //frequency
z = sin(2*%pi*f*X);
imshow(z, []);

Figure 2. 2D sinusoids with frequencies f = 1, 4, and 16 and their corresponding FTs.

Figure 3 shows how the Fourier Transform of the resulting sinusoids would look like if a constant bias (upper row) or another sinusoid of low frequency (lower row) is added to the original sinusoids shown in Figure 2. The frequency order is the same as in Figure 2.

Figure 3. Fourier Transform of a sinusoid with a constant bias (upper row) or with an added low frequency sinusoid (lower row.)

Adding a constant bias to the original sinusoids resulted to the addition of a peak (which is the bias) at f = 0 to the respective Fourier Transforms of the sinusoids. If another sinusoid is added to a sinusoid the FT will show the frequencies of the two sinusoids like if a sinusoid with f=4 is added with a sinusoid with f=1 then their FT will show a peak at f=4 and f=1.

Rotating the sinusoids will also rotate their FTs as shown in Figure 4 for theta = 30, 45, and 60. Then we combine these rotated sinusoids to make patterns.

Figure 4. Rotated sinusoids (upper row) with f = 4 and the FTs (lower row) at theta = 0, 45, and 60 degrees respectively.

Now if we combine a sinusoid running in the x direction with another running in the y direction by addition or multiplication we get patterns as shown by Figure 5 below. Their FTs are just the superposition of their individual FTs. The code that generated these combinations are the following:

z_sum = sin(2*%pi*4*X) + sin(2*%pi*4*Y);
z_prod = sin(2*%pi*4*X) * sin(2*%pi*4*Y);
z_prod2 = sin(2*%pi*4*X). * sin(2*%pi*4*Y);

Figure 5. Combinations of a vertical and horizontal sinusoid using the equations shown above and their FTs.

If we add series of arbitrary rotated sinusoids to the patterns in Figure 5 we get patterns that looks like twisted checkered patterns as shown in Figure 6 below. The FTs of the patterns below is the same as the predicted results except for the matrix multiplication (middle row) where its FT is the same as the original pattern. It can be observed that the pattern itself has little deviation from the original pattern. As for the addition and the element by element multiplication, their FTs is the same as predicted which will be again, a superposition of the FTs of the original pattern and the added rotated patterns. If the added rotated sinusoids have rotations that scan from 0 t0 90 degrees in small increments then we expect a circle to form in the FT of that image which is partially shown by the FTs for the addition and the element by element multiplication.

Figure 6. Combinations of a vertical and horizontal sinusoids added with arbitrary rotated sinusoids and their corresponding FTs.

I give myself a 10 for this activity since the FTs produced were correct and they matched what was expected.

Activity 5 | Fourier Transform Model of Image Formation

noreply@blogger.com (Anonymous) — Tue, 07 Jul 2009 10:31:00 +0000

The Fourier Transform is one of the basic transforms used in imaging and is implemented in a fast and efficient way by the Fast Fourier Transform or FFT which is also mainly used in signal processing. The FFT is available in many signal and image processing software like the one used in this activity which is Scilab. The fft2() function in Scilab performs the Fast Fourier Transform for any two-dimensional signal or for this activity, images. Analyzing and processing images is explored using the the Fast Fourier Transform, convolution used in edge detection and correlation used in template matching.

5A - Familiarization with Discrete FFT

In the first part of the activity we get to familiarize the built-in function fft2() by applying it to a 128x128 image of a white circle on black background and the letter A that is constructed using Paint. Then we apply fftshift() to get the Fourier Transform of the image. The Fourier Transforms obtained agree with the analytical Fourier Transform of a circle which is an airy disk that becomes a airy point when the circle is large while for the letter A, it is like the Fourier Transform of a triangle that has three lines crossing each other at the center.

If we take the FFT of the resulting FFT of the images we get the intensity image or the original image that is inverted as shown by the last row of images. What is expected after getting the Fourier transform of the Fourier transform of the original image is the original image itself but inverted since an fftshift() is applied after the first FFT.

Original image
FFT of image
Shifted FFT image
FFT of FFT of image

5B - Simulation of an Imaging Device

In this activity we simulate an imaging device - the lens - by using FFT and convolution where a function is operated on another function to get a new function. The object is a 128x128 image of the letters VIP in Arial which will serve as the first function and the aperture of the lens is a 128x128 image of a circle and its transfer function will be the second function. The convolution of these two functions will give the image produced by this system. The Fourier Transform is the lens in this system such that by doing a convolution between the Fourier Transforms of the object and the aperture you get the produced image of the system. Below are the images used and produced in this activity.

Different sizes of aperture are used as shown by the different sizes of circle images. It can be seen that as the aperture size becomes larger, the produced image becomes clearer and more defined. This agrees with the fact that small, finite apertures can't get or capture all light coming from the object thus a blurring is observed in the resulting image.

5C - Template Matching Using Correlation

Another method used in signal and image processing is correlation that is similar to convolution such that it is also related to the complex conjugation of the Fourier Transforms of the desired signals. Correlation measures the degree of similarity between two signals thus making it useful for checking patterns like for this activity where an image containing a phrase is correlated with another image containing the letter A. This letter A will be the template or the pattern.

The resulting image after the correlation shows the highest values, with black=0 and white=1, are at areas where the letter A can be found.

5D - Edge Detection Using the Convolution Integral

Edge detection is also like template matching only a pattern is used and that pattern is convolved with the VIP image using the built-in function imcorrcoef() in Scilab SIP Toolbox. Below are the resulting images using different patterns such as vertical, horizontal, diagonal, back diagonal, and spot pattern.

The images at the right side show the areas that are highly similar to the respective pattern. Say for example for the vertical and horizontal patterns, in the resulting image, the most pronounced areas are the edges that are vertical and horizontal respectively. The horizontal and vertical lines especially in the letters I and P are gone respectively for the vertical and horizontal pattern. Lastly, among the five patterns used, the spot pattern is the one that gave the best edge detection since it does not suggest any specific direction and it is like tracing the edge pixels of the letters V, I, and P.

I'd like to give myself a grade of 10 for this activity because it was fairly easy since the codes are already given but I give myself a grade of 9 for this activity because I did not finish the blog on time. Also I had a hard time exporting quality images from my results since I did not know anything about the command imwrite() but I thank Mr. Miguel Sison for mentioning it so I was able to apply it to the last part of the activity. It is a lot easier and the image quality is better.

Reference - UPNIP - AP186 - Activity Sheet 5 - Fourier Transform of Image Model.

Activity 4 | Histogram Backprojection

noreply@blogger.com (Anonymous) — Tue, 07 Jul 2009 00:49:00 +0000

Images are comprised of pixels with values that represent a certain color. Getting the histogram of these pixel values give us information on the spread of the image's color from say black '0' to white '1' like grayscale images. In the previous activity, the histogram of a grayscale image was used to enhance the image by thresholding such that the desired region is well separated from the background.

In this activity, image enhancement is once again performed by tweaking the cumulative distribution function or the normalized histogram of a grayscale image into functions like the linear, parabolic, and logarithmic functions. Let us examine the PDF and CDF of a sample grayscale image as shown in Figure 1.

Figure 1. Original image with PDF and CDF respectively.

Performing histogram backprojection on the image using a linear CDF shown in Figure 2, we get a new image, its PDF and CDF as shown in Figure 3. From the image, PDF and CDF of the new image we can see the presence of higher values making the image brighter and more even.

Figure 2. Linear CDF.

Figure 3. New image with PDF and linear CDF.

Performing histogram backprojection on the image using a parabolic CDF shown in Figure 4, we get another new image, its PDF and CDF as shown in Figure 5. Because of the shape of the parabolic CDF, the majority of the pixels have high values thus producing an image with the very bright areas that are not needed as seen in the new x-ray image making it saturated.

Figure 4. Parabolic CDF.

Figure 5. New image with PDF and parabolic CDF.

Now the human eye has a nonlinear response, specifically a logarithmic response so using a logarithmic CDF shown in Figure 6, we get a new image, its PDF and CDF as shown in Figure 7. Because of the logarithmic trend, the resulting image has high contrast than the linear CDF. The logarithmic response is repeated on another image as shown in Figure 8.

Figure 6. Logarithmic CDF.

Figure 7. Using a logarithmic CDF.

Figure 8. Simulation of human eye response using a logarithmic CDF with original image (left) and new image (right) using logarithmic CDF.

I give myself a grade of 9 for this activity since I was able to do the histogram back projection using different CDFs and was able to produce the PDFs and CDFs of the new images correctly.

Activity 3 | Image Types and Basic Image Enhancement

noreply@blogger.com (Anonymous) — Sun, 05 Jul 2009 18:23:00 +0000

Digitized images have four types. We can obtain sample images for each type by searching over the internet. The following four images obtained from the internet (Figure 1-4) show an example for each image type together with their properties obtained using the SIP Toolbox function imfinfo().

BINARY IMAGE

File Source : http://www.fmwconcepts.com/imagemagick/morphology/images/logo2_fuzz15_mask_binary_open_1.gif
Format: GIF | Size : 951 | Width: 240 | Height: 240
Depth: 8 //bits per pixel
Storage Type: Indexed
Number of Colors: 2
Resolution Unit: centimeter
X&Y Resolution: 0

GRAYSCALE IMAGE

File Source : http://mangs.multiply.com/photos/album/13/hellsing_volume_1_chapter_3#3
Format: JPEG | Size : 52017 | Width: 352 | Height: 499
Depth: 8 //bits per pixel
Storage Type: Indexed
Number of Colors: 256
Resolution Unit: inch
X&Y Resolution: 0

TRUECOLOR IMAGE

File Source : http://www.dcercado.tk
Format: JPEG | Size : 38668 | Width: 500 | Height: 350
Depth: 8 //bits per pixel
Storage Type: Truecolor
Number of Colors: 0
Resolution Unit: inch
X&Y Resolution: 0

INDEXED IMAGE

File Source : http://mangs.multiply.com/photos/album/37/album_covers#21
Format: GIF | Size : 7550 | Width: 100 | Height: 100
Depth: 8 //bits per pixel
Storage Type: Indexed
Number of Colors: 256
Resolution Unit: centimeter
X&Y Resolution: 0

Now knowing the four basic image types let us do some basic image enhancement of a grayscale image by thresholding to separate the background from the Region Of Interest or ROI. To do this we need to know the grayscale distribution which can help in knowing the gray values of the ROI and obtain it by thresholding.

Figures 4 and 5 shows the scanned images to be used for basic image enhancement and below them are their image properties again using imfinfo(). Similar properties of both images are, Format: BMP, Depth: 8bits/pixel, Storage Type: Indexed, No. of Colors: 256, Resolution Unit: centimeter, and X&Y Resolution: - 1.718D+09.

From the grayscale images we can get the histogram of the gray values by using the Scilab Statistics function tabul() which returns a matrix of the pixel values and the corresponding frequency of each value in an image. To get the Probability Density Function of PDF of the image we normalize the histogram by the total number of pixels in the image. Figures 6 and 7 shows the corresponding PDFs for HEART and BATMAN.

Figure 6. HEART PDF

Figure 7. BATMAN PDF

The PDFs shows clearly the pixel values of the background and the ROI. Based from both histograms we can decide what threshold value to use to get the ROI and compute its area just like what was done previously in Activity 2. The threshold value used for both images is 0.5 but the threshold value for both images can be anything between the pixel values between the two distinct peaks that has a frequency value of zero. Anything in this range separates the pixel values for the background and the ROI. Figures 8 and 9 show the corresponding black and white images for HEART and BATMAN.

Figure 8. HEART with threshold at 0.5

Figure 9. BATMAN with threshold at 0.5

Now let us apply what we have learned from Activity 2 and compute the area of the ROI for each of our images. Figure 10 shows the obtained contour of the ROI. The computed area of the ROI using Green's Theorem in pixels for HEART is 8418px while for BATMAN is 12552.5px. But summing up the number of pixels with values equal to 1 the total area in pixels for HEART is 8598px and for BATMAN, 12885px. The physical area can be computed since the images were taken along with a ruler beside them. By computing the physical area of one pixel, we could estimate the physical area of the ROI. Using GIMP we observed that 1mm covers 3px and thus we can estimate the physical area of the ROI. A summary on the area measurements is provided below the ROI contours.

Figure 10. HEART and BATMAN contours.

HEART | Green's: 8418px > 935.33mm² | PxCount: 8598px > 955.33mm²
BATMAN | Green's: 12552.5px > 1394.72mm² | PxCount: 12885px > 1431.66mm²

From the obtained area measurements there is a great difference observed. Possible sources of error are the contour used as mentioned in Activity 2, the thresholding and the estimation of the area of 1 pixel. Still I give myself a 9 or a 10 for this activity since I have fully understood the concepts used and was able to get estimated area measurements of scanned images of irregularly shaped ROIs.

I thank Jay Samuel Combinido, Jica Monsanto and Miguel Sison for providing the scanned images that I used in this activity. I also thank Jica Monsanto for referring me to the function tabul() in getting the histogram of the gray values for the scanned images.

Activity 2 | Area Estimation for Images with Defined Edges

noreply@blogger.com (Anonymous) — Thu, 02 Jul 2009 01:07:00 +0000

Computing the areas of images is a very important tool in different technologies nowadays. Here in this activity the area of a binary image is computed by using Green's Theorem on computing the area of an image by using the contour of the image. From the activity manual itself, using Green's Theorem is getting the 'Area from edge.'

Green's Theorem ^[1] relates a double integral to a line integral as shown by the following equation where F1 and F2 are continuous functions with continuous partial derivatives that encloses the desired region or image.

This equation is then reduced into a simple summation that uses the contour of the desired region or image and is given by the next equation.

Below is generated and used code in this activity using Scilab. The procedure on what and how it is done is also shown here.

1. Load image.

rect = imread('C:\Users\Cindy\Documents\physics\ap186\A2\rectangle.jpg');
//circle = imread('C:\Users\Cindy\Documents\physics\ap186\A2\circle.jpg');

2. Convert Image from RGB to binary format using the SIP Toolbox function im2bw. Two binary images are used in this activity where one is an image of a rectangle and another is the image of a circle. Both images are created using Paint. Their analytical area is computed from the pixel dimensions given by Paint.

bw = im2bw(rect, 0.5);

3. Get the contour of the image using the SIP Toolbox function follow

[x,y] = follow(bw);

4. Implement GREEN'S THEOREM using the contour obtained.

len = length(x); //length of contour array
Aa=51012; //analytical area for rectangle
//Aa=53820; //analytical area for circle
Ac=0; //computed area

for i=1:len-1
a = (x(i)*y(i+1)) - (x(i+1)*y(i));
Ac = Ac+a;
end

5. But since the generated contour is not closed as shown in #3, we add the first value of the contour to close it to a loop.

a = (x(len)*y(1)) - (x(1)*y(len));
Ac = (Ac+a)/2

6. Check if the contour fits the original image by superimposing the contour on the original image.

imshow(rect)
plot(x,y)

7. COMPUTE PERCENT ERROR

perr = (abs(Aa - Ac)/Aa)*100

The results for each image is shown below. The analytical and computed area using Green's Theorem is given below each image.

Image 1: Rectangle
Analytical Area: 51012 px
Computed Area : 49220 px
Percent Error: 3.51%

Image 2: Circle
Analytical Area: 47143.52 px
Computed Area : 45575 px
Percent Error: 3.33%

Based on the results obtained, we can say that the method used gave a good approximation in computing the area of a region in an image. One possible source of error is the contour used which was generated by a built-in function of the SIP Toolbox. As seen from the superimposed images, the contour does not exactly fit the outline of the polygon leaving out spaces that should be included in the area.

I give myself a grade of 9 for this activity since the computed area is a close approximation of the analytically obtained area of the image as shown also by the small percent error. It is obvious enough that a major source of error is the contour used in the computation.

I thank Ms. Kaye Vergel for helping me on the basic functions of the SIP Toolbox such as the help function, imread, follow, im2bw. :D :D :D Thank you also for pointing me to the SIP Functions Reference.

Reference:
[1] UPNIP - AP186 - Activity Sheet 2 - Area Estimation for Images with Defined Edges.

Activity 1 | Digital Scanning

noreply@blogger.com (Anonymous) — Thu, 18 Jun 2009 02:29:00 +0000

Knowing the numerical values of a hand-drawn plot is a very tedious job when done manually. But with the help of technology and a bit of knowledge about image properties and a spreadsheet application, we can easily determine the numerical values of a hand-drawn plot.

To do this, the hand-drawn plot is digitally scanned (see Figure1 below) and the scanned image is used to obtain the image positions (in pixels) of desired points in the graph using Paint. The recorded positions are then plotted using OpenOffice.org Calc, but since the origin or the pixel position (0,0) is located at the top-left for images, the y-axis values are subtracted from the height of the image (in pixels) to change the origin position from top-left to lower-left in order to produce a plot similar to the image. Figure 2 shows the plot produced using OpenOffice.org Calc.

Figure1. Digitally scanned image of hand-drawn plot.

The x and y pixel values picked from the plot image obtained using Paint is shown below in Table1 while the corrected position values are shown in Table2.

Figure2. Produced plot from .

To make sure that the original hand-drawn plot was reproduced digitally, the plotted values are superimposed on the scanned plot image by making the plot image into a background of the chart produced in OpenOffice.org Calc. Figure3 shows the digitally-scanned plot with the original plot image.

Figure3. Produced digitally-scanned plot superimposed on the original plot image.

By superimposing the produced digitally-scanned plot with the original plot image we can check if the plot is reproduced successfully by digital scanning. As seen from Figure3, it can be seen that the plot is successfully reproduced meaning the values obtained by digital scanning are good approximations of the values picked from the original plot.

I give myself a grade of 9 for this activity since the digitally-scanned plot produced approximately coincides with the original plot.

I thank Mr. Gilbert Gubatan for pointing me to an image editor for rotating and cropping the desired plot from a scanned image of a page of a book. The computer I used in the classroom did not have any image editors except paint which does not have the features I need to get an upright and exact image of the plot.

I thank Mr. Rafael Santos for sharing how to superimpose the digitally-scanned plot on the original plot image by making the original plot image a background of the produced digitally-scanned plot using OpenOffice Calc.