Computer Vision for Estimating Cooper Density by Optical Microscope Images

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Figure 2. An imaging unit

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Figure 3. RGB images of copper mineral

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Optical microscope is a cost-effective and gives valuable qualitative knowledge about the material studied. Figure 3 shows RGB images of copper minerals. This image is 550x710 pixels. It is clear that shiny particles' are copper minerals and dark background is other minerals like sulphur, iron and etc. particles.

To get a statistically valid quantitative dataset for a material containing several minerals, digital image analysis should be used. Image analysis has an advantage compared to indirect bulk analysis because of its ability to visualize the microstructures and to quantify how the minerals occur in free and composite particles.

2. Discrete Fourier Transform

The Discrete Fourier Transform (DFT) converts a finite list of equally spaced pixels of images into the list of coefficient of a finite combination of complex sinusoids, ordered by their frequencies, that has those same samples values. It can be said to convert the sampled function from its original domain (location in image) to the frequency domain. In image processing, the samples can be the values of pixels along a row or column of a raster image.

Figure 4 shows sinusoids wave, which uses for image pixels converting. For MxN image, 2D DFT is calculated as:

(1)

where I(x,y) is the images function. Figure 5 shows Fourier transform of copper soil. Fourier analysis is as a mathematical technique for transforming our view of the image from location-based to frequency-based.

Figure 4. Sinusoid wave

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For many images, Fourier analysis is extremely useful because the image’s frequency content is of great importance. A serious drawback in transforming to the frequency domain, location information is lost. When looking at a Fourier transform of an image, it is impossible to tell where a particular event took place ^[6].

If the images properties do not change much over place that is, if it is what is called a stationary image, this drawback isn’t very important. However, most interesting images, like copper soil image in this paper, contain numerous no stationary or transitory characteristics: drift, trends, abrupt changes, and beginnings and ends of particles. These characteristics are often the most important part of the image, and Fourier analysis is not suited to detecting them.

3. Wavelet Transform

Wavelet analysis allows the use of long time intervals where we want more precise low-frequency information, and shorter regions where we want high-frequency information. One major advantage afforded by wavelets is the ability to perform local analysis, that is, to analyze a localized area of a larger signal.

Wavelet analysis is capable of revealing aspects of data that other image analysis techniques miss, aspects like trends, breakdown points, discontinuities in higher derivatives, and self-similarity. Furthermore, because it affords a different view of data than those presented by traditional techniques, wavelet analysis can often compress or de-noise an image without appreciable degradation ^[5].

Figure 5. Fourier transform of mineral image

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Figure 6. Sinusoid versus wavelet mother wave

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A wavelet is a waveform of effectively limited duration that has an average value of zero. Compare wavelets with sine waves, which are the basis of Fourier analysis.

Sinusoids, which used for Fourier transform, do not have limited duration. They extend from minus to plus infinity. And where sinusoids are smooth and predictable, wavelets tend to be irregular and asymmetric. Fourier analysis consists of breaking up a image into sine waves of various frequencies. Similarly, wavelet analysis is the breaking up of a signal into shifted and scaled versions of the original (or mother) wavelet.

Figure 6 depicts sinusoids and wavelet wave difference. Just looking at pictures of wavelets and sine waves, you can see intuitively that signals with sharp changes might be better analyzed with an irregular wavelet than with a smooth sinusoid, just as some foods are better handled with a fork than a spoon ^[6].

It also makes sense that local features can be described better with wavelets that have local extent. The mother wavelet function is given as:

(2)

Where a, b∈R, a>0 and R is the wavelet space. Parameters ‘a’ and ‘b’ are the scaling factor and the shifting factor, respectively, since choosing a prototype function as the mother wavelet should always satisfy the admissibility condition (Equation 3):

(3)

Where is the Fourier transform of . For any image with MxN pixels, discrete wavelet transform is calculated as equation (4).

(4)

where I(x,y) is the pixel of image at x and y coordination, and g(x,y) is the wavelet wave. U and V indicate the wavelet space. We can use difference wavelet wave for this transform. Figure 7 exhibits the symlets wave. The symlets are nearly symmetrical wavelets proposed by Daubechies as modifications to the db family. The properties of the two wavelet families are similar ^{[7, 8]}.

Figure 7. Symlet wave

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Figure 8. Decompose soil image to red, green and blue

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After wavelet transform, two type of coefficient is produced; approximation Coefficients Matrix and details coefficients matrix. Approximation coefficient shows low frequency content. It is what gives the images its identity. Details coefficient matrix depicts high frequency content. The high-frequency content, on the other hand, imparts flavor or nuance in the images.

The approximations are the high-scale, low-frequency components of the signal and the details are the low-scale, high-frequency components.

3.1. Continuous Wavelet Transform

First of all, each image is decomposed to red, green and blue images. Color of an image can carry much more information than gray level. Color is the most prominent feature of any image. Extracting color information from any image has many applications related to computer vision algorithms. This work shows in Figure 8. Then continuous wavelet transform is calculated:

(5)

Figure 9 shows 2D plot of soil image wavelet transform and Figure 10 shows 3D plot of this transform.

Continuous analysis is often easier to interpret, since its redundancy tends to reinforce the features and makes all information more visible. When the image is taken in discrete space (like this images), it is apparent that discrete wavelet transform could be more useful than continuous wavelet transform.

Figure 9. 2D plot of soil image continuous wavelet transform

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Figure 10. 3D plot of soil image continuous wavelet transform

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3.2. Discrete Wavelet Transform

Figure 11. Wavelet Approximation Matrix

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Figure 11 presents approximation matrix and Figure 12 presents vertical, horizontal and diagonal details coefficient matrix of discrete wavelet transform of soil image. At approximation coefficient matrix, white and semi-white pixels depict high value of coefficient and it mains that edge and sharp changes are detected in image.

At details coefficient matrix, red and semi-red pixels depict high value of coefficient and it mains that edge and sharp changes are detected in image. So it is obvious that edge and sharp change in the image have a considerable wavelet coefficient and vice versa.

3.3. Discrete Energy of Each Coefficient

The wavelet energy provides information about the relative energy associated with different frequency bands presents in the image and its corresponding degree of importance. Energy of each coefficient can calculate with equation (6) and (7):

(6)

(7)

Figure 12. Wavelet Details Matrix

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Figure 13. Percentage of energy for approximation wavelet coefficient

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Figure 14. Percentage of energy for horizontal detail wavelet coefficient

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