Figure 1 is a two-dimensional image of the attenuation coefficients of a plane from the reconstruction of the X-ray tomography data. Note that there are a wide variety of features present in the image. The light areas represent cement paste, the somewhat darker areas are sand grains and the darkest areas are air voids. Inside the air voids are crystalline growths, which most likely are calcium hydroxide or possibly gypsum or ettringite. There is also a crack resulting from the sulfate attack. Notice also the existence of concentric rings, which are an artifact of the tomographical data collection process. Also, there is some noise which may, in part, be due to the local inhomogeneity of the component materials.
Figure 1: Original two-dimensional image obtained from tomography data set.
For this preliminary study we decided to subdivide the image into three separate phases: sand, cement paste and air voids. Although there are many other features, as described above, these three phases make the largest volumetric contribution and are clearly (at least to the eye) distinguishable. Once the decision is made to convert the original attenuation coefficients into three phases, a procedure must be developed to decide what voxel (volume element) corresponds to what phase. Due to the noise present in the original image it is not easy to systematically separate the phases. For instance a simple thresholding of the attenuation coefficients, which range in value over two decades, produces an unrealistic image.
In order to produce a more realistic image, a four step image processing procedure was performed. The first step was intended to remove the ringing artifact and some of the local variation from the image. We chose a technique based on the discrete wavelet transform [5]. In many ways the wavelet transform is similar to the Fourier transform. It consists of representing the image as a weighted sum of basis elements. With the Fourier transform, the basis elements are the sine and cosine functions of different frequencies. The wavelet basis elements also have different frequencies. However, unlike the Fourier transform, a wavelet basis element is nonzero only in a finite region. Therefore, high frequency noise can be removed locally instead of globally as in the case of the Fourier transform. As a result, an advantage of the wavelet transform over the Fourier transform is that it is capable of preserving boundaries better. The particular wavelet technique employed is based on the thresholding idea of Donoho [6].
In Fig. 2 we show a comparison of attenuation profiles before and after the wavelet processing.
Figure 2: Attenuation profiles of Figure 1 along a vertical line before (top) and after (bottom) wavelet filtering.
Clearly plateaus are beginning to form that correspond to the different phases. Also the boundaries are generally preserved. For instance the rapid fluctuation of attenuation coefficients around voxels 190 to 200 has not been diminished. This fluctuation corresponds to the two small air voids in the original image.
Next thresholding was preformed on the image to segment it into the three phases. We found there was still some spurious noise which was cleared up with use of a median filter [7].
The final stage removed the unphysical presence of a band of sand surrounding air voids. This is due to the interpolation of the attenuation coefficient in regions where air voids border the cement paste. To remove the bordering of spurious sand, the regions near the sand-air void boundary were dilated (i.e., sand regions neighboring air-void were converted to air-void). Figure 3 shows the final processed image.
Figure 3: Final processed image. Black corresponds to air voids, grey to sand and white to cement paste.
Once all the images had been processed, we then determined the ratio of sand to cement in the three dimensional reconstruction of the mortar and found it was in excellent agreement with the mix design (i.e., 55 volume % sand, 45 volume % paste). We are currently studying the sand grain size distribution. Here, there is the difficulty that one must carefully separate touching sand grains. However, our preliminary studies, employing watershed segmentation techniques [8] are encouraging.