Digitized images, added to signal, belong to data types that are unstruc-tured and heavily rich with information hidden in. That’s why, in general, tasks related to image processing are relatively complicated and working on such cases requires deep work toward several directions such as descriptors extraction and classifiers as already explained in the previous sections. The representation space remains as well an interesting area to explore.
Images are originally given in the space domain where the pixels are referenced with their locations and intensities. Researchers among years illustrated that
1.3 Images transformed to frequency domain
moving from space domain to the the frequency domain is meaningfully useful in image understanding tasks. In this context, a multitude of well performing transforms are explored, yet Fourier Transform, Gabor filters and Wavelets remain the most known and commonly used.
1.3.1 Fourier transform
Fourier Transform is a valuable tool among image processing domain. it is the-oretically based on decomposing the concerned image into its sine and cosine items. This transformation results into a new depiction of the initial input in the frequency domain (also called Fourier domain) corresponding to the initial spatial domain. Each point of an image in the Fourier domain refers to a specific frequency (pixel intensity) that appears "somewhere" in the spatial domain image.
Figure 1.6: Images with different edges contained with their FT transform below [11]
Actually, experiments in favor of Fourier transform proved that it is usually convenient to characterize several image processing operations at the expense of how do they behave toward the frequencies appearing in the image. In fact, non-theoretically and from a rather conceptual point of view, Fourier transform has the role of telling us what is happening in the original image representation (where shapes, forms colours and edges appear) in terms of fre-quencies. In this context, associating what is happening in the image to pixels intensities behaviour is back to the dependency of what we see in the original image on the values taken by our data frequency as well as its distribution.
for example whenever we need to blur an image we just have to eliminate high frequencies. However, when we eliminate low frequencies we get closer
to edges contained in the image. Finally, if we enhance high frequencies while maintaining low frequencies we are considered as sharpening the image.
The figure 1.6 shows Fourier transform of two images containing different va-rieties of edges. Noticing the obvious periodic texture, in the vertical direction in the bricks captures image (top left), explains the close appearance of the horizontal components contained in the corresponding FT (bottom left). Sim-ilarly, the FT associated to the lightened cubes image, shows a bright lines flowing to high frequencies and aligned with perpendicular direction to the relevant edges contained in the space domain image.
Hence, whenever we’re in an area of the image where we have a strong con-trast sharp edge, the gray intensities alter very speedily. It gets associated to plenty of high frequency power in order to be able to pursue such an edge.
That’s what justifies the existence of those bright lines in the corresponding magnitude spectrum. Fourier transform is a performing image processing and it has wide range of applications where it can be explored from which we can list, image filtering, image analysis, image reconstruction and especially image compression.
However, this does not deny that it has a non neglectful limit which is the lack of information it gives about the location indexes of intensities values. In other words, Fourier Transform doesn’t give us any idea about where did a behaviour of particular frequency took place in the spatial domain. That’s to say, it is important to know that our input image contains edges for example but it is also very useful to understand where did those edges appear exactly in our image. Gabor filters is one of the methods that goes through solving this issue.
1.3.2 Gabor transform
Gabor filters, also called Gaussian filters shown in figure 1.7 belong to linear filters class with the particularity of being oriented. They make highlighting textures as well as homogeneous zones of an image possible.
Thanks to the Gaussian form of the Gabor filter, the envelopes of the filtered images bring a local spectral information in each pixel. Furthermore, they provide information on the energy content of the image in the direction of the image in the direction of the chosen filter. The figure 1.8 shows the output obtained when implementing Gabor filters in different directions on an input face images.
1.3 Images transformed to frequency domain
Figure 1.7: Images with different edges contained with their FT transform below.
Gabor filters started to usefully assign the frequency information to localities however the size of the used windows in this method being non flexible and non adaptable to the analyzed signals frequency variation remains a weakness point of this transform that will be solved within Wavelets, subject of the next section.
Figure 1.8: Output of the implementation of Gabor filters with different ori-entations [12]
1.3.3 wavelet transform
The formalism of the 1D M-band wavelet transform has been developed for continuous signals. This transform is characterized given a scale function and an M-1 wavelet functions, each of them able to be subject of translation with respect to a real location parameter as well as dilation/contraction by a
strictly positive scale factor [13]. The fact of having a local treatment to data and of being able to apply flexible wavelets in terms of dilation/contraction and translation, this transform overcomes somehow the limits of Fourier and Gabor transforms mentioned in the two previous sections.
Figure 1.9: Different Wavelet shapes [14]
The figure 1.9 shows different shapes of wavelets that might be used in the domain transform operation. Wavelet transform has a wild range use in the features extraction component of classification systems. It is employed in several application to build a consistent features vector (figure1.10that serves lately to the classifier’s training.
Figure 1.10: Building the Feature Vector based on the Wavelet transform [4]
In this section, concerning domains transforming and its methods and going form Fourier to Wavelet transform we sufficed with a small overview free of mathematical details. In fact, we will not be in need to get into their very detailed theories nevertheless we should understand the basic definitions of the frequency-domain data that we will be dealing with in next chapters.