Image compression using Principal Component Analysis

The file size of images can be reduced by performing image compression. When we capture images using our cameras, they are usually compressed immediately as JPEG files. Most digital single lens reflex cameras (DSLR's for short) allows the user to retain all the information received by the camera sensor (either CMOS or CCD) by providing a raw image. A high resolution image from my DSLR, which is in JPEG has a typical size of 4 to 6 MB. At the same time, a 20 to 40 MB raw file is saved. Discrete cosine transform (which I have mentioned way back) is performed on the image to achieve a smaller file size.

Image compression can also be performed using the concept of principal components. An image can be decomposed into a set of principal components. To visualize how it works, think of a point in a 3-dimensional space. Its distance from the origin can be described by its location along the x-, y-, and z-axis. Furthermore, if we have an multi-dimensional space, the location can be described with respect to the vectors that are orthogonal with each other. However, some of these vectors constitute a small portion on the measurement of the location of one point on the multi-dimensional space. If we neglect them, we can still be able to have a rough estimate of the location.

The same goes for image compression. This image:

Figure 1: Image that will be used for compression. (Photo credit: Norman Mascariñas)

Figure 2. Gray scale image of the red component of Figure 1

after using the pca() function of Scilab, produces a set of images like this (obtained from facpr of the function pca):

Figure 3: Principal components

The original image (800x500 pixels) was split into 10x10 sub-images. This is why we have principal components that are 10x10 pixels each. As mentioned earlier, these components (the eigenvectors of the image) will become the bases for the reconstruction of each sub-image. You will see in Figure 3that the first component (top left) have larger clustering than the other ones, with the last component (bottom right) having the smallest or finest pixels. Choosing the number of principal components depends on the contribution of each of them (which can be obtained from lambda of the function pca). This can be shown as:

Figure 4. Cumulative contribution of the components

You'll see that there is a steep increase in the contribution from the first four components. This means that we adding one component at a time will give a noticeable change in the overall reconstruction of the image. Beyond the 40th component, the contribution no longer changes, which gives us an idea that the reconstruction will not have any difference, whatever number of components we select.

To reconstruct the image, I used the first M components and get the corresponding values in comprinc for each sub-image. The function pca() computes individually per sub-image, so we need to retrieve the "intensity" of each component for every sub-image.

Figure 5. Comparison of image reconstruction, with the number beside each indicating the number of
components used for reconstruction

As expected, the first few reconstructions are more "pixelated", meaning they look like they have chunks of pixels.. Zooming in on the first reconstruction:

Figure 6. Reconstruction using only the first component

It really is a bad reconstruction, and looks like the usual images compressed using DCT at low compression rate. Increasing the number of components, as seen in Figure 5, produces a sharper and smoother image.

Figure 7. Reconstruction with 20 components

Figure 8. Reconstruction with 90 components

Comparing Figures 7 and 8, we see that there are no visible differences between the two. With this, we can just select the right number of component, which would roughly give around 99% of the image. Both of them (Figures 7 and 8) have cumulative contributions greater than 99%.

For this activity, I will give myself a grade of 10/10.

Reference:
A13-Image compression. M. Soriano. 2013