Project One - Image Compression

* j * mike * kristen * angi *
(group team)

* What's this "image compression" thing? *

Image compression is used to minimize the amount of memory needed to represent an image. Images often require a large number of bits to represent them, and if the image needs to be transmitted or stored, it is impractical to do so without somehow reducing the number of bits. The problem of transmitting or storing an image affects all of us daily. TV and fax machines are both examples of image transimission, and digital video players and web pictures of Catherine Zeta-Jones are examples of image storage.

Three techniques of image compression that we have discussed in lecture are pixel coding, predictive coding, and transform coding. The idea behind pixel coding, is to encode each pixel independently. The pixel values that occur more frequently are assigned shorter code words (fewer bits), and those pixel values that are more rare are assigned longer code words. This makes the average code word length decrease.

Predictive coding is based upon the principle that images are most likely smooth, so if pixel b is physically close to pixel a, the value of pixel b will be similar to the value of pixel a. When compressing an image using predictive coding, quantized past values are used to predict future values, and only the new info (or more specifically, the error between the value of pixels a and b) is coded.

The image compression technique most often used is transform coding. A typical image's energy often varies significantly throughout the image, which makes compressing it in the spatial domain difficult; however, images tend to have a compact representation in the frequency domain packed around the low frequencies, which makes compression in the frequency domain more efficient and effective. Transform coding is an image compression technique that first switches to the frequency domain, then does it's compressing. The transform coefficients should be decorrelated, to reduce redundancy and to have a maximum amount of information stored in the smallest space. These coefficients are then coded as accurately as possible to not lose information. In this project, we will use transform coding.

* What're you guys doing, eh? *

The choice of which transform to use in a transform coder has no clear answer since the performance of a transform compression scheme depends on both the transform and the images being compressed. This project studies the Haar Discrete Wavelet Transform. The Haar DWT is more computationally efficient than the sinusoidal based discrete transforms, but this quality is a tradeoff with decreased energy compaction compared to the DCT.

The Haar transform operates as a square matrix of length N = some integral power of 2. Implementing the discrete Haar transform consists of acting on a matrix row-wise finding the sums and differences of consecutive elements. If the matrix is split in half from top to bottom the sums are stored in one side and the differences in the other. Next operation occurs column-wise, splitting the image in half from left to right, and storing the sums on one half and the differences in the other. The process is repeated on the smaller square, power-of-two matrix resulting in sums of sums. The number of times this process occurs can be thought of as the depth of the transform. In our project, we worked with depth four transforms changing a 256x256 images to a new 256x256 image with a 16x16 purely sums region in the upper-lefthand corner.

Mathematically speaking, this sums and differences approach comes from the Haar functions which are scaled and shifted versions of a single basis function, specifically the odd rectangular pulse pair. Since the Haar transform initiates from a single basis function, the Haar transform identifies edges and lines more directly than sinusoidal based transforms, since the basis function resembles an edge or a line.

Finally, since the vast majority of the energy of an image is stored in the upper-lefthand corner of the transformed matrix, pixel coding can be used to reduce the number of bits necessary to represent the image. Simply use many bits for the high energy region, then fewer and fewer as you progress through the different levels of depth of the transform.

Of course, the proof is in the pudding so the rest of this report will further detail our actions and display the results of our efforts in this area.

* Matlab? What's Matlab? *

Figure 1 - Original image of boy.256

In order to implement the image compression algorithm we chose, we divided the process into various steps:

The first two steps are accomplished using simple loops in Matlab. Specifically, we wrote two different functions - rowthing for caluclating sums and differences of individual rows and colthing for calculating sums and differences of individual columns.

In order to keep the energy of the image the same, we multiplied each sum and difference by a factor of 1/sqrt(2). Performing these two operations once will result in an image that is split into four parts, with the upper left hand quadrant being the sums of sums region (see Figure 2).

Figure 2 - Sums and differences after one iteration

The third step involves repeating the previous two steps until we get down to a small enough final image. The function squisher accomplishes this task. This function performs the sums and differences of rows and columns, in alternating order, until the final sums of sums image is of size 16x16. The resulting image can be seen in the following figure (due to the normalized display, the various quadrants are not too visible, but contain various edge information):

Figure 3 - Final sums and differences matrix

The next step is quantization. This is performed during the writing to a binary file (see proceeding discussion of writing to a file); however, we wrote a distinct quantization function to analyze this step statistically (MSE and PSNR -- see the Results section), as well as for our own educational benefit(!). The quantization function, quant, is also called in squisher. Our quantization scheme simply assigns different numbers of bits to different regions, using masks (for example, [b16, b32, b64, b128, b256], where b16 is the quantization level for the upper left 16x16 matrix, b32 is for the next 32x32 matrix surrounding the first one, etc.). We used a number of different bit allocation masks (see next section) in order to determine which scheme is better.

The quantization function takes in as arguments not only the input matrix, but also the mask, i.e. the number of bits to be used to represent each region in the compression scheme. This allows us to modify and test out difference bit allocation algorithms easily. The quantized image looks similar to Figure 3, although depending on the mask used, the level of details may vary.

Once we have generated the compressed matrix, we are ready to transfer it to a binary file for storage (and, in the process, quantize it). We use the fwrite Matlab command, specifying the number of bits with which to quantize. The function bit performs this operation.

Once the file is saved, we can use the file-size information to evaluate the success of the compression technique. Additionally, a further use of the lossless gzip command in Unix can show how thoroughly the image could really be stored. The next section also shows some of these results.

With compression complete, we are ready to decompress and examine the errors introduced by compression. To reverse our compression scheme, we take the following steps:

Quantizing the matrix back to its original levels is easy: just quantize the compressed image back to eight bits, or bitshift 8-(first quantization level). The function iquant performs this operation.

The inverse sums and differences calculations are made by adding and subtracting various pairs of numbers in the matrix and dividing by two. These values are renormalized, and placed in their appropriate positions in the target matrix. The two functions rowthingi and colthingi accomplishe this task.

These operations are performed first on the columns, and then on the rows of succeedingly larger sections of the matrix until the entire 256x256 matrix has been decompressed. This process is done with the function unsquisher. Once this is done, we have reconstructed our image. Figure 4 shows the reconstructed image using the mask [8 6 4 2 0].

Figure 4 - Reconstructed image using mask [8 6 4 2 0]

* So? Did it work? *

We tested our wavelet transform coder on a few different images to see if some images would compress better (i.e. with less error) than others. In addition to the boy.256 image, which is relatively smooth and does not have too many edges, we also used the following three images:

Figure 5 - Image with sharp edges and high contrast

Figure 6 - Blurry satellite image

Figure 7 - Image with low contrast

We ran our coder with various different masks (thus resulting in different numbers of bits per pixel) on each of these images. For each run, we determined the mean-square error (MSE) as well as the peak signal-to-noise ratio (PSNR). The following two graphs show the results of these tests:

Figure 8 - Mean-square error plot

Figure 9 - Peak signal-to-noise ratio plot

As we can see, the boy.256 image seem to perform the best out of the four while the madrill.256 image seem to perform the worst. We show a nice linear increase in PSNR with additional bits-per-pixel. Note we get a PSNR of greater than 20 for all images even with less than 1 bit per pixel.

In addition to the MES and PSNR results, we also analyzed the results of the compression (since that is what the assignment is called). For this analysis, we used the mask [8 6 4 2 0] (the same mask used in the representative images above) in the coding scheme because it gave us really good approximations of the original image. For comparison purposes, it is useful to note that the JPEG compression standard has a compression ratio of 40:1.

(Note: All numbers in units of bytes.)
image original file after our coder compression ratio gzipped final ratio
boy.256 65536 5440 12.1:1 822 79.7:1
mandrill.256 65536 5440 12.1:1 670 97.8:1
bridge.256 65536 5440 12.1:1 972 67.4:1
urban.256 65536 5441 12.0:1 855 76.7:1

* Authors' Note *

To show our respect for Ingrid Daubechies, we have refrained from using the image lenna.256 throughout our web page. We feel it is highly inappropriate and degrading to haberdashers everywhere.

We also wish to voice our complete support for the Rice Men's Swimming Team! Go Water Owls!