Knee & Chest Images

Resized images

In order to make calculations less intense, I scaled and padded the images to 512 x 512 pixels. I scaled the largest side of each image down and then padded the short side up. The chest images are 8 bits per pixel, whereas the knee images are 16 bits per pixel. However, the knee images are either significantly underexposed, or aren't using all 16 bits per pixel. Since the histograms show no values over 4095, I think they're using 12 bits out of the 16. All resized images have 8 bits per pixel. This change was necessary to calculate mutual information.

00006517	00007157
chest_00_8bit-cropped	chest_11qq_8bit-cropped
chest_22qq_8bit-cropped	chest_33qq_8bit-cropped

Fourier Transform

Here are the Fourier transforms of the images. Note that I used the 2-D Fast Fourier Transform that I implemented in Scheme. Since the Fourier Transform of an image with real-value pixels is an image with complex-value pixels, the complex images are rendered with the angle of each pixel value corresponding to hue, and the log of the magnitude of each pixel value corresponding to intensity (saturation for all pixels is set to 100%).

00006517	00007157
chest_00_8bit-cropped	chest_11qq_8bit-cropped
chest_22qq_8bit-cropped	chest_33qq_8bit-cropped

Daubechies Wavelet Transform

Here are the results of applying the two-dimensional x - y separable Daubechies 4 wavelet transform. The image is constructed by placing the low-low, low-high, and high-low wavelet subbands in three quadrants, then recursively applying the transform to the high-high subband. The quadrants representing each of the wavelet subbands are normalized to the range [0-255] with grey level 0 mapped to grey level 128 to euqalize contrast.

00006517	00007157
chest_00_8bit-cropped	chest_11qq_8bit-cropped
chest_22qq_8bit-cropped	chest_33qq_8bit-cropped

Mutual information

The mutual information between like images is much higher than that between different images. Not surprisingly, since the fourth chest image (chest_33qq_8bit-cropped) is closer to square than the other chest images, it has fewer black padding rows beneath the actual image, and therefore the mutual information between the first and fourth chest image is less than the mutual information between the first and second chest images. Note that even with this setback, a threshold of .6 bits of mutual information segments the pairings into same and different.

Image 1	Image 2	Mutual Information
00006517	00007157	1.2594311845815556
00006517	chest_00_8bit-cropped	0.4325962599691735
00006517	chest_11qq_8bit-cropped	0.5249847803288219
00006517	chest_22qq_8bit-cropped	0.5539400368061251
00006517	chest_33qq_8bit-cropped	0.4237036607653524
00007157	chest_00_8bit-cropped	0.3667725264752004
00007157	chest_11qq_8bit-cropped	0.47275702036728084
00007157	chest_22qq_8bit-cropped	0.44786624096007177
00007157	chest_33qq_8bit-cropped	0.4247918819897354
chest_00_8bit-cropped	chest_11qq_8bit-cropped	1.3377034992728074
chest_00_8bit-cropped	chest_22qq_8bit-cropped	1.3420743247768163
chest_00_8bit-cropped	chest_33qq_8bit-cropped	0.6804222800666713
chest_11qq_8bit-cropped	chest_22qq_8bit-cropped	1.463441133712287
chest_11qq_8bit-cropped	chest_33qq_8bit-cropped	0.7940866041083128
chest_22qq_8bit-cropped	chest_33qq_8bit-cropped	0.6735144508609565