IsmailPeters

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Background

CFA Interpolation

A digital camera captures color images by recording data in different channels of the visible spectrum (red, green, blue). Most cameras accomplish this by using a single type of CCD or CMOS sensor at each pixel. This sensor measures the intensity of light that is focused on it, but cannot distinguish between colors. Therefore, a color filter is placed over each sensor, and the sensor only records the intensity of a specific color at that pixel. The arrangement of these color filters over an image is known as a Color Filter Array. Since each color channel is only tallied at specific coordinates, the remaining pixels in that channel must be estimated in some way. There are many different estimation methods - the most simple ones incorporating the nearest neighbors in the same channel. Regardless of the type of interpolation used in a given image, the estimated pixels should exhibit a strong correlation, or dependence, on their surrounding pixels. If one can categorize each pixel as either correlated or independent with respect to other pixels, one should see a periodic pattern that mimics the CFA used to construct the image. Even if an altered image does not have any visual cues that point to its forgery, inspecting the correlation of pixels to one another can potentially expose which parts of an image were altered.

The way a digital camera captures an image is as follow, the scene under observation is focused through a number of lens onto a small slap of semiconductor material, typically silicon. The material is divided into a number of sub blocks known as pixels. As the light strikes the surface, the material within a particular pixel will begin to develop a charge proportional the the intensity of light hitting it. This charges is later collected and measured to determine the values necessary to properly recreate the image at a later date. This device is commonly referred to as a Charge Coupled Devices(CCD). This process work great for a gray scale image, however the CCD is blind to the individual wavelengths that are inducing the charge in the pixel. In order to capture the color image, the light must be filtered before striking the surface of the CCD, this filter is known as a Color Filter Array(CFA)
Shown below is a common CFA configuration known as a Bayer Array. It consists of a periodic pattern of green, red and blue pixels. The number of green pixels often doubles that of the blue or red due to our eyes sensitivity to the green part of the spectrum. Now when looking at a particular pixel, lets say green, if the pixel exhibits a high number of charges after capturing an image, it means that a majority of the light striking the surface must have been green; likewise a small number of charges implies low levels of green light on that particular pixel. This is repeated for both the red and blue pixel. In order to complete determine the proper levels of the remaining two colors at that pixel, a weighted average of the adjacent pixels in the image is used, this is known as CFA interpolation.

Below is another example of a reinotopic map in a different subject.
Figure 2

Once you upload the images, they look like this. Note that you can control many features of the images, like whether to show a thumbnail, and the display resolution.

Figure 3

MNI space

MNI is an abbreviation for Montreal Neurological Institute.

Methods

Measuring retinotopic maps

Retinotopic maps were obtained in 5 subjects using Population Receptive Field mapping methods Dumoulin and Wandell (2008). These data were collected for another research project in the Wandell lab. We re-analyzed the data for this project, as described below.

Subjects

Subjects were 5 healthy volunteers.

MR acquisition

Data were obtained on a GE scanner. Et cetera.

MR Analysis

The MR data was analyzed using mrVista software tools.

Pre-processing

All data were slice-time corrected, motion corrected, and repeated scans were averaged together to create a single average scan for each subject. Et cetera.

PRF model fits

PRF models were fit with a 2-gaussian model.

MNI space

After a pRF model was solved for each subject, the model was trasnformed into MNI template space. This was done by first aligning the high resolution t1-weighted anatomical scan from each subject to an MNI template. Since the pRF model was coregistered to the t1-anatomical scan, the same alignment matrix could then be applied to the pRF model.
Once each pRF model was aligned to MNI space, 4 model parameters - x, y, sigma, and r^2 - were averaged across each of the 6 subjects in each voxel.

Et cetera.

Results - What you found

Retinotopic models in native space

Some text. Some analysis. Some figures.

Retinotopic models in individual subjects transformed into MNI space

Some text. Some analysis. Some figures.

Retinotopic models in group-averaged data on the MNI template brain

Some text. Some analysis. Some figures. Maybe some equations.

Equations

If you want to use equations, you can use the same formats that are use on wikipedia.
See wikimedia help on formulas for help.
This example of equation use is copied and pasted from wikipedia's article on the DFT.

The sequence of N complex numbers x₀, ..., x_N−1 is transformed into the sequence of N complex numbers X₀, ..., X_N−1 by the DFT according to the formula:

$X_k=\sum _{n=0}^{N-1}{x}_n{e}^{-\frac{2\pi i}{N}kn}k=0,\dots ,N-1$

where i is the imaginary unit and $e^{\frac{2\pi i}{N}}$ is a primitive N'th root of unity. (This expression can also be written in terms of a DFT matrix; when scaled appropriately it becomes a unitary matrix and the X_k can thus be viewed as coefficients of x in an orthonormal basis.)

The transform is sometimes denoted by the symbol $ℱ$, as in $𝐗=ℱ\left\{𝐱\right\}$ or $ℱ\left(𝐱\right)$ or $ℱ𝐱$.

The inverse discrete Fourier transform (IDFT) is given by

$x_n=\frac{1}{N}\sum _{k=0}^{N-1}{X}_k{e}^{\frac{2\pi i}{N}kn}n=0,\dots ,N-1.$

Retinotopic models in group-averaged data projected back into native space

Some text. Some analysis. Some figures.

Conclusions

Here is where you say what your results mean.

References - Resources and related work

References

Software

Appendix I - Code and Data

Code

File:CodeFile.zip

Data

zip file with my data

Appendix II - Work partition (if a group project)

Brian and Bob gave the lectures. Jon mucked around on the wiki.