View source for Hampapur

= Background =

The goal of this project is to determine if and by how much the blue pigment (color) present in the Sellaio face painting has degraded. To this end, I attempt to use the hyperspectral data of said image. 

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= Methods =
The key constraint with this project is that the ground truth is not available i.e we lack information as to how the artist intended the painting to look like. Consequently, all methods attempted employed analysis and heuristics with the hope that we might perhaps make a cogent argument on how the painting could be restored, if it should need restoring. Methods were employed both in the spectral domain and in CIELAB space. [[http://en.wikipedia.org/wiki/Lab_color_space]]. Furthermore, this problem is further complicated by the fact that over time, owing to sunlight, moisture, etc, the spectra could change in undefined ways. This is mostly attributed to the fact that the color pigments along with the material over which they're painted, degrade in undefined ways.

== Methods in the spectral domain ==

=== Principal Component Analysis ===
   An attempt was made to look at the principal components across the various spectra and determine the number of principal components to preserve. However, truncating the number and magnitude of the components proved to yield no useful information. Hence, this method was not pursued further.[[http://en.wikipedia.org/wiki/Principal_component_analysis]]

=== Amplitude Boosting  ===
 The key idea under this method was the assumption that over time, the only change in the spectra was the attenuation of the amplitude associated with said color. Studies of other paintings have shown that the original painting, as intended by the artist, were meant to be garish and bright as opposed to subtle and smooth. Selectively boosting the spectra for blue would perhaps yield the intended result.   

=== Clustering ===
 This employs state-of-the-art unsupervised learning algorithm detailed in [[http://www.stanford.edu/class/ee378B/papers/newman-community.pdf]]. The idea that I wished to pursue in this section is that even if the spectra of the image changed in undefined ways, it should be possible to aggregate like colors together. To this end I frame an adjacency matrix A (or similarity matrix as defined in the paper) based on two criteria - one, the spectral proximity and two,the energy content in said spectral bands. So, for instance, if there were four wavelengths at 300 nm, 700 nm, 720 nm, 730 nm at 20, 25, 23, 50 energy units, the adjacency matrix would claim that the 720 nm and 700 nm are related (by putting in a 1 in the corresponding spot in the matrix) and say that the 300 nm, 730 nm and the group of 700 nm and 720 nm are not related/similar by putting in a 0 in the corresponding positions. Note that the 730 nm is not classified as related to the 700 nm and 720 nm because of the large difference in energy levels. Of course, a particular wavelength is related to itself,so all the diagonal entries would be 1. This is then fed into the clustering algorithm. At the end of the algorithm, I expected to get a cluster of the key colors making up the painting which I would then use to construct the painting.

== Methods in the CIELAB space ==
  The CIELAB space is constructed to reflect the similarity in the perception of the human visual system. Therefore, by working in this space , the hope was to be able to determine the direction and magnitude of selectively moving the blue pigments so as to achieve maximum visual appeasement. Unfortunately, this method also lacked a key data metric: the delta E [[http://en.wikipedia.org/wiki/Color_difference]] values for the color pigments used in the creation of the blue color in the painting.
=== Moving from spectral data to LAB space ===
  Ideally, we would first begin by changing the illuminant in the scene to D65. Unfortunately, owing to memory constraints, ISET refused to do this transformation.We then move from spectral data to XYZ space. From this space, we get the white point of light and then move into LAB space. It is here that the grid search and selective changing of the pigments in blue is achieved.Then the space is moved back to XYZ and then moved into sRGB where we get the final result and evaluate the effect of performing the changes. 

=== Grid search ===
  For lack of practical data detailing the changes in the delta E values of the blue pigment, I had to improvise and use similar data. In particular, I used the degradation data detailed for the red pigment (also found in other paintings) as a baseline estimate for the delta E values for blue. In particular, as detailed in Kirby and Saunders' paper, I bound the upper value of delta E value by 2 and 6. Owing to computational constraints, I ended up using a delta E value of 2 distributed over a logarithmic spacing of values. Furthermore, for each of the values of blue in the LAB space, this method generates a sphere of possible values. As noted by the delta E values, the small change in delta E value should be indicative of the (perhaps) limited change in the perception of the painting. 

== Some practical experiments ==
  This was one of the first attempts in trying to be able to determine how the spectra of plant based pigments change under natural conditions. I ground raw spinach and brinjal to get green and purple/blue color pigments. I dried the colors using paper towels and then applied the colors to white cotton cloth and left outside to outside conditions. Over a period of 48 hours, the green color had degraded to yellow , perhaps owing to the death of the chlorophyll in the pigments and the blue/purple had degraded to black. The goal was to determine if there was a particular direction in the spectum in which plant based colors moved upon natural degradation.

= 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. <br> 
''See wikimedia help on  [http://meta.wikimedia.org/wiki/Help:Displaying_a_formula formulas] for help.'' <br>
This example of equation use is copied and pasted from [http://en.wikipedia.org/wiki/Discrete_Fourier_transform wikipedia's article on the DFT]. 

The [[sequence]] of ''N'' [[complex number]]s ''x''<sub>0</sub>, ..., ''x''<sub>''N''−1</sub> is transformed into the  sequence of ''N'' complex numbers ''X''<sub>0</sub>, ..., ''X''<sub>''N''−1</sub> by the DFT according to the formula:

:<math>X_k = \sum_{n=0}^{N-1} x_n e^{-\frac{2 \pi i}{N} k n} \quad \quad k = 0, \dots, N-1</math>   
            
where i is the imaginary unit and <math>e^{\frac{2 \pi i}{N}}</math>  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''<sub>''k''</sub> can thus be viewed as coefficients of ''x'' in an [[orthonormal basis]].)

The transform is sometimes denoted by the symbol <math>\mathcal{F}</math>, as in <math>\mathbf{X} = \mathcal{F} \left \{ \mathbf{x} \right \} </math> or <math>\mathcal{F} \left ( \mathbf{x} \right )</math> or <math>\mathcal{F} \mathbf{x}</math>.  

The '''inverse discrete Fourier transform (IDFT)''' is given by 

:<math>x_n = \frac{1}{N} \sum_{k=0}^{N-1} X_k e^{\frac{2\pi i}{N} k n} \quad \quad n = 0,\dots,N-1.</math>

== Retinotopic models in group-averaged data projected back into native space ==
Some text. Some analysis. Some figures.