# Multidimentional data - Matrices and Images


```python
%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt

from scipy import linalg
```


```python
plt.style.use('ggplot')
plt.rc('axes', grid=False)   # turn off the background grid for images
```

Let us work with the matrix:
$
\left[
\begin{array}{cc}
1 & 2 \\
1 & 1
\end{array}
\right]
$


```python
my_matrix = np.array([[1,2],[1,1]])

print(my_matrix.shape)
```


```python
print(my_matrix)
```


```python
my_matrix_transposed = np.transpose(my_matrix)

print(my_matrix_transposed)
```


```python
my_matrix_inverse = linalg.inv(my_matrix)

print(my_matrix_inverse)
```

### numpy matrix multiply uses the `dot()` function:


```python
my_matrix_inverse.dot(my_matrix)
```

### Caution the `*` will just multiply the matricies on an element-by-element basis:


```python
my_matrix_inverse * my_matrix_inverse
```

### Solving system of linear equations

$$
\begin{array}{c}
x + 2y = 4 \\
x + y = 3 \\
\end{array}
\hspace{2cm}
\left[
\begin{array}{cc}
1 & 2 \\
1 & 1 \\
\end{array}
\right]
\left[
\begin{array}{c}
x\\
y
\end{array}
\right]
=
\left[
\begin{array}{c}
4\\
3\\ 
\end{array}
\right]
\hspace{2cm}
{\bf A}x = {\bf b}
\hspace{2cm}
\left[
\begin{array}{c}
x\\
y
\end{array}
\right]
=
\left[
\begin{array}{cc}
1 & 2 \\
1 & 1 \\
\end{array}
\right]^{-1}
\left[
\begin{array}{c}
4\\
3\\ 
\end{array}
\right]
=
\left[
\begin{array}{c}
2\\
1
\end{array}
\right]
$$


```python
A = np.array([[1,2],[1,1]])

print(A)
```


```python
b = np.array([[4],[3]])

print(b)
```


```python
# Solve by inverting A and then mulitply by b

linalg.inv(A).dot(b) 
```


```python
# Cleaner looking

linalg.solve(A,b)
```

### System of 3 equations example (Numpy):

$$
\begin{array}{c}
x + 3y + 5z = 10 \\
2x + 5y + z = 8 \\
2x + 3y + 8z = 3 \\
\end{array}
\hspace{3cm}
\left[
\begin{array}{ccc}
1 & 3 & 5 \\
2 & 5 & 1 \\
2 & 3 & 8 
\end{array}
\right]
\left[
\begin{array}{c}
x\\
y\\
z 
\end{array}
\right]
=
\left[
\begin{array}{c}
10\\
8\\
3 
\end{array}
\right]
$$


```python
A = np.array([[1,3,5],[2,5,1],[2,3,8]])
b = np.array([[10],[8],[3]])

print(linalg.inv(A))

print(linalg.solve(A,b))
```

### System of 3 equations example (SymPy) - Python's Symbolic Math Package


```python
import sympy as sym

AA = sym.Matrix([[1,3,5],[2,5,1],[2,3,8]])
bb = sym.Matrix([[10],[8],[3]])

print(AA**-1)

print(AA**-1 * bb)
```

### SymPy is slower than NumPy


```python
%timeit AA**-1 * bb
%timeit linalg.solve(A,b)
```

# Images are just 2-d arrays - `imshow` will display 2-d arrays as images


```python
print(A)

plt.imshow(A, interpolation='nearest', cmap=plt.cm.Blues);
```

### Read in some data


```python
I = np.load("test_data.npy")    # load in a saved numpy array
```


```python
I.ndim, I.shape, I.dtype
```


```python
print("The minimum value of the array I is {0:.2f}".format(I.min()))
print("The maximum value of the array I is {0:.2f}".format(I.max()))
print("The mean value of the array I is {0:.2f}".format(I.mean()))
print("The standard deviation of the array I is {0:.2f}".format(I.std()))
```


```python
#flatten() collapses n-dimentional data into 1-d

plt.hist(I.flatten(),bins=30);
```

### Math on images applies to every value (pixel)


```python
II = I + 8

print("The minimum value of the array II is {0:.2f}".format(II.min()))
print("The maximum value of the array II is {0:.2f}".format(II.max()))
print("The mean value of the array II is {0:.2f}".format(II.mean()))
print("The standard deviation of the array II is {0:.2f}".format(II.std()))
```

### Show the image represenation of `I` with a colorbar


```python
plt.imshow(I, cmap=plt.cm.gray)
plt.colorbar();
```

### Colormap reference: http://matplotlib.org/examples/color/colormaps_reference.html


```python
fig, ax = plt.subplots(1,5,sharey=True)

fig.set_size_inches(12,6)

fig.tight_layout()

ax[0].imshow(I, cmap=plt.cm.viridis)
ax[0].set_xlabel('viridis')

ax[1].imshow(I, cmap=plt.cm.hot)
ax[1].set_xlabel('hot')

ax[2].imshow(I, cmap=plt.cm.magma)
ax[2].set_xlabel('magma')

ax[3].imshow(I, cmap=plt.cm.spectral)
ax[3].set_xlabel('spectral')

ax[4].imshow(I, cmap=plt.cm.gray)
ax[4].set_xlabel('gray')
```

## WARNING! Common image formats DO NOT preserve dynamic range of original data!!
- Common image formats: jpg, gif, png, tiff
- Common image formats will re-scale your data values to [0:1]
- Common image formats are **NOT** suitable for scientific data!


```python
plt.imsave('Splash.png', I, cmap=plt.cm.gray)     # Write the array I to a PNG file

Ipng = plt.imread('Splash.png')                   # Read in the PNG file

print("The original data has a min = {0:.2f} and a max = {1:.2f}".format(I.min(), I.max()))

print("The PNG file has a min = {0:.2f} and a max = {1:.2f}".format(Ipng.min(), Ipng.max()))
```

## Creating images from math


```python
X = np.linspace(-5, 5, 500)
Y = np.linspace(-5, 5, 500)

X, Y = np.meshgrid(X, Y)     # turns two 1-d arrays (X, Y) into one 2-d grid

Z = np.sqrt(X**2+Y**2)+np.sin(X**2+Y**2)

Z.min(), Z.max(), Z.mean()
```

### Fancy Image Display


```python
from matplotlib.colors import LightSource

ls = LightSource(azdeg=0,altdeg=40)
shadedfig = ls.shade(Z,plt.cm.copper)

fig, ax = plt.subplots(1,3)

fig.set_size_inches(12,6)

fig.tight_layout()

ax[0].imshow(shadedfig)

contlevels = [1,2,Z.mean()]

ax[1].axis('equal')
ax[1].contour(Z,contlevels)

ax[2].imshow(shadedfig)
ax[2].contour(Z,contlevels);
```

### Reading in images (`imread`) - Common Formats


```python
I2 = plt.imread('doctor5.png')

print("The image I2 has a shape [height,width] of {0}".format(I2.shape))
print("The image I2 is made up of data of type {0}".format(I2.dtype))
print("The image I2 has a maximum value of {0}".format(I2.max()))
print("The image I2 has a minimum value of {0}".format(I2.min()))
```


```python
plt.imshow(I2,cmap=plt.cm.gray);
```

## Images are just arrays that can be sliced. 

- ### For common image formats the origin is the upper left hand corner


```python
fig, ax = plt.subplots(1,4)
fig.set_size_inches(12,6)

fig.tight_layout()

# You can show just slices of the image - Rememeber: The origin is the upper left corner

ax[0].imshow(I2, cmap=plt.cm.gray)
ax[0].set_xlabel('Original')

ax[1].imshow(I2[0:300,0:100], cmap=plt.cm.gray)
ax[1].set_xlabel('[0:300,0:100]')                 # 300 rows, 100 columns

ax[2].imshow(I2[:,0:100], cmap=plt.cm.gray)       # ":" = whole range
ax[2].set_xlabel('[:,0:100]')                     # all rows, 100 columns

ax[3].imshow(I2[:,::-1], cmap=plt.cm.gray);
ax[3].set_xlabel('[:,::-1]')                      # reverse the columns
```


```python
fig, ax = plt.subplots(1,2)
fig.set_size_inches(12,6)

fig.tight_layout()

CutLine = 300

ax[0].imshow(I2, cmap=plt.cm.gray)
ax[0].hlines(CutLine, 0, 194, color='b', linewidth=3)

ax[1].plot(I2[CutLine,:], color='b', linewidth=3)
ax[1].set_xlabel("X Value")
ax[1].set_ylabel("Pixel Value")
```

## Simple image manipulation


```python
from scipy import ndimage
```


```python
fig, ax = plt.subplots(1,5)
fig.set_size_inches(14,6)

fig.tight_layout()

ax[0].imshow(I2, cmap=plt.cm.gray)

I3 = ndimage.rotate(I2,45,cval=0.75)               # cval is the value to set pixels outside of image
ax[1].imshow(I3, cmap=plt.cm.gray)                 # Rotate and reshape

I4 = ndimage.rotate(I2,45,reshape=False,cval=0.75) # Rotate and do not reshape
ax[2].imshow(I4, cmap=plt.cm.gray)

I5 = ndimage.shift(I2,(10,30),cval=0.75)           # Shift image      
ax[3].imshow(I5, cmap=plt.cm.gray)

I6 = ndimage.gaussian_filter(I2,5)                # Blur image
ax[4].imshow(I6, cmap=plt.cm.gray);
```

### `ndimage` can do much more: http://scipy-lectures.github.io/advanced/image_processing/

---

## FITS file (Flexible Image Transport System) - Standard Astro File Format
- **FITS format preserves dynamic range of data**
- FITS format can include lists, tables, images, and combunations of different types of data


```python
import astropy.io.fits as fits
```


```python
x = fits.open('bsg01.fits')

x.info()
```


```python
x[0].header
```


```python
xd = x[0].data

print("The image x has a shape [height,width] of {0}".format(xd.shape))
print("The image x is made up of data of type {0}".format(xd.dtype))
print("The image x has a maximum value of {0}".format(xd.max()))
print("The image x has a minimum value of {0}".format(xd.min()))
```


```python
fig, ax = plt.subplots(1,2)

fig.set_size_inches(12,6)

fig.tight_layout()

ax[0].imshow(xd,cmap=plt.cm.gray)

ax[1].hist(xd.flatten(),bins=20);
```

## You can use masks on images


```python
CopyData = np.copy(xd)

CutOff = 40

mask = np.where(CopyData > CutOff)
CopyData[mask] = 50                   # You can not just throw data away, you have to set it to something.

fig, ax = plt.subplots(1,2)

fig.set_size_inches(12,6)

fig.tight_layout()

ax[0].imshow(CopyData,cmap=plt.cm.gray)

ax[1].hist(CopyData.flatten(),bins=20);
```

## You can add and subtract images


```python
fig, ax = plt.subplots(1,2)
fig.set_size_inches(12,6)

fig.tight_layout()

ax[0].imshow(xd, cmap=plt.cm.gray)

# Open another file 'bsg02.fits'

y = fits.open('bsg02.fits')
yd = y[0].data

ax[1].imshow(yd, cmap=plt.cm.gray);
```

### The two images above may look the same but they are not! Subtracting the two images reveals the truth.


```python
fig, ax = plt.subplots(1,3)
fig.set_size_inches(12,6)

fig.tight_layout()

ax[0].imshow(xd, cmap=plt.cm.gray)
ax[1].imshow(yd, cmap=plt.cm.gray)

z = xd - yd                          # Subtract the images pixel by pixel

ax[2].imshow(z, cmap=plt.cm.gray);
```

## FITS Tables - An astronomical example

* Stellar spectra data from the [ESO Library of Stellar Spectra](http://www.eso.org/sci/facilities/paranal/decommissioned/isaac/tools/lib.html)


```python
S = fits.open('SolarSpectra.fits')

S.info()
```


```python
Data = S[0].data
```


```python
Head = S[0].header
Head
```


```python
# The FITS header has the information to make an array of wavelengths

Start = Head['CRVAL1']
Number = Head['NAXIS1']
Delta  = Head['CDELT1']

End = Start + (Number * Delta)

Wavelength = np.arange(Start,End,Delta)
```


```python
fig, ax = plt.subplots(2,1)
fig.set_size_inches(11,8.5)

fig.tight_layout()

# Full spectra

ax[0].plot(Wavelength, Data, color='b')
ax[0].set_ylabel("Flux")
ax[0].set_xlabel("Wavelength [angstroms]")

# Just the visible range with the hydrogen Balmer lines

ax[1].set_xlim(4000,7000)
ax[1].set_ylim(0.6,1.2)
ax[1].plot(Wavelength, Data, color='b')
ax[1].set_ylabel("Flux")
ax[1].set_xlabel("Wavelength [angstroms]")

H_Balmer = [6563,4861,4341,4102,3970,3889,3835,3646]

ax[1].vlines(H_Balmer,0,2, color='r', linewidth=3, alpha = 0.25)
```

# Pseudocolor - All color astronomy images are fake.

### Color images are composed of three 2-d images: 

### JPG images are 3-d, even grayscale images


```python
redfilter = plt.imread('sphereR.jpg')

redfilter.shape,redfilter.dtype
```

### We just want to read in one of the three channels


```python
redfilter = plt.imread('sphereR.jpg')[:,:,0]

redfilter.shape,redfilter.dtype
```


```python
plt.imshow(redfilter,cmap=plt.cm.gray);
```


```python
greenfilter = plt.imread('sphereG.jpg')[:,:,0]
bluefilter = plt.imread('sphereB.jpg')[:,:,0]
```


```python
fig, ax = plt.subplots(1,3)
fig.set_size_inches(12,3)

fig.tight_layout()

ax[0].set_title("Red Filter")
ax[1].set_title("Green Filter")
ax[2].set_title("Blue Filter")

ax[0].imshow(redfilter,cmap=plt.cm.gray)
ax[1].imshow(greenfilter,cmap=plt.cm.gray)
ax[2].imshow(bluefilter,cmap=plt.cm.gray);
```

### Need to create a blank 3-d array to hold all of the images


```python
rgb = np.zeros((480,640,3),dtype='uint8')

print(rgb.shape, rgb.dtype)

plt.imshow(rgb,cmap=plt.cm.gray);
```

## Fill the array with the filtered images


```python
rgb[:,:,0] = redfilter
rgb[:,:,1] = greenfilter
rgb[:,:,2] = bluefilter
```


```python
fig, ax = plt.subplots(1,4)
fig.set_size_inches(14,3)

fig.tight_layout()

ax[0].set_title("Red Filter")
ax[1].set_title("Green Filter")
ax[2].set_title("Blue Filter")
ax[3].set_title("All Filters Stacked")

ax[0].imshow(redfilter,cmap=plt.cm.gray)
ax[1].imshow(greenfilter,cmap=plt.cm.gray)
ax[2].imshow(bluefilter,cmap=plt.cm.gray)
ax[3].imshow(rgb,cmap=plt.cm.gray);
```


```python
print("The image rgb has a shape [height,width] of {0}".format(rgb.shape))
print("The image rgb is made up of data of type {0}".format(rgb.dtype))
print("The image rgb has a maximum value of {0}".format(rgb.max()))
print("The image rgb has a minimum value of {0}".format(rgb.min()))
```


```python
rgb[:,:,0] = redfilter * 1.5

plt.imshow(rgb)
```


```python

```

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Jupyter Notebook
07_Images_In_Python.ipynb
UWashington-Astro300/Astro300-A16
bb3ee938c905035dfd1ac123036a140a117e6b95
MIT
07_Images_In_Python.ipynb
UWashington-Astro300/Astro300-A16
bb3ee938c905035dfd1ac123036a140a117e6b95
MIT
07_Images_In_Python.ipynb
UWashington-Astro300/Astro300-A16
bb3ee938c905035dfd1ac123036a140a117e6b95
MIT
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