def calc_gradients(image, dx, dg, lambda0, peak, zT, downsample=3):
"""This function performs Fourier fringe analysis of the Talbot image,
and downsamples the gradient by selecting an ROI around the Fourier peak
Arguments:
image -- Talbot image (NxM)
dx -- pixel size in Talbot image (m)
dg -- grating period (m)
lambda0 -- wavelength (m)
mag -- Talbot magnification (approximate)
peak -- peak location in Fourier space for subtraction of linear phase
zT -- Talbot distance, distance between grating and detector (m)
downsample -- amount to downsample (power of 2) (default by factor of 8)
Returns:
h_grad -- downsampled gradient
v_grad -- downsampled gradient
zero_order -- downsampled zero order
x1, y1 -- downsampled coordinates
mid_peak -- average Fourier location of horizontal and vertical peaks
p0 -- base second order coefficient
"""
# calculate second order coefficient corresponding to peak shift that will be applied
# R2 is the distance from focus to detector corresponding to this peak
R2 = zT / (1 - dg * peak)
p0 = np.pi / lambda0 / R2
print(zT)
mag = R2/(R2-zT)
print('magnification: %.1f' % mag)
# get image dimensions
N, M = np.shape(image)
# fourier transform
start = time.time()
fourier_plane = Beam.NFFT(image)
end = time.time()
print(end-start)
#plt.figure()
#plt.imshow(np.abs(fourier_plane))
# spatial frequencies
fxmax = 1.0/(dx*2)
dfx = 2.*fxmax/M
fx = np.linspace(-M/2.,M/2.-1,M)*dfx
dfy = 2.*fxmax/N
fy = np.linspace(-N/2.,N/2.-1,N)*dfy
fx, fy = np.meshgrid(fx,fy)
# spatial frequency of grating (m^-1)
fG = 1.0/dg
# mask off first order peaks in fourier space
v_mask = fx**2 + (fy-fG/mag)**2 < (fG/mag/4)**2
h_mask = (fx-fG/mag)**2 + fy**2 < (fG/mag/4)**2
# mask off zero order peak in Fourier space
zero_mask = fx**2 + fy**2 < (fG/mag/4)**2
# multiply fourier plane by mask
v_mask = fourier_plane*v_mask
h_mask = fourier_plane*h_mask
#plt.figure()
#plt.imshow(np.abs(v_mask))
#plt.figure()
#plt.imshow(np.abs(h_mask))
#plt.show()
# multiply fourier plane by zero order mask
zero_fourier = fourier_plane*zero_mask
# find peak location for both horizontal and vertical
# project along each dimension
# thresholding of masked Fourier peaks to calculate peak location
h_2 = beam_threshold(h_mask,.2)
v_2 = beam_threshold(v_mask,.2)
# set up coordinates (Talbot image plane)
xp = np.linspace(-M/2,M/2-1,M)
yp = np.linspace(-N/2,N/2-1,N)
xp,yp = np.meshgrid(xp,yp)
x1 = xp*dx
y1 = yp*dx
# find peaks in Fourier space
h_peak = np.sum(h_2*fx)/np.sum(np.abs(h_2))
v_peak = np.sum(v_2*fy)/np.sum(np.abs(v_2))
h_mask = (fx-h_peak)**2 + fy**2<(fG/mag/4)**2
v_mask = fx**2 + (fy-v_peak)**2<(fG/mag/4)**2
h_mask = fourier_plane*h_mask
v_mask = fourier_plane*v_mask
# thresholding of masked Fourier peaks to calculate peak location
h_2 = beam_threshold(h_mask,.2)
v_2 = beam_threshold(v_mask,.2)
# find peaks in Fourier space
h_peak = np.sum(h_2*fx)/np.sum(np.abs(h_2))
v_peak = np.sum(v_2*fy)/np.sum(np.abs(v_2))
# find peak widths in Fourier space
h_width = np.sqrt(np.sum(h_2*(fx-h_peak)**2)/np.sum(np.abs(h_2)))
v_width = np.sqrt(np.sum(v_2*(fy-v_peak)**2)/np.sum(np.abs(v_2)))
# calculate average position of horizontal/vertical peaks
mid_peak = (v_peak+h_peak)/2.
peak = mid_peak
#R2x = zT / (1 - dg * h_peak)
#p0x = np.pi / lambda0 / R2x
#R2y = zT / (1 - dg * v_peak)
#p0y = np.pi / lambda0 / R2y
p0x = -np.pi/lambda0/zT * dg * h_peak
p0y = -np.pi/lambda0/zT * dg * v_peak
R2 = zT / (1 - dg * mid_peak)
p0 = np.pi / lambda0 / R2
# define linear phase related to approximate peak location
#h_grating = np.exp(-1j*2.*np.pi*h_peak*x1)
#v_grating = np.exp(-1j*2.*np.pi*v_peak*y1)
h_grating = np.exp(-1j*2.*np.pi*h_peak*x1)
v_grating = np.exp(-1j*2.*np.pi*v_peak*y1)
# Fourier transform back to real space, and multiply by linear phase
h_grad = np.conj(Beam.INFFT(h_mask)*h_grating)
v_grad = np.conj(Beam.INFFT(v_mask)*v_grating)
# back to Fourier space, now peaks have been shifted to zero
h_fourier = Beam.NFFT(h_grad)
v_fourier = Beam.NFFT(v_grad)
# crop out center of Fourier pattern to downsample
down = (2**downsample)*2
v_fourier = v_fourier[N/2-N/down:N/2+N/down,M/2-M/down:M/2+M/down]
h_fourier = h_fourier[N/2-N/down:N/2+N/down,M/2-M/down:M/2+M/down]
zero_fourier = zero_fourier[N/2-N/down:N/2+N/down,M/2-M/down:M/2+M/down]
# downsampled array size
N2,M2 = np.shape(v_fourier)
# downsampled image coordinates
xp = np.linspace(-M2/2,M2/2-1,M2)
yp = np.linspace(-N2/2,N2/2-1,N2)
xp,yp = np.meshgrid(xp,yp)
x1 = xp*dx*M/M2
y1 = yp*dx*N/N2
# back to real space, now downsampled
h_grad = Beam.INFFT(h_fourier)
v_grad = Beam.INFFT(v_fourier)
# calculate zero order (downsampled)
zero_order = np.abs(Beam.INFFT(zero_fourier))
params = {}
params['zero_order'] = zero_order
params['x1'] = x1
params['y1'] = y1
params['h_peak'] = h_peak
params['v_peak'] = v_peak
params['h_width'] = h_width
params['v_width'] = v_width
params['p0x'] = p0x
params['p0y'] = p0y
params['p0'] = p0
params['fourier'] = fourier_plane
# output
return h_grad, v_grad, params
def get_amplitude(image, dx, dg, mag):
"""This function performs Fourier fringe analysis of the Talbot image,
and downsamples the gradient by selecting an ROI around the Fourier peak
Arguments:
image -- Talbot image (NxM)
dx -- pixel size in Talbot image (m)
dg -- grating period (m)
lambda0 -- wavelength (m)
mag -- Talbot magnification (approximate)
peak -- peak location in Fourier space for subtraction of linear phase
zT -- Talbot distance, distance between grating and detector (m)
downsample -- amount to downsample (power of 2) (default by factor of 8)
Returns:
h_grad -- downsampled gradient
v_grad -- downsampled gradient
zero_order -- downsampled zero order
x1, y1 -- downsampled coordinates
mid_peak -- average Fourier location of horizontal and vertical peaks
p0 -- base second order coefficient
"""
# get image dimensions
N, M = np.shape(image)
# fourier transform
fourier_plane = Beam.NFFT(image)
# spatial frequencies
fxmax = 1.0 / (dx * 2)
dfx = 2. * fxmax / M
fx = np.linspace(-M / 2., M / 2. - 1, M) * dfx
dfy = 2. * fxmax / N
fy = np.linspace(-N / 2., N / 2. - 1, N) * dfy
fx, fy = np.meshgrid(fx, fy)
# spatial frequency of grating (m^-1)
fG = 1.0 / dg
# mask off first order peaks in fourier space
v_mask = fx ** 2 + (fy - fG / mag) ** 2 < (fG / mag / 2) ** 2
h_mask = (fx - fG / mag) ** 2 + fy ** 2 < (fG / mag / 2) ** 2
# mask off zero order peak in Fourier space
zero_mask = fx ** 2 + fy ** 2 < (fG / mag / 2) ** 2
# multiply fourier plane by mask
v_mask = fourier_plane * v_mask
h_mask = fourier_plane * h_mask
# multiply fourier plane by zero order mask
zero_fourier = fourier_plane * zero_mask
# find peak location for both horizontal and vertical
# project along each dimension
# thresholding of masked Fourier peaks to calculate peak location
h_2 = beam_threshold(h_mask, .2)
v_2 = beam_threshold(v_mask, .2)
# set up coordinates (Talbot image plane)
xp = np.linspace(-M / 2, M / 2 - 1, M)
yp = np.linspace(-N / 2, N / 2 - 1, N)
xp, yp = np.meshgrid(xp, yp)
x1 = xp * dx
y1 = yp * dx
# find peaks in Fourier space
h_peak = np.sum(h_2 * fx) / np.sum(np.abs(h_2))
v_peak = np.sum(v_2 * fy) / np.sum(np.abs(v_2))
# calculate average position of horizontal/vertical peaks
mid_peak = (v_peak + h_peak) / 2.
# Fourier transform back to real space, and multiply by linear phase
h_grad = np.conj(Beam.INFFT(h_mask))
v_grad = np.conj(Beam.INFFT(v_mask))
params = {}
params['x1'] = x1
params['y1'] = y1
params['mid_peak'] = mid_peak
zero_order = (np.abs(h_grad) + np.abs(v_grad)) / 2
# output
return zero_order, params
