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transparency.py
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transparency.py
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from __future__ import division, print_function
import numpy as np
from imageio import imread, imsave
# from matplotlib import pyplot as plt
def saveGrayscale(filename, data):
image = data - np.min(data)
image *= 255 / np.max(image)
np.round(image, out = image)
np.clip(image, 0, 255, out = image)
imsave(filename, image.astype(np.uint8))
im1 = imread("in1.png").astype(float) / 255
im2 = imread("in2.png").astype(float) / 255
# Align images
def getTotalArea(*images):
area = 0
for image in images:
area += image.shape[0] * image.shape[1]
return area
def getFilterLength(filterRadius, sigmasInFrame):
idealLength = filterRadius * sigmasInFrame # filterRadius = 2 * sigma
oddIntegerLength = 2 * int(round(0.5 * (idealLength + 1))) - 1
return oddIntegerLength
def getGaussian(length, sigmasInFrame):
x = np.linspace(-sigmasInFrame, sigmasInFrame, length, endpoint = True)
np.square(x, out = x)
x *= -0.5
np.exp(x, out = x)
x *= sigmasInFrame / (0.5 * length * np.sqrt(2 * np.pi))
return x
def getSpotlight(shape, sigmasInFrame):
column = getGaussian(shape[0], sigmasInFrame)
row = getGaussian(shape[1], sigmasInFrame)
spotlight = column[:, None] * row
return spotlight
def getEdgeFilter(filterSize, sigmasInFrame):
assert filterSize % 2 == 1
edgeFilter = getGaussian(filterSize, sigmasInFrame)
edgeFilter[filterSize // 2:] *= -1
edgeFilter[filterSize // 2] = 0
return edgeFilter
def getHalfOverhang(fixedShape, movingShape):
fixedShape = np.asarray(fixedShape[:2])
movingShape = np.asarray(movingShape[:2])
smallShape = np.minimum(fixedShape, movingShape)
overhang = movingShape - (smallShape + 1) // 2
return overhang
def getConvolvedShape(fixedShape, movingShape, overhang):
fixedShape = np.asarray(fixedShape[:2])
movingShape = np.asarray(movingShape[:2])
overhang = np.asarray(overhang[:2])
convolvedShape = tuple(fixedShape - movingShape + 1 + 2 * overhang)
return convolvedShape
def convolve(fixed, moving, overhang):
wrappedShape = tuple(np.asarray(fixed.shape) + np.asarray(overhang))
wrapped = np.fft.irfft2(
np.fft.rfft2(fixed, s = wrappedShape) \
* np.fft.rfft2(moving[::-1, ::-1], s = wrappedShape),
s = wrappedShape)
resultShape = getConvolvedShape(fixed.shape, moving.shape, overhang)
result = wrapped[-resultShape[0]:, -resultShape[1]:]
return result
def getConvolvedLength(fixedLength, movingLength, overhang):
convolvedLength = fixedLength - movingLength + 1 + 2 * overhang
return convolvedLength
def convolve1D(fixed, moving, overhang, axis):
wrappedLength = fixed.shape[axis] + overhang
moreDimensions = (slice(None),) + (None,) * (len(fixed.shape) - axis - 1)
wrapped = np.fft.irfft(
np.fft.rfft(fixed, n = wrappedLength, axis = axis) \
* np.fft.rfft(moving[::-1], n = wrappedLength)[moreDimensions],
n = wrappedLength, axis = axis)
resultLength = getConvolvedLength(fixed.shape[axis], moving.size, overhang)
result = wrapped[(slice(None),) * axis + (slice(-resultLength, None),)]
return result
def getAlignment(alignmentScores, overhang):
alignment = np.unravel_index(np.argmax(alignmentScores),
alignmentScores.shape) - overhang
return alignment
def crop(fixed, moving, alignment):
alignment = np.asarray(alignment)
fixedStart = np.maximum(alignment, 0)
fixedStop = np.minimum(alignment + moving.shape[:2], fixed.shape[:2])
croppedFixed = fixed[fixedStart[0]:fixedStop[0],
fixedStart[1]:fixedStop[1]]
movingStart = fixedStart - alignment
movingStop = fixedStop - alignment
croppedMoving = moving[movingStart[0]:movingStop[0],
movingStart[1]:movingStop[1]]
return croppedFixed, croppedMoving
grayIm1 = np.mean(im1, axis = 2)
grayIm2 = np.mean(im2, axis = 2)
filterRadius = np.sqrt(getTotalArea(im1, im2)) * 0.01
print("Filter radius:", filterRadius)
filterLength = getFilterLength(filterRadius, sigmasInFrame = 3)
edgeFilter = getEdgeFilter(filterLength, sigmasInFrame = 3)
# Add the filtered images for each axis together so that we only have to
# perform one 2D convolution. Unfortunately this doesn't make it much faster
# because filtering, not 2D convolution, is the performance bottleneck for some
# reason. I will probably switch to OpenCV's Sobel method.
filteredIm1 = (
np.abs(convolve1D(grayIm1, edgeFilter, filterLength // 2, axis = 0)) \
+ np.abs(convolve1D(grayIm1, edgeFilter, filterLength // 2, axis = 1))) \
* getSpotlight(grayIm1.shape, sigmasInFrame = 3)
del grayIm1
filteredIm2 = (
np.abs(convolve1D(grayIm2, edgeFilter, filterLength // 2, axis = 0)) \
+ np.abs(convolve1D(grayIm2, edgeFilter, filterLength // 2, axis = 1))) \
* getSpotlight(grayIm2.shape, sigmasInFrame = 3)
del grayIm2
overhang = getHalfOverhang(filteredIm1.shape, filteredIm2.shape)
alignmentScores = convolve(filteredIm1, filteredIm2, overhang)
# saveGrayscale("filteredIm1.png", filteredIm1)
# saveGrayscale("filteredIm2.png", filteredIm2)
del filteredIm1, filteredIm2
alignment = getAlignment(alignmentScores, overhang)
print("Alignment:", alignment)
saveGrayscale("alignmentScores.png", alignmentScores)
del alignmentScores
alignedIm1, alignedIm2 = crop(im1, im2, alignment)
del im1, im2
# Calculate hub axis
def getHubAxis(im1, im2):
imDiff = im2 - im1
imDiff -= np.mean(imDiff, axis = (0, 1))
xxT = np.einsum("ijk,ijl", imDiff, imDiff)
eigVals, eigVecs = np.linalg.eig(xxT)
eigIndex = np.argmax(eigVals)
hubAxis = eigVecs[:, eigIndex]
return hubAxis
def getFlatError(im1, im2, hubAxis):
imDiff = im2 - im1
imDiff -= np.mean(imDiff, axis = (0, 1))
xxT = np.einsum("ijk,ijl", imDiff, imDiff)
xTx = np.dot(np.ravel(imDiff), np.ravel(imDiff))
flatError = xTx - np.dot(hubAxis, np.dot(xxT, hubAxis))
return flatError
hubAxis = getHubAxis(alignedIm1, alignedIm2)
print("Hub Axis:", hubAxis)
flatError = getFlatError(alignedIm1, alignedIm2, hubAxis)
print("Flat Error:", flatError)
# Calculate hub center and cutoff values
imMean = alignedIm1 + alignedIm2
imMean *= 0.5
bgDiff = np.dot(alignedIm2 - alignedIm1, hubAxis)
del alignedIm1, alignedIm2
def extrapolateVarianceFromPreviousElementsTakingEachElementAsMean(x, weights):
middleTerm = x * weights
np.cumsum(middleTerm, out = middleTerm)
middleTerm *= x
middleTerm *= -2
variance = middleTerm
variance += np.cumsum(np.square(x) * weights)
variance /= np.cumsum(weights)
variance += np.square(x)
return variance
def getMeanOfFirstGaussian(x, weights):
cost = extrapolateVarianceFromPreviousElementsTakingEachElementAsMean(
x, weights)
cost /= np.cumsum(weights)
endOfNoise = np.argmax(cost)
bestIndex = endOfNoise + np.argmin(cost[endOfNoise:])
# plt.plot(x[1:], 0.2 * np.log(cost[1:]))
return x[bestIndex]
def getTransparentDiff(sortedBgDiff, weights):
if np.sum(sortedBgDiff) >= 0:
sortedBgDiff = sortedBgDiff[::-1]
weights = weights[::-1]
return getMeanOfFirstGaussian(sortedBgDiff, weights)
def getOpaqueDiff(sortedBgDiff, weights):
if np.sum(sortedBgDiff) < 0:
sortedBgDiff = sortedBgDiff[::-1]
weights = weights[::-1]
return getMeanOfFirstGaussian(sortedBgDiff, weights)
def getMountainWeights(bgDiff, mountainCenter, halfLife = 0.1):
mountainWeights = bgDiff - mountainCenter
k = np.log(2) / halfLife
mountainWeights *= k
np.abs(mountainWeights, out = mountainWeights)
np.negative(mountainWeights, out = mountainWeights)
np.exp(mountainWeights, out = mountainWeights)
return mountainWeights
def getHubCenter(imMean, weights):
weightedImMean = imMean * weights[:, :, None]
hubCenter = np.sum(weightedImMean, axis=(0, 1)) / np.sum(weights)
return hubCenter
# TODO: Consider weighting the pixels by flattened error when calculating
# opaqueDiff and transparentDiff
# TODO: Stop assuming transparentDiff is constant throughout the image
bgDiffOrder = np.argsort(np.ravel(bgDiff))
spotlight = getSpotlight(bgDiff.shape, sigmasInFrame = 3)
sortedSpotlight = np.ravel(spotlight)[bgDiffOrder]
del spotlight
sortedBgDiff = np.ravel(bgDiff)[bgDiffOrder]
del bgDiffOrder
# plt.hist(sortedBgDiff, weights = sortedSpotlight, bins = 200, density = True, rwidth = 1, linewidth = 0)
transparentDiff = getTransparentDiff(sortedBgDiff, sortedSpotlight)
print("Transparent Difference:", transparentDiff)
opaqueDiff = getOpaqueDiff(sortedBgDiff, sortedSpotlight)
print("Opaque Difference:", opaqueDiff)
# plt.show()
del sortedBgDiff
transparentWeights = getMountainWeights(bgDiff, transparentDiff)
hubCenter = getHubCenter(imMean, transparentWeights)
print("Hub Center:", hubCenter)
del transparentWeights
# Calculate transparency values
# TODO: Smooth the difference between the images before using it to find alpha
def getAlpha(bgDiff, opaqueDiff, transparentDiff):
alpha = bgDiff - transparentDiff
alpha /= opaqueDiff - transparentDiff
np.clip(alpha, 0, 1, out = alpha)
# Remove negative zero so that we don't get negative infinity from division
np.abs(alpha, out = alpha)
return alpha
alpha = getAlpha(bgDiff, opaqueDiff, transparentDiff)
del bgDiff
# Calculate true colors
def getTrueColors(imMean, hubCenter, alpha):
centeredImMean = imMean - hubCenter
oldSettings = np.seterr(divide = "ignore")
colorMultiplier = np.reciprocal(alpha)
perChannelCutoff = np.sign(centeredImMean)
perChannelCutoff += 1
perChannelCutoff *= 0.5
perChannelCutoff -= hubCenter
perChannelCutoff /= centeredImMean
np.seterr(**oldSettings)
perChannelCutoff = np.min(perChannelCutoff, axis = 2)
colorMultiplier = np.minimum(colorMultiplier, perChannelCutoff,
out = colorMultiplier)
# TODO: Test an example where this is necessary
colorMultiplier[np.all(centeredImMean == 0, axis = 2)] = 1
assert np.all(np.isfinite(colorMultiplier))
# TODO: Test an example where this is necessary
np.maximum(colorMultiplier, 1, out = colorMultiplier)
trueColors = centeredImMean
trueColors *= colorMultiplier[:, :, None]
trueColors += hubCenter
# Even though we never scale colors past the color space to compensate for
# alpha, some pixels move outside the color space during the initial color
# adjustment
np.clip(trueColors, 0, 1, out = trueColors)
return trueColors
# It's possible to choose better values for color and alpha in terms of
# reproducing the original images, but that would require more complicated math
# and it wouldn't be as good at preserving fully transparent areas.
def getTransparentImage(trueColors, alpha):
transparentImage = np.concatenate((trueColors, alpha[:, :, None]),
axis = 2)
return transparentImage
trueColors = getTrueColors(imMean, hubCenter, alpha)
transparentImage = getTransparentImage(trueColors, alpha)
def imageToBytes(image):
return np.round(image * 255).astype(np.uint8)
imsave("alpha.png", imageToBytes(alpha))
imsave("transparentImage.png", imageToBytes(transparentImage))