SOCR ≫

DSPA ≫

Topics ≫

Data Science and Predictive Analytics (UMich HS650)

Appendix 6: Image Convolution, Filtering, Fourier Transform using EBImage

SOCR/MIDAS (Ivo Dinov)

In this DSPA Appendix, we present examples of image processing, spectral manipulation, and filtering.

## Warning: package 'knitr' was built under R version 4.1.2

1 Getting started

r Biocpkg(“EBImage”) is an R package distributed as part of the Bioconductor project. To install the package, start R and enter:

install.packages("BiocManager")
BiocManager::install("EBImage")

Once the R package(“EBImage”) is installed, it can be loaded by the following command.

library("EBImage")

• Note: If you can’t install the EBImage package for any reason (e.g., operating system or version incompatibility, this is not a problem as you should be able to use the jpeg package instead.

# install.packages("jpeg")
library(jpeg)

2 Reading, displaying and writing images

Basic R package(“EBImage”) functionality includes reading, writing, and displaying of images. Images are read using the function readImage, which takes as input a filename or an URL. To start off, let us load a sample picture distributed with the package.

f = system.file("images", "sample.png", package="EBImage")
img = readImage(f)

The R package (“EBImage”) currently supports three image file formats: jpeg, png and tiff. Additional image formats can be imported using the R GitHubPkg (“aoles/RBioFormats”), which adds support for a wide range of file formats including proprietary microscopy image data and metadata.

Imported images can be visualized by the function display().

display(img, method="browser")

In interactive R sessions, display opens the image in a JavaScript viewer in a web browser tab. Mouse or keyboard shortcuts allow zooming in and out of the image, panning, and cycling through multiple image frames. Images can also be displayed using core R plotting methods, which allows for combining images with other plotting functionality, e.g., adding text labels on top of the image.

display(img, method="raster")
text(x = 20, y = 20, label = "Colorful Parrots", adj = c(0,1), col = "orange", cex = 2)

The graphics displayed in an R device can be saved using R package(“base”) R functions dev.print or dev.copy. For example, let’s save our annotated image as a JPEG file and verify its size on disk.

filename = "parrots.jpg"
dev.print(jpeg, filename = filename , width = dim(img)[1], height = dim(img)[2])

png 
  2 

file.info(filename)$size

[1] 34311

The default behavior of display can be globally changed by setting the "options("EBImage.display") to either“browser”or“raster”`. This is useful, for example, to preview images inside RStudio.

It is also possible to read and view color images,

imgcol = readImage(system.file("images", "sample-color.png", package="EBImage"))
display(imgcol)

Images may contain multiple frames, in which case they can be displayed all at once in a grid arrangement by specifying the function argument all = TRUE.

nuc = readImage(system.file("images", "nuclei.tif", package="EBImage"))
display(nuc, method = "raster", all = TRUE)

Alternatively, single frames can be displayed.

In addition to importing images, images can be exported (saved) to files using the EBImage::writeImage(). The image that we loaded was a PNG file. To save this image as a JPEG file, the JPEG format allows setting a quality value between 1 and 100 to reflect the desired level of image compression. The default value of the quality argument of writeImage is 100. Smaller values yield images of smaller size, but worse resolution (less detail).

writeImage(imgcol, "sample.jpeg", quality = 85)

Similarly, we could have saved the image as a TIFF file and set which compression algorithm we want to use. For a complete list of available parameters see ?writeImage.

3 Image data representation

EBImage uses a package-specific class Image to store and process images. It extends the R base class array, and all R package(“EBImage”) functions can also be called directly on matrices and arrays. You can find out more about this class by typing ?Image. Let us peek into the internal structure of an Image object.

str(img)

Formal class 'Image' [package "EBImage"] with 2 slots
  ..@ .Data    : num [1:768, 1:512] 0.447 0.451 0.463 0.455 0.463 ...
  ..@ colormode: int 0
  ..$ dim: int [1:2] 768 512

The .Data slot contains a numeric array of pixel intensities. We see that in this case the array is two-dimensional, with 768 times 512 elements, and corresponds to the pixel width and height of the image. These dimensions can be accessed using the dim function, just like for regular arrays.

dim(img)

[1] 768 512

Image data can be accessed as a plain R array using the imageData accessor.

imageData(img)[1:3, 1:6]

          [,1]      [,2]      [,3]      [,4]      [,5]      [,6]
[1,] 0.4470588 0.4627451 0.4784314 0.4980392 0.5137255 0.5294118
[2,] 0.4509804 0.4627451 0.4784314 0.4823529 0.5058824 0.5215686
[3,] 0.4627451 0.4666667 0.4823529 0.4980392 0.5137255 0.5137255

The as.array() method can be used to coerce an Image to an array.

is.Image( as.array(img) )

[1] FALSE

The distribution of pixel intensities can be plotted in a histogram, and their range inspected using the range function.

hist(img)

range(img)

[1] 0 1

A useful summary of Image objects is also provided by the show method, which is invoked if we simply type the object’s name.

img

Image 
  colorMode    : Grayscale 
  storage.mode : double 
  dim          : 768 512 
  frames.total : 1 
  frames.render: 1 

imageData(object)[1:5,1:6]
          [,1]      [,2]      [,3]      [,4]      [,5]      [,6]
[1,] 0.4470588 0.4627451 0.4784314 0.4980392 0.5137255 0.5294118
[2,] 0.4509804 0.4627451 0.4784314 0.4823529 0.5058824 0.5215686
[3,] 0.4627451 0.4666667 0.4823529 0.4980392 0.5137255 0.5137255
[4,] 0.4549020 0.4666667 0.4862745 0.4980392 0.5176471 0.5411765
[5,] 0.4627451 0.4627451 0.4823529 0.4980392 0.5137255 0.5411765

For a more compact representation without the preview of the intensities array use the print method with the argument short set to TRUE.

print(img, short=TRUE)

Image 
  colorMode    : Grayscale 
  storage.mode : double 
  dim          : 768 512 
  frames.total : 1 
  frames.render: 1 

Color images are based on 3 channels (RBG).

print(imgcol, short=TRUE)

Image 
  colorMode    : Color 
  storage.mode : double 
  dim          : 768 512 3 
  frames.total : 3 
  frames.render: 1 

They differ from gray-scale images by the property colorMode and the number of dimensions, 3 (B&W) vs. 4 (color). The colorMode slot turns out to be convenient when dealing with stacks of images. If it is set to gray-scale, then the third and all higher dimensions of the array are considered as separate image frames corresponding, for instance, to different z-positions, time points, replicates, etc. On the other hand, if colorMode is Color, then the third dimension is assumed to hold different color channels, and only the fourth and higher dimensions are used for multiple image frames. imgcol contains three color channels, which correspond to the red, green and blue intensities of the photograph. However, this does not necessarily need to be the case, and the number of color channels is arbitrary.

The “frames.total” and “frames.render” fields shown by the object summary correspond to the total number of frames contained in the image, and to the number of rendered frames. These numbers can be accessed using the function numberOfFrames by specifying the type argument.

numberOfFrames(imgcol, type = "render")

[1] 1

numberOfFrames(imgcol, type = "total")

[1] 3

Image frames can be extracted using getFrame and getFrames. getFrame returns the i-th frame contained in the image y. If type is "total", the function is unaware of the color mode and returns an xy-plane. For type="render" the function returns the i-th image as shown by the display function. While getFrame returns just a single frame, getFrames retrieves a list of frames which can serve as input to lapply-family functions. See the “Global thresholding” section for an illustration of this approach.

Let’s look at nuclear/cellular imaging data, which contains 4 total frames that correspond to the 4 separate gray-scale images, as indicated by “frames.render”.

nuc

Image 
  colorMode    : Grayscale 
  storage.mode : double 
  dim          : 510 510 4 
  frames.total : 4 
  frames.render: 4 

imageData(object)[1:5,1:6,1]
           [,1]       [,2]       [,3]       [,4]       [,5]       [,6]
[1,] 0.06274510 0.07450980 0.07058824 0.08235294 0.10588235 0.09803922
[2,] 0.06274510 0.05882353 0.07843137 0.09019608 0.09019608 0.10588235
[3,] 0.06666667 0.06666667 0.08235294 0.07843137 0.09411765 0.09411765
[4,] 0.06666667 0.06666667 0.07058824 0.08627451 0.08627451 0.09803922
[5,] 0.05882353 0.06666667 0.07058824 0.08235294 0.09411765 0.10588235

4 Color management

As described in the previous section, the class Image extends the base class array and uses colorMode to store how the color information of the multi-dimensional data should be handled.

The function colorMode can be used to access and change this property, modifying the rendering mode of an image. For example, if we take a Color image and change its mode to gray-scale, then the image won’t display as a single color image anymore but rather as three separate gray-scale frames corresponding to the red, green and blue channels. The function colorMode does not change the actual content of the image but only changes the way the image is rendered by R package(“EBImage”).

colorMode(imgcol) = Grayscale
display(imgcol, all=TRUE)

Color space conversions between gray-scale and Color images are performed using the function channel. It has a flexible interface which allows to convert either way between the modes, and can be used to extract color channels. Unlike colorMode, channel changes the pixel intensity values of the image.

Color to gray-scale conversion modes include taking a uniform average across the RGB channels, and a weighted luminescence preserving conversion mode better suited for display purposes.

The asred, asgreen and asblue modes convert a gray-scale image or array into a color image of the specified hue.

The convenience function toRGB promotes a gray-scale image to RGB color space by replicating it across the red, green and blue channels, which is equivalent to calling channel with mode set to rgb. When displayed, this image doesn’t look different from its gray-scale origin, which is expected because the information between the color channels is the same. To combine three gray-scale images into a single rgb image use the function rgbImage.

The function Image can be used to construct a color image from a character vector or array of named R colors (as listed by colors()) and/or hexadecimal strings of the form “#rrggbb” or “#rrggbbaa”.

colorMat = matrix(rep(c("red","green", "#0000ff"), 25), 5, 5)
colorImg = Image(colorMat)
colorImg

Image 
  colorMode    : Color 
  storage.mode : double 
  dim          : 5 5 3 
  frames.total : 3 
  frames.render: 1 

imageData(object)[1:5,1:5,1]
     [,1] [,2] [,3] [,4] [,5]
[1,]    1    0    0    1    0
[2,]    0    1    0    0    1
[3,]    0    0    1    0    0
[4,]    1    0    0    1    0
[5,]    0    1    0    0    1

display(colorImg, interpolate=FALSE)

5 Manipulating images

Being numeric arrays, images can be conveniently manipulated by any of R’s arithmetic operators. For example, we can produce a negative image by simply subtracting the image from its maximum value.

img_neg = max(img) - img
display( img_neg )

We can also increase the brightness of an image through addition, adjust the contrast through multiplication, and apply gamma correction through exponentiation.

img_comb = combine(
  img,
  img + 0.3,
  img * 2,
  img ^ 0.5
)

display(img_comb, all=TRUE)

In the example above we have used combine to merge individual images into a single multi-frame image object.

Furthermore, we can crop and threshold images with standard matrix operations.

img_crop = img[366:749, 58:441]
img_thresh = img_crop > .5
display(img_thresh)

The thresholding operation returns an Image object with binarized pixel values. The R data type used to store such an image is logical.

img_thresh

Image 
  colorMode    : Grayscale 
  storage.mode : logical 
  dim          : 384 384 
  frames.total : 1 
  frames.render: 1 

imageData(object)[1:5,1:6]
      [,1]  [,2]  [,3]  [,4]  [,5]  [,6]
[1,] FALSE FALSE FALSE FALSE FALSE FALSE
[2,] FALSE FALSE FALSE FALSE FALSE FALSE
[3,] FALSE FALSE FALSE FALSE FALSE FALSE
[4,] FALSE FALSE FALSE FALSE FALSE FALSE
[5,] FALSE FALSE FALSE FALSE FALSE FALSE

For image transposition, use transpose rather than R’s R package(“base”) function t. This is because the former one works also on color and multiframe images by swapping its spatial dimensions.

img_t = transpose(img)
display( img_t )

6 Spatial transformations

We just saw one type of spatial transformation, transposition, but there are many more, for example translation, rotation, reflection and scaling. The translate() method moves the image plane by the specified two-dimensional vector in such a way that pixels that end up outside the image region are cropped, and pixels that enter into the image region are set to background.

img_translate = translate(img, c(100,-50))
display(img_translate)

The background color can be set using the argument bg.col common to all relevant spatial transformation functions. The default sets the value of background pixels to zero which corresponds to black. Let us demonstrate the use of this argument with rotate which rotates the image clockwise by the given angle.

img_rotate = rotate(img, 30, bg.col = "white")
display(img_rotate)

To scale an image to desired dimensions use resize. If you provide only one of either width or height, the other dimension is automatically computed keeping the original aspect ratio.

img_resize = resize(img, w=256, h=256)
display(img_resize )

The functions flip and flop reflect the image around the image horizontal and vertical axis, respectively.

img_flip = flip(img)
img_flop = flop(img)

display(combine(img_flip, img_flop), all=TRUE)

Spatial linear transformations are implemented using the general affine transformation. It maps image pixel coordinates px using a 3x2 transformation matrix m in the following way: cbind(px, 1) %*% m. For example, horizontal sheer mapping can be applied by

m =  matrix(c(1, -.5, 128, 0, 1, 0), nrow=3, ncol=2)
img_affine = affine(img, m)
display( img_affine )

7 Filtering

7.1 Linear filters

A common preprocessing step involves cleaning up the images by removing local artifacts or noise through smoothing. An intuitive approach is to define a window of a selected size around each pixel and average the values within that neighborhood. After applying this procedure to all pixels, the new, smoothed image is obtained. Mathematically, this can be expressed as \[ f'(x,y) = \frac{1}{N} \sum_{s=-a}^{a}\sum_{t=-a}^{a} f(x+s, y+t), \] where \(f(x,y)\) is the value of the pixel at position \((x, y)\), and \(a\) determines the window size, which is \(2a+1\) in each direction. \(N=(2a+1)^2\) is the number of pixels averaged over, and \(f'\) is the new, smoothed image.

More generally, we can replace the moving average by a weighted average, using a weight function \(w\), which typically has the highest value at the window midpoint (\(s=t=0\)) and then decreases towards the edges. \[ (w * f)(x,y) = \sum_{s=-\infty}^{+\infty} \sum_{t=-\infty}^{+\infty} w(s,t)\, f(x+s, y+s). \] For notational convenience, we let the summations range from \(-\infty\) to \(+\infty\), even if in practice the sums are finite and \(w\) has only a finite number of non-zero values. In fact, we can think of the weight function \(w\) as another image, and this operation is also called the convolution of the images \(f\) and \(w\), indicated by the the symbol \(*\).

Convolution is a linear operation in the sense that \(w*(c_1f_1+c_2f_2)=c_1w*f_1 + c_2w*f_2\) for any two images \(f_1\), \(f_2\) and numbers \(c_1\), \(c_2\).

In r Biocpkg(“EBImage”), the 2-dimensional convolution is implemented by the function filter2, and the auxiliary function makeBrush can be used to generate the weight function. In fact, filter2 does not directly perform the summation indicated in the equation above. Instead, it uses the Fast Fourier Transformation in a way that is mathematically equivalent but computationally more efficient.

w = makeBrush(size = 31, shape = 'gaussian', sigma = 5)
plot(w[(nrow(w)+1)/2, ], ylab = "w", xlab = "", cex = 0.7)

img_flo = filter2(img, w)
display(img_flo)

Here we have used a Gaussian filter of width 5 given by sigma. Other available filter shapes include "box" (default), "disc", "diamond" and "line", for some of which the kernel can be binary; see ?makeBrush for details.

If the filtered image contains multiple frames, the filter is applied to each frame separately. For convenience, images can be also smoothed using the wrapper function gblur which performs Gaussian smoothing with the filter size automatically adjusted to sigma.

nuc_gblur = gblur(nuc, sigma = 5)
display(nuc_gblur, all=TRUE )

In signal processing the operation of smoothing an image is referred to as low-pass filtering. High-pass filtering is the opposite operation which allows to detect edges and sharpen images. This can be done, for instance, using a Laplacian filter.

fhi = matrix(1, nrow = 3, ncol = 3)
fhi[2, 2] = -8
img_fhi = filter2(img, fhi)
display(img_fhi)

7.2 Median filter

Another approach to perform noise reduction is to apply a median filter, which is a non-linear technique as opposed to the low pass convolution filter described in the previous section. Median filtering is particularly effective in the case of speckle noise, and has the advantage of removing noise while preserving edges.

The local median filter works by scanning the image pixel by pixel, replacing each pixel by the median of its neighbors inside a window of specified size. This filtering technique is provided in R package(“EBImage”) by the function medianFilter. We demonstrate its use by first corrupting the image with uniform noise, and reconstructing the original image by median filtering.

l = length(img)
n = l/10
pixels = sample(l, n)
img_noisy = img
img_noisy[pixels] = runif(n, min=0, max=1)
display(img_noisy)

img_median = medianFilter(img_noisy, 1)
display(img_median)

7.3 Morphological operations

Binary images are images which contain only two sets of pixels, with values, say 0 and 1, representing the background and foreground pixels. Such images are subject to several non-linear morphological operations: erosion, dilation, opening, and closing. These operations work by overlaying a mask, called the structuring element, over the binary image in the following way:

• erosion: For every foreground pixel, put the mask around it, and if any pixel covered by the mask is from the background, set the pixel to background.

• dilation: For every background pixel, put the mask around it, and if any pixel covered by the mask is from the foreground, set the pixel to foreground.

shapes = readImage(system.file('images', 'shapes.png', package='EBImage'))
logo = shapes[110:512,1:130]
display(logo)

kern = makeBrush(5, shape='diamond')
display(kern, interpolate=FALSE)

logo_erode= erode(logo, kern)
logo_dilate = dilate(logo, kern)

display(combine(logo_erode, logo_dilate), all=TRUE)

Opening and closing are combinations of the two operations above: opening performs erosion followed by dilation, while closing does the opposite, i.e, performs dilation followed by erosion. Opening is useful for morphological noise removal, as it removes small objects from the background, and closing can be used to fill small holes in the foreground. These operations are implemented by opening and closing.

8 Thresholding

8.1 Global thresholding

In the “Manipulating images” section we have already demonstrated how to set a global threshold on an image. There we used an arbitrary cutoff value. For images whose distribution of pixel intensities follows a bi-modal histogram a more systematic approach involves using the Otsu’s method. Otsu’s method is a technique to automatically perform clustering-based image thresholding. Assuming a bi-modal intensity distribution, the algorithm separates image pixels into foreground and background. The optimal threshold value is determined by minimizing the combined intra-class variance.

Otsu’s threshold can be calculated using the function otsu. When called on a multi-frame image, the threshold is calculated for each frame separately resulting in an output vector of length equal to the total number of frames in the image.

threshold = otsu(nuc)
threshold

[1] 0.3535156 0.4082031 0.3808594 0.4121094

nuc_th = combine( mapply(function(frame, th) frame > th, getFrames(nuc), threshold, SIMPLIFY=FALSE) )
display(nuc_th, all=TRUE)

Note the use of getFrames to split the image into a list of individual frames, and combine to merge the results back together.

8.2 Adaptive thresholding

The idea of adaptive thresholding is that, compared to straightforward thresholding from the previous section, the threshold is allowed to be different in different regions of the image. In this way, one can anticipate spatial dependencies of the underlying background signal caused, for instance, by uneven illumination or by stray signal from nearby bright objects.

Adaptive thresholding works by comparing each pixel’s intensity to the background determined from a local neighborhood. This can be achieved by comparing the image to its smoothed version, where the filtering window is bigger than the typical size of objects we want to capture.

disc = makeBrush(31, "disc")
disc = disc / sum(disc)
offset = 0.05
nuc_bg = filter2( nuc, disc )
nuc_th = nuc > nuc_bg + offset
display(nuc_th, all=TRUE)

This technique assumes that the objects are relatively sparsely distributed in the image, so that the signal distribution in the neighborhood is dominated by background. While for the nuclei in our images this assumption makes sense, for other situations you may need to make different assumptions. The adaptive thresholding using a linear filter with a rectangular box is provided by thresh, which uses a faster implementation compared to directly using filter2.

display( thresh(nuc, w=15, h=15, offset=0.05), all=TRUE )

9 Fourier Transform

9.1 FFT Shift Function

# FFT SHIFT
#' This function is useful for visualizing the Fourier transform with the zero-frequency 
#' component in the middle of the spectrum.
#' 
#' @param img_ff A Fourier transform of a 1D signal, 2D image, or 3D volume.
#' @param dim Number of dimensions (-1, 1, 2, 3).
#' @return A properly shifted FT of the array.
#' 
fftshift <- function(img_ff, dim = -1) {

  rows <- dim(img_ff)[1]    
  cols <- dim(img_ff)[2]
  # planes <- dim(img_ff)[3]

  swap_up_down <- function(img_ff) {
    rows_half <- ceiling(rows/2)
    return(rbind(img_ff[((rows_half+1):rows), (1:cols)], img_ff[(1:rows_half), (1:cols)]))
  }

  swap_left_right <- function(img_ff) {
    cols_half <- ceiling(cols/2)
    return(cbind(img_ff[1:rows, ((cols_half+1):cols)], img_ff[1:rows, 1:cols_half]))
  }
  
  #swap_side2side <- function(img_ff) {
  #  planes_half <- ceiling(planes/2)
  #  return(cbind(img_ff[1:rows, 1:cols, ((planes_half+1):planes)], img_ff[1:rows, 1:cols, 1:planes_half]))
  #}

  if (dim == -1) {
    img_ff <- swap_up_down(img_ff)
    return(swap_left_right(img_ff))
  }
  else if (dim == 1) {
    return(swap_up_down(img_ff))
  }
  else if (dim == 2) {
    return(swap_left_right(img_ff))
  }
  else if (dim == 3) {
    # Use the `abind` package to bind along any dimension a pair of multi-dimensional arrays
    # install.packages("abind")
    library(abind)
    
    planes <- dim(img_ff)[3]
    rows_half <- ceiling(rows/2)
    cols_half <- ceiling(cols/2)
    planes_half <- ceiling(planes/2)
    
    img_ff <- abind(img_ff[((rows_half+1):rows), (1:cols), (1:planes)], 
                    img_ff[(1:rows_half), (1:cols), (1:planes)], along=1)
    img_ff <- abind(img_ff[1:rows, ((cols_half+1):cols), (1:planes)], 
                    img_ff[1:rows, 1:cols_half, (1:planes)], along=2)
    img_ff <- abind(img_ff[1:rows, 1:cols, ((planes_half+1):planes)], 
                    img_ff[1:rows, 1:cols, 1:planes_half], along=3)
    return(img_ff)
  }
  else {
    stop("Invalid dimension parameter")
  }
}

str(img)

Formal class 'Image' [package "EBImage"] with 2 slots
  ..@ .Data    : num [1:768, 1:512] 0.447 0.451 0.463 0.455 0.463 ...
  ..@ colormode: int 0
  ..$ dim: int [1:2] 768 512

imgMat <- matrix(img, nrow = dim(img)[1], ncol = dim(img)[2])
dim(imgMat)

[1] 768 512

9.2 FT

ft_imgMat <- fft(imgMat)  # fftw2d # Display Re(FT):
display(fftshift(ft_imgMat))

Warning in storage.mode(x) <- "double": imaginary parts discarded in coercion

9.3 Magnitude and Phase

mag_ft_imgMat <- sqrt(Re(ft_imgMat)^2+Im(ft_imgMat)^2)
display(mag_ft_imgMat)

# Phase  <- atan(Im(img_ff)/Re(img_ff))
phase_ft_imgMat  <- atan2(Im(ft_imgMat), Re(ft_imgMat))
# display(phase_ft_imgMat)

# Display FT
EBImage::display(log(fftshift(ft_imgMat)), title="FT Magnitude (Cyrillic Alphabet)")

Warning in storage.mode(x) <- "double": imaginary parts discarded in coercion

9.4 IFT Reconstruction

# IFT Reconstruction: Magnitude=mag_ft_imgMat   AND  Phase=phase_ft_imgMat
Real = mag_ft_imgMat * cos(phase_ft_imgMat)
Imaginary = mag_ft_imgMat * sin(phase_ft_imgMat)
ift_ft_imgMat = Re(fft(Real+1i*Imaginary, inverse = T)/length(mag_ft_imgMat))
EBImage::display(ift_ft_imgMat, method = "raster", title="(IFT o FT) Magnitude=Img | Phase=Img")

9.5 Low-pass frequency filtering

Remove Fourier coefficients indexed above \(k_{x,max/2}\) and \(k_{y,max/2}\).

mag_ft_imgMat_1 <- mag_ft_imgMat
for (i in 1:dim(mag_ft_imgMat)[1]) {
  for (j in 1:dim(mag_ft_imgMat)[2]) {
    if (abs(i-(dim(mag_ft_imgMat)[1])/2) > (dim(mag_ft_imgMat)[1])/4 | 
        abs(j-(dim(mag_ft_imgMat)[2])/2) > (dim(mag_ft_imgMat)[2])/4) {
      mag_ft_imgMat_1[i,j] <- 0
    }
  }
}

EBImage::display(mag_ft_imgMat_1, method = "raster", title="(IFT o LowPassFilter o FT) Magnitude=Img | Phase=Img")

Real = mag_ft_imgMat_1 * cos(phase_ft_imgMat)
Imaginary = mag_ft_imgMat_1 * sin(phase_ft_imgMat)
ift_ft_imgMat_1 = Re(fft(Real+1i*Imaginary, inverse = T)/length(mag_ft_imgMat_1))
EBImage::display(100*ift_ft_imgMat_1, method = "raster", title="(IFT o LowPassFilter o FT) Magnitude=Img | Phase=Img")

9.6 High-pass frequency filtering

Remove the Fourier coefficients indexed below \(k_{x,max/4}\) and \(k_{y,max/4}\).

# High-pass frequency filtering, remove Fourier coeff's indexed below k_{x,max/4} and k_{y,max/4}
mag_ft_imgMat_2 <- mag_ft_imgMat
for (i in 1:dim(mag_ft_imgMat)[1]) {
  for (j in 1:dim(mag_ft_imgMat)[2]) {
    if (abs(i-(dim(mag_ft_imgMat)[1])/2) < (dim(mag_ft_imgMat)[1])/4 & 
        abs(j-(dim(mag_ft_imgMat)[2])/2) < (dim(mag_ft_imgMat)[2])/4) {
      mag_ft_imgMat_2[i,j] <- 0
    }
  }
}

EBImage::display(mag_ft_imgMat_2, method = "raster", title="(IFT o HighPassFilter o FT) Magnitude=Img | Phase=Img")

Real = mag_ft_imgMat_2 * cos(phase_ft_imgMat)
Imaginary = mag_ft_imgMat_2 * sin(phase_ft_imgMat)
ift_ft_imgMat_2 = Re(fft(Real+1i*Imaginary, inverse = T)/length(mag_ft_imgMat_2))
EBImage::display(ift_ft_imgMat_2, method = "raster", title="(IFT o HighPassFilter o FT) Magnitude=Img | Phase=Img")

10 Image segmentation

Image segmentation performs partitioning of an image, and is typically used to identify objects in an image. Non-touching connected objects can be segmented using the function bwlabel, while watershed and propagate use more sophisticated algorithms able to separate objects which touch each other.

bwlabel finds every connected set of pixels other than the background, and relabels these sets with a unique increasing integer. It can be called on a thresholded binary image in order to extract objects.

logo_label = bwlabel(logo)
table(logo_label)

logo_label
    0     1     2     3     4     5     6     7 
42217  1375  2012   934  1957  1135  1697  1063 

The pixel values of the logo_label image range from 0 corresponding to background to the number of objects it contains, which is given by

max(logo_label)

[1] 7

To display the image we normalize it to the (0,1) range expected by the display function. This results in different objects being rendered with a different shade of gray.

display( normalize(logo_label) )

The horizontal gray-scale gradient which can be observed reflects the way bwlabel() scans the image and labels the connected sets: from left to right and from top to bottom. Another way of visualizing the segmentation is to use the colorLabels function, which color codes the objects by a random permutation of unique colors.

display( colorLabels(logo_label) )

10.1 Watershed

Some of the nuclei in nuc are quite close to each other and get merged into one big object when thresholded, as seen in nuc_th. bwlabel would incorrectly identify them as a single object. The watershed transformation allows us to overcome this issue. The watershed algorithm treats a gray-scale image as a topographic relief, or height-map. Objects that stand out of the background are identified and separated by flooding an inverted source image. In case of a binary image its distance map can serve as the input height-map. The distance map, which contains for each pixel the distance to the nearest background pixel, can be obtained by distmap.

nmask = watershed( distmap(nuc_th), 2 )
display(colorLabels(nmask), all=TRUE)

Let’s try the watershed algorithm on structural MRI data.

library(brainR)
#load the sMRI data (same as in Problem 1)
# 3D sMRI data: http://socr.umich.edu/HTML5/BrainViewer/data/ABIDE_MRI_MPRAGE_peds_defaced.nii.gz
brainURL <- "http://socr.umich.edu/HTML5/BrainViewer/data/ABIDE_MRI_MPRAGE_peds_defaced.nii.gz"
brainFile <- file.path(tempdir(), "ABIDE_MRI_MPRAGE_peds_defaced.nii.gz")
download.file(brainURL, dest=brainFile, quiet=TRUE)
brainVolume <- readNIfTI(brainFile, reorient=FALSE)

brainVolDims <- dim(brainVolume); brainVolDims

[1] 256 182 256

# get the mid-axial slice index
midZ = brainVolDims[3] / 2; midZ

[1] 128

# plot 2D sMRI slice
image(brainVolume[,, midZ], asp=1, col = hcl.colors(12, "YlOrRd", rev = TRUE)) # Hot-metal

# Apply watershed image segmentation.
sMRI_Watershed_seg = watershed( brainVolume[,, midZ], ext=2 )
display(normalize(sMRI_Watershed_seg), method="raster")

10.2 Voronoi tessellation

Voronoi tessellation is useful when we have a set of seed points (or regions) and want to partition the space that lies between these seeds in such a way that each point in the space is assigned to its closest seed. This function is implemented in the R package(“EBImage”) by the function propagate. Let us illustrate the concept of Voronoi tessellation with a basic example. We use the nuclei mask nmask as seeds and partition the space between them.

The 2D Interactive Voronoi Tessellation webapp provides a hands-on example of dynamic 2D Voronoi tessellation.

voronoiExamp = propagate(seeds = nmask, x = nmask, lambda = 100)
voronoiPaint = colorLabels (voronoiExamp)
display(voronoiPaint)

Only the first frame of the image stack is displayed.
To display all frames use 'all = TRUE'.

The basic definition of Voronoi tessellation, which we have given above, allows for two generalizations:

• By default, the space that we partition is the full, rectangular image area, but indeed we could restrict ourselves to any arbitrary subspace. This is akin to finding the shortest distance from each point to the next seed not in a simple flat landscape, but in a landscape that is interspersed by lakes and rivers (which you cannot cross), so that all paths need to remain on the land. propagate allows for this generalization through its mask argument.

• By default, we think of the space as flat – but in fact it could have hills and canyons, so that the distance between two points in the landscape not only depends on their x- and y-positions but also on the ascents and descents, up and down in z-direction, that lie in between. You can specify such a landscape to propagate through its x argument.

Mathematically, we can say that instead of the simple default case (a flat rectangle image with an Euclidean metric), we perform the Voronoi segmentation on a Riemann manifold, which can have an arbitrary shape and an arbitrary metric. Let us use the notation \(x\) and \(y\) for the column and row coordinates of the image, and \(z\) for the elevation of the landscape. For two neighboring points, defined by coordinates \((x, y, z)\) and \((x+dx, y+dy, z+dz)\), the distance between them is given by

\[ ds = \sqrt{ \frac{2}{\lambda+1} \left[ \lambda \left( dx^2 + dy^2 \right) + dz^2 \right] }. \] For \(\lambda=1\), this reduces to \(ds = ( dx^2 + dy^2 + dz^2)^{1/2}\). Distances between points further apart are obtained by summing \(ds\) along the shortest path between them. The parameter \(\lambda\ge0\) has been introduced as a convenient control of the relative weighting between sideways movement (along the \(x\) and \(y\) axes) and vertical movement. Intuitively, if you imagine yourself as a hiker in such a landscape, by choosing \(\lambda\) you can specify how much you are prepared to climb up and down to overcome a mountain, versus sideways walking around it.

When \(\lambda\) is large, the expression becomes equivalent to \(ds = \sqrt{dx^2 + dy^2}\), i. e., the importance of \(dz\) becomes negligible. This is what we did when we used lambda = 100 in our propagate example.

A more advanced application of propagate to the segmentation of cell bodies is presented in the “Cell segmentation example” section.

Let’s demonstrate Voronoi tessellation using the 2D sMRI image.

# Try Voronoi tessellation

nmask = thresh(brainVolume[,, midZ], w=100, h=50, offset=0.05)
nmask = opening(nmask, makeBrush(5, shape='disc'))
nmask = fillHull(nmask)
nmask = bwlabel(nmask)
display(nmask, all=TRUE)

sMRI_Watershed_seg = watershed( brainVolume[,, midZ], ext=2 )
brain <- sMRI_Watershed_seg * nmask
display(normalize(brain), method="raster")

voronoi_sMRI = propagate(seeds = brain, x = brain, lambda = 100)
voronoiPaint = colorLabels (voronoi_sMRI)
display(voronoiPaint, method="raster")

11 Object manipulation

11.1 Object removal

The R package(“EBImage”) defines an object mask as a set of pixels with the same unique integer value. Typically, images containing object masks are the result of segmentation functions such as bwlabel, watershed, or propagate. Objects can be removed from such images by rmObject, which deletes objects from the mask simply by setting their pixel values to 0. By default, after object removal all the remaining objects are relabeled so that the highest object ID corresponds to the number of objects in the mask. The reenumerate argument can be used to change this behavior and to preserve original object IDs.

objects = list(
    seq.int(from = 2, to = max(logo_label), by = 2),
    seq.int(from = 1, to = max(logo_label), by = 2)
    )
logos = combine(logo_label, logo_label)
z = rmObjects(logos, objects, reenumerate=FALSE)
display(z, all=TRUE)

In the example above we demonstrate how the object removal function can be applied to a multi-frame image by providing a list of object indices to be removed from each frame. Additionally we have set reenumerate to FALSE keeping the original object IDs.

showIds = function(image) lapply(getFrames(image), function(frame) unique(as.vector(frame)))

showIds(z)

[[1]]
[1] 0 1 3 5 7

[[2]]
[1] 0 2 4 6

Recall that 0 stands for the background. If at some stage we decide to relabel the objects, we can use for this the standalone function reenumarate.

showIds( reenumerate(z) )

[[1]]
[1] 0 1 2 3 4

[[2]]
[1] 0 1 2 3

11.2 Filling holes and regions

Holes in object masks can be filled using the function fillHull.

filled_logo = fillHull(logo)
display(filled_logo)

floodFill fills a region of an image with a specified color. The filling starts at the given point, and the filling region is expanded to a connected area in which the absolute difference in pixel intensities remains below tolerance. The color specification uses R color names for Color images, and numeric values for gray-scale images.

rgblogo = toRGB(logo)
points = rbind(c(50, 50), c(100, 50), c(150, 50))
colors = c("red", "green", "blue")
rgblogo = floodFill(rgblogo, points, colors)
display( rgblogo )

display(floodFill(img, rbind(c(200, 300), c(444, 222)), col=0.2, tolerance=0.2))

11.3 Highlighting objects

Given an image containing object masks, the function paintObjects() can be used to highlight the objects from the mask in the target image provided in the tgt argument. Objects can be outlined and filled with colors of given opacities specified in the col and opac arguments, respectively. If the color specification is missing or equals NA it is not painted.

d1 = dim(img)[1:2]
overlay = Image(dim=d1)
d2 = dim(logo_label)-1

offset = (d1-d2) %/% 2

overlay[offset[1]:(offset[1]+d2[1]), offset[2]:(offset[2]+d2[2])] = logo_label

img_logo = paintObjects(overlay, toRGB(img), col=c("red", "yellow"), opac=c(1, 0.3), thick=TRUE)

display( img_logo )

In the example above we have created a new mask overlay matching the size of our target image img, and copied the mask containing the “EBImage” logo into that overlay mask. The output of paintObjects retains the color mode of its target image, therefore in order to have the logo highlighted in color it was necessary to convert img to an RGB image first, otherwise the result would be a gray-scale image. The thick argument controls the object contour drawing: if set to FALSE, only the inner one-pixel wide object boundary is marked; if set to TRUE, also the outer boundary gets highlighted resulting in an increased two-pixel contour width.

12 Cell segmentation example

We conclude our vignette by applying the functions described before to the task of segmenting cells. Our goal is to computationally identify and qualitatively characterize the cells in the sample fluorescent microscopy images. Even though this by itself may seem a modest goal, this approach can be applied to collections containing thousands of images, and that need is no longer a modest aim!

We start by loading the images of nuclei and cell bodies. To visualize the cells we overlay these images as the green and the blue channel of a false-color image.

nuc = readImage(system.file('images', 'nuclei.tif', package='EBImage'))
cel = readImage(system.file('images', 'cells.tif', package='EBImage'))

cells = rgbImage(green=1.5*cel, blue=nuc)
display(cells, all = TRUE)

First, we segment the nuclei using thresh, fillHull, bwlabel and opening.

nmask = thresh(nuc, w=10, h=10, offset=0.05)
nmask = opening(nmask, makeBrush(5, shape='disc'))
nmask = fillHull(nmask)
nmask = bwlabel(nmask)

display(nmask, all=TRUE)

Next, we use the segmented nuclei as seeds in the Voronoi segmentation of the cytoplasm.

ctmask = opening(cel>0.1, makeBrush(5, shape='disc'))
cmask = propagate(cel, seeds=nmask, mask=ctmask)

display(ctmask, all=TRUE)

To visualize our segmentation we use paintObject.

segmented = paintObjects(cmask, cells, col='#ff00ff')
segmented = paintObjects(nmask, segmented, col='#ffff00')

display(segmented, all=TRUE)

13 Learning Activity

Let’s use the protocol below to gain some hands-on experience processing a 2D brain image using basic functions, without relying on the EBImage package. Of course, you can also apply all of the methods presented above on this 2D (JPEG) brain image.

# Install and load the JPEG package
# install.packages("jpeg")
library(jpeg)

# Read Image in and display it
# Either download the JPG image locally from https://umich.instructure.com/courses/38100/files/1627149/download?download_frd=1 and then read it in
# mri <- readJPEG("C:/Users/Dinov/Desktop/MRI_ImageHematoma.jpg")

# Or directly load it from the web
pathToMRI <- tempfile(, fileext = ".jpg")
download.file("https://umich.instructure.com/courses/38100/files/1627149/download?download_frd=1", mode = "wb", pathToMRI)
mri <- readJPEG(pathToMRI)
# display(mri)
plot.new(); rasterImage(mri,0,0,1,1)

# Threshold the image
# display( thresh(mri, offset=0.05), all=TRUE )
# plot.new(); rasterImage(EBImage::thresh(mri, offset=0.05),0,0,1,1)
mri_thresh = mri[,,1] > 0.35
mri_thresh = mri[,,1] * mri_thresh
plot.new(); rasterImage(mri_thresh,0,0,1,1)

# Histogram
hist(mri)

# Affine transformation
m = matrix(c(1, -.5, 128, 0, 1, 0), nrow=3, ncol=2)
img_affine = EBImage::affine(mri, m)
# display( img_affine )
plot.new(); rasterImage(img_affine,0,0,1,1)

# Sheer Affine Transformation and translation
m = matrix(c(1, -.5, 128, 0, 1, 0), nrow=3, ncol=2)
img_affine = EBImage::affine(EBImage::translate(mri, c(-50, 0)), m)
# display( img_affine )
plot.new(); rasterImage(img_affine,0,0,1,1)

# Gaussian Kernel smoothing (Convolution)
kernel = EBImage::makeBrush(size = 31, shape = 'gaussian', sigma = 5)
mri_smooth = EBImage::filter2(mri, kernel)
# display(mri_smooth)
plot.new(); rasterImage(mri_smooth,0,0,1,1)

# FFT
ft_mriMat <- fft(mri[,,1])  # fftw2d # Display Re(FT):
EBImage::display(fftshift(ft_mriMat))

# FFT Magnitude
mag_ft_mriMat <- sqrt(Re(ft_mriMat)^2+Im(ft_mriMat)^2)
display(mag_ft_mriMat)

#plot.new(); rasterImage(normalize(mag_ft_mriMat),0,0,1,1)

# IFT
phase_ft_mriMat <- atan2(Im(ft_mriMat), Re(ft_mriMat)); Real = mag_ft_mriMat * cos(phase_ft_mriMat)
Imaginary = mag_ft_mriMat * sin(phase_ft_mriMat)
ift_ft_mag_ft_mriMat = Re(fft(Real+1i*Imaginary, inverse = T)/length(mag_ft_mriMat)); 
display(t(ift_ft_mag_ft_mriMat))

# Low- and High-pass Fourier domain filtering

## Low-pass frequency filtering
## Remove Fourier coefficients indexed above $k_{x,max/2}$ and $k_{y,max/2}$.
mag_ft_mriMat_1 <- mag_ft_mriMat
for (i in 1:dim(mag_ft_mriMat)[1]) {
  for (j in 1:dim(mag_ft_mriMat)[2]) {
    if (abs(i-(dim(mag_ft_mriMat)[1])/2) > (dim(mag_ft_mriMat)[1])/4 | 
        abs(j-(dim(mag_ft_mriMat)[2])/2) > (dim(mag_ft_mriMat)[2])/4) {
      mag_ft_mriMat_1[i,j] <- 0
    }
  }
}

# EBImage::display(mag_ft_mriMat_1, method = "raster", title="(IFT o LowPassFilter o FT) Magnitude=Img | Phase=Img")
plot.new(); rasterImage(normalize(mag_ft_mriMat_1),0,0,1,1)

Real = mag_ft_mriMat_1 * cos(phase_ft_mriMat)
Imaginary = mag_ft_mriMat_1 * sin(phase_ft_mriMat)
ift_ft_mriMat_1 = Re(fft(Real+1i*Imaginary, inverse = T)/length(mag_ft_mriMat_1))
EBImage::display(t(400*ift_ft_mriMat_1), method = "raster", title="(IFT o LowPassFilter o FT) Magnitude=MRI | Phase=MRI")

# plot.new(); rasterImage(normalize(ift_ft_mriMat_1),0,0,1,1)

## High-pass frequency filtering
## Remove the Fourier coefficients indexed below $k_{x,max/4}$ and $k_{y,max/4}$.
# High-pass frequency filtering, remove Fourier coeff's indexed below k_{x,max/4} and k_{y,max/4}
mag_ft_mriMat_2 <- mag_ft_mriMat
for (i in 1:dim(mag_ft_mriMat)[1]) {
  for (j in 1:dim(mag_ft_mriMat)[2]) {
    if (abs(i-(dim(mag_ft_mriMat)[1])/2) < (dim(mag_ft_mriMat)[1])/4 & 
        abs(j-(dim(mag_ft_mriMat)[2])/2) < (dim(mag_ft_mriMat)[2])/4) {
      mag_ft_mriMat_2[i,j] <- 0
    }
  }
}

EBImage::display(mag_ft_mriMat_2, method = "raster", title="(IFT o HighPassFilter o FT) Magnitude=Img | Phase=Img")

# plot.new(); rasterImage(normalize(mag_ft_mriMat_2),0,0,1,1)

Real = mag_ft_mriMat_2 * cos(phase_ft_mriMat)
Imaginary = mag_ft_mriMat_2 * sin(phase_ft_mriMat)
ift_ft_mriMat_2 = Re(fft(Real+1i*Imaginary, inverse = T)/length(mag_ft_mriMat_2))
# EBImage::display(t(ift_ft_mriMat_2), method = "raster", title="(IFT o HighPassFilter o FT) Magnitude=Img | Phase=Img")
plot.new(); rasterImage(normalize(ift_ft_mriMat_2),0,0,1,1)

14 Art, Image Processing and Spectral Representation of Signals

Let’s try to interrogate some artworks from UMMA’s 2020 Curriculum-Collection. This museum collection exhibits a variety of University of Michigan courses that use artwork to enhance the curricula of graduate and undergraduate courses. Each course examines a set of artworks and objects to address the nature of reality, imagination, and vision relative to the current environments and scientific advances. The art exhibit demonstrates diverse and creative examples of learning across the art, science, technology, and biosocial disciplines.

For instance, the painting by Khaled al-Saa’i, “Winter in Ann Arbor”, constructed with natural ink, tempera and gouache on paper, may reveal interesting patterns, shapes, topological structure, and calligraphy. We can interrogate the artwork using the following image-processing protocol:

• Load the image from UMMA CC archive
• Display the native image
• Examine the image intensity histogram (for all 3 RGB channels)
• Generate negative image
• Optionally crop and threshold the image
• Apply low-pass filtering (smoothing)
• Apply high-pass filtering (Laplace filter) for edge detection
• Denoise the image
• Apply watershed segmentation
• Segment and Voronoi tessellate the image

Further studies may include text recognition, identification of obfuscated structures, or graphical representation of 2D scenes. The readers are encouraged to expand this protocol and apply it to some of the other artworks in these UMMA collection 1 and collection 2.

# UMMA HS650 CC: https://umma.umich.edu/curriculum-collection/health-sciences-650
library(EBImage)

# Load the image from CC
# CC_WhitePoemFigure_Dill <- readImage("https://exchange.umma.umich.edu/media/W1siZiIsIjIwMjEvMDIvMTkvZGJ3dWk2cTVjX2RlZmF1bHQuanBnIl0sWyJwIiwidGh1bWIiLCIxMDAweDEwMDAiXV0?sha=bf87973a898145f8", type="jpeg")

CC_WhitePoemFigure_Dill <- readImage("https://exchange.umma.umich.edu/media/W1siZiIsIjIwMjEvMDkvMDMvNXZjaXdkOWdpal9kZWZhdWx0LmpwZyJdLFsicCIsInRodW1iIiwiMTAwMHgxMDAwIl1d?sha=9b8e04fa0a73cbb8", type="jpeg")

# Display raw image
display(CC_WhitePoemFigure_Dill, method = "raster", all = TRUE)

# Intensity Histogram
hist(CC_WhitePoemFigure_Dill)

# Generate negative image
img_neg <- max(CC_WhitePoemFigure_Dill) - CC_WhitePoemFigure_Dill
display( img_neg )

# dim: 633 1000    3

# crop and threshold
img_crop <- CC_WhitePoemFigure_Dill  # [200:578, 250:750, ]
img_thresh <- img_crop > 0.55
display(img_thresh)

# Low-pass Filter/smooth
w <- makeBrush(size = 31, shape = 'gaussian', sigma = 5)
img_flo <- filter2(CC_WhitePoemFigure_Dill, w)
display(img_flo)

# High-pass filtering Laplace Filter/Edge detection
fhi <- matrix(1, nrow = 3, ncol = 3)
fhi[2, 2] <- -7
img_fhi <- filter2(CC_WhitePoemFigure_Dill, fhi)
display(img_fhi)

# Denoising
img_median <- medianFilter(CC_WhitePoemFigure_Dill, size=5, cacheSize=20000)
display(img_median)

# Segmentation
grayimage <- channel(CC_WhitePoemFigure_Dill,"gray")
threshold <- otsu(grayimage)
CC_th <- combine( mapply(function(frame, th) frame > th, 
                         getFrames(CC_WhitePoemFigure_Dill), 
                         threshold, SIMPLIFY=FALSE) )
display(CC_th, all=TRUE)

disc <- makeBrush(51, "disc")
disc <- disc/sum(disc)
offset <- 0.0001
CC_bg <- filter2(CC_WhitePoemFigure_Dill, disc)
CC_th <- CC_WhitePoemFigure_Dill < CC_bg + offset
display(CC_th, all=TRUE)

# Watershed segmentation
nmask <- watershed(distmap(CC_th), tolerance=5, ext=20)
display(colorLabels(nmask), all=TRUE)

# Voronoi image tessellation
voronoiExamp <- propagate(x = nmask, seeds = nmask, lambda = 0.001)
voronoiPaint <- colorLabels (voronoiExamp)
display(voronoiPaint, all=TRUE)

CC <- rgbImage(green=2.0*CC_bg, blue=img_median, red=voronoiPaint[ , , , 1])
display(CC, all = TRUE)

ctmask = opening(img_fhi<0.6, makeBrush(5, shape='disc'))
cmask = propagate(img_fhi, seeds=nmask, mask=ctmask)
display(ctmask, all=TRUE)

ctmask = opening(img_fhi>0.1, makeBrush(5, shape='disc'))
cmask = propagate(img_fhi, seeds=nmask, mask=ctmask)
display(ctmask, all=TRUE)

# segmented <- paintObjects(channel(CC_th,"gray"), CC_WhitePoemFigure_Dill, col='#ff00ff')
# segmented <- paintObjects(channel(nmask,"gray"), segmented, col='#ffff00')
# display(segmented, all=TRUE)
segmented = paintObjects(channel(cmask,"gray"), CC_WhitePoemFigure_Dill, col='#ff00ff')
segmented = paintObjects(channel(ctmask,"gray"), segmented, col='#ffff00')
display(segmented, all=TRUE)

15 Reference

• This module is based on Bioconductor/EBImage documentation and Spacekime Analytics/TCIU materials.

SOCR Resource Visitor number