# Image Processing with Matlab

## Pascal Getreuer

This tutorial discusses how to use Matlab for image processing. Some familiarity with Matlab is assumed (you should know how to use matrices and write an M-file).

It is helpful to have the Matlab Image Processing Toolbox, but fortunately, no toolboxes are needed for most operations. Commands requiring the Image Toolbox are indicated with ** [Image Toolbox]**.

## Image representation

There are five types of images in Matlab.

**Grayscale.**A grayscale image \(M\) pixels tall and \(N\) pixels wide is represented as a matrix of double datatype of size \(M\times N\). Element values (e.g.,`MyImage(m, n)`

) denote the pixel grayscale intensities in [0, 1] with 0=black and 1=white.**Truecolor RGB.**A truecolor red-green-blue (RGB) image is represented as a three-dimensional \(M\times N \times 3\) double matrix. Each pixel has red, green, blue components along the third dimension with values in [0, 1], for example, the color components of pixel (m, n) are`MyImage(m, n, 1)`

= red,`MyImage(m, n, 2)`

= green,`MyImage(m, n, 3)`

= blue.**Indexed.**Indexed (palette based) images are represented with an index matrix of size \(M\times N\) and a colormap matrix of size \(K\times 3\). The colormap holds all colors used in the image and the index matrix represents the pixels by referring to colors in the colormap. For example, if the 22nd color is magenta`MyColormap(22, :)`

=`[1, 0, 1]`

, then`MyImage(m, n) = 22`

is a magenta-colored pixel.**Binary.**A binary image is represented by an \(M\times N\) logical matrix where pixel values are 1 (true) or 0 (false).**uint8.**This type uses less memory and some operations compute faster than with double types. For simplicity, this tutorial does not discuss uint8 further.

When possible, it is convenient to work with grayscale format for image processing. In cases requiring color, an RGB color image can be decomposed and handled as three separate grayscale images. Indexed images must be converted to grayscale or RGB for most operations.

Below are some common manipulations and conversions. A few commands require the Image Toolbox and are indicated with ** [Image Toolbox]**.

```
% Display a grayscale or binary image
image(MyGray * 255);
axis image
colormap(gray(256));
% Display an RGB image (error if any element outside of [0, 1])
image(MyRGB);
axis image
% Display an RGB image (clips elements to [0, 1])
image(min(max(MyRGB, 0), 1));
axis image
% Display an indexed image
image(MyIndexed);
axis image
colormap(MyColormap);
% Separate the channels of an RGB image
MyRed = MyRGB(:, :, 1);
MyGreen = MyRGB(:, :, 2);
MyBlue = MyRGB(:, :, 3);
% Put the channels back together
MyRGB = cat(3, MyRed, MyGreen, MyBlue);
% Convert grayscale to RGB
MyRGB = cat(3, MyGray, MyGray, MyGray);
% Convert RGB to grayscale using simple average
MyGray = mean(MyRGB, 3);
% Convert RGB to grayscale using NTSC weighting [Image Toolbox]
MyGray = rgb2gray(MyRGB);
% Convert RGB to grayscale using NTSC weighting
MyGray = 0.299*MyRGB(:, :, 1) + 0.587*MyRGB(:, :, 2) + 0.114*MyRGB(:, :, 3);
% Convert indexed image to RGB [Image Toolbox]
MyRGB = ind2rgb(MyIndexed, MyColormap);
% Convert indexed image to RGB
MyRGB = reshape(cat(3, MyColormap(MyIndexed, 1), MyColormap(MyIndexed, 2),...
MyColormap(MyIndexed, 3)), size(MyIndexed, 1), size(MyIndexed, 2), 3);
% Convert an RGB image to indexed using K colors [Image Toolbox]
MyIndexed, MyColormap] = rgb2ind(MyRGB, K);
[
% Convert binary to grayscale
MyGray = double(MyBinary);
% Convert grayscale to binary
MyBinary = (MyGray > 0.5);
```

## Reading and writing image files

Matlab can read and write images with the `imread`

and `imwrite`

commands. Although a fair number of file formats are supported, some are not. Use `imformats`

to see what your installation supports:

```
>> imformats
EXT ISA READ WRITE ALPHA DESCRIPTION
--------------------------------------------------------------------------
bmp isbmp readbmp writebmp 0 Windows Bitmap (BMP)
gif isgif readgif writegif 0 Graphics Interchange Format (GIF)
pbm ispbm readpnm writepnm 0 Portable Bitmap (PBM)
pcx ispcx readpcx writepcx 0 Windows Paintbrush (PCX)
pgm ispgm readpnm writepnm 0 Portable Graymap (PGM)
png ispng readpng writepng 1 Portable Network Graphics (PNG)
pnm ispnm readpnm writepnm 0 Portable Any Map (PNM)
ppm isppm readpnm writepnm 0 Portable Pixmap (PPM)
...
```

When reading images, an inconvenience is that `imread`

returns the image data in uint8 datatype, which must be converted to double and rescaled before use. So instead of calling `imread`

directly, I use the following M-file function to read and convert images:

`getimage.m`

```
function Img = getimage(Filename)
%GETIMAGE Read an image given a filename
% V = GETIMAGE(FILENAME) where FILENAME is an image file. The image is
% returned either as an MxN double matrix for a grayscale image or as an
% MxNx3 double matrix for a color image, with elements in [0,1].
% Pascal Getreuer 2008-2009
% Read the file
Img, Map, Alpha] = imread(Filename);
[Img = double(Img);
if ~isempty(Map) % Convert indexed image to RGB.
Img = Img + 1;
Img = reshape(cat(3, Map(Img, 1), Map(Img, 2), Map(Img, 3)), ...
size(Img, 1), size(Img, 2), 3);
else
Img = Img / 255; % Rescale to [0, 1].
end
```

Save this code as getimage.m to use this M-function. If image baboon.png is in the current directory (or somewhere in the Matlab search path), you can read it with `MyImage = getimage('baboon.png')`

. You can also use partial paths, for example if the image is in `<current directory>/images/`

with `getimage('images/baboon.png')`

.

To write a grayscale or RGB image, use

`imwrite(MyImage, 'myimage.png');`

Take care that `MyImage`

is a double matrix with elements in [0, 1]—if improperly scaled, the saved file will probably be blank.

When writing image files, I highly recommend using the PNG file format. This format is a reliable choice since it is lossless, supports truecolor RGB, and compresses pretty well. Use other formats with caution.

## Basic operations

Below are some basic operations on a grayscale image `u`

. Commands requiring the Image Toolbox are indicated with ** [Image Toolbox]**.

```
% Statistics
uMax = max(u(:)); % Compute the maximum value
uMin = min(u(:)); % Minimum
uPower = sum(u(:).^2); % Power
uAvg = mean(u(:)); % Average
uVar = var(u(:)); % Variance
uMed = median(u(:)); % Median
hist(u(:), linspace(0, 1, 256)); % Plot histogram
% Basic manipulations
uClip = min(max(u, 0), 1); % Clip elements to [0,1]
uPad = u([1, 1:end, end], [1, 1:end, end]); % Pad with one-pixel margin
uPad = padarray(u, [k, k], 'replicate'); % Pad of size k [Image Toolbox]
uCrop = u(RowStart:RowEnd, ColStart:ColEnd); % Crop image
uFlip = flipud(u); % Flip up/down
uFlip = fliplr(u); % Flip left/right
uResize = imresize(u, ScaleFactor); % Resize [Image Toolbox]
uRot = rot90(u, k); % Rotate by k*90 degrees
uRot = imrotate(u, Angle); % Rotate [Image Toolbox]
uc = (u - min(u(:))/(max(u(:)) - min(u(:))); % Stretch contrast to [0, 1]
uq = round(u * (K - 1))/(K - 1); % Quantize to K graylevels
% Add white Gaussian noise of standard deviation sigma.
uNoisy = u + randn(size(u)) * sigma;
% Simluate salt and pepper noise.
uNoisy = u; uNoisy(rand(size(u)) < p) = round(rand(size(u)));
% Debugging
any(~isfinite(u(:))) % Are any elements are infinite or NaN?
nnz(u > 0.5) % Count elements satisfying some condition.
```

*(Note: For any array, the syntax u(:) means “unroll u into a column vector.” For example, if u = [1,5;0,2], then u(:) is [1;0;5;2].)*

For example, image signal power is used in computing signal-to-noise ratio (SNR) and peak signal-to-noise ratio (PSNR). Given clean image `uclean`

and noise-contaminated image `u`

,

```
% Compute SNR
snr = -10 * log10( sum((uclean(:) - u(:)).^2) / sum(uclean(:).^2) );
% Compute PSNR (where the maximum possible value of uclean is 1)
psnr = -10 * log10( mean((uclean(:) - u(:)).^2) );
```

**Be careful with norm:** the behavior is

`norm(v)`

on vector `v`

computes `sqrt(sum(v.^2))`

, but `norm(A)`

on matrix `A`

computes the induced \(L^2\) matrix norm,`norm(A) = sqrt(max(eig(A' * A))) % gaah!`

So `norm(A)`

is certainly not `sqrt(sum(A(:).^2))`

. It is nevertheless an easy mistake to use `norm(A)`

where it should have been `norm(A(:))`

.

## Linear filters

Linear filtering is the cornerstone technique of signal processing. To briefly introduce, a linear filter is an operation where at every pixel \(x_{m,n}\) of an image, a linear function is evaluated on the pixel and its neighbors to compute a new pixel value \(y_{m,n}\).

A linear filter in two dimensions has the general form \[ y_{m,n} = \sum_j \sum_k h_{j,k} x_{m-j,n-k} \] where \(x\) is the input, \(y\) is the output, and \(h\) is the filter impulse response. Different choices of \(h\) lead to filters that smooth, sharpen, and detect edges, to name a few applications. The right-hand side of the above equation is denoted concisely as \(h*x\) and is called the “convolution of \(h\) and \(x\).”

### Spatial-domain filtering

Two-dimensional linear filtering is implemented in Matlab with `conv2`

. Unfortunately, `conv2`

can only handle filtering near the image boundaries by zero-padding, which for images is usually inappropriate. To work around this, we can pad the input image and then use the `'valid'`

option when calling `conv2`

. The following M-function does this.

`conv2padded.m`

```
function x = conv2padded(varargin)
%CONV2PADDED Two-dimensional convolution with padding.
% Y = CONV2PADDED(X,H) applies 2D filter H to X with constant
% extension padding.
%
% Y = CONV2PADDED(H1,H2,X) first applies 1D filter H1 along the rows
% and then applies 1D filter H2 along the columns.
%
% If X is a 3D array, filtering is done separately on each channel.
% Pascal Getreuer 2009
if nargin == 2 % Function was called as "conv2padded(x,h)"
x = varargin{1};
h = varargin{2};
top = ceil(size(h, 1)/2) - 1;
bottom = floor(size(h, 1)/2);
left = ceil(size(h, 2)/2) - 1;
right = floor(size(h, 2)/2);
elseif nargin == 3 % Function was called as "conv2padded(h1,h2,x)"
h1 = varargin{1};
h2 = varargin{2};
x = varargin{3};
top = ceil(length(h1)/2) - 1;
bottom = floor(length(h1)/2);
left = ceil(length(h2)/2) - 1;
right = floor(length(h2)/2);
else
error('Wrong number of arguments.');
end
% Pad the input image
xPadded = x([ones(1, top), 1:size(x, 1), size(x, 1) + zeros(1, bottom)],...
ones(1, left), 1:size(x, 2), size(x, 2) + zeros(1, right)], :);
[
% Since conv2 cannot handle 3D inputs, do filtering channel by channel
for p = 1:size(x, 3)
if nargin == 2
x(:, :, p) = conv2(xPadded(:, :, p), h, 'valid'); % Call conv2
else
x(:, :, p) = conv2(h1, h2, xPadded(:, :, p), 'valid'); % Call conv2
end
end
```

Save this code as `conv2padded.m`

to use this M-function. Here are some examples:

```
% A light smoothing filter
h = [0, 1, 0;
1, 4, 1;
0, 1, 0];
h = h / sum(h(:)); % Normalize the filter
uSmooth = conv2padded(u, h);
% A sharpening filter
h = [0, -1, 0;
-1, 8, -1;
0, -1, 0];
h = h / sum(h(:)); % Normalize the filter
uSharp = conv2padded(u, h);
% Sobel edge detection
hx = [1, 0, -1;
2, 0, -2;
1, 0, -1];
hy = rot90(hx, -1);
u_x = conv2padded(u, hx);
u_y = conv2padded(u, hy);
EdgeStrength = sqrt(u_x.^2 + u_y.^2);
% Moving average
WindowSize = 5;
h1 = ones(WindowSize, 1) / WindowSize;
uSmooth = conv2padded(h1, h1, u);
% Gaussian filtering
sigma = 3.5;
FilterRadius = ceil(4 * sigma); % Truncate the Gaussian at 4*sigma
h1 = exp(-(-FilterRadius:FilterRadius).^2 / (2 * sigma^2));
h1 = h1 / sum(h1); % Normalize the filter
uSmooth = conv2padded(h1, h1, u);
```

A 2D filter h is said to be *separable* if it can be expressed as the outer product of two 1D filters h1 and h2, that is, `h = h1(:) * h2(:)'`

. It is faster to pass h1 and h2 than h, as is done above for the moving average window and the Gaussian filter. In fact, the Sobel filters `hx`

and `hy`

in the above code are also separable—what are h1 and h2?

### Fourier-domain filtering

For large filters, spatial-domain filtering with `conv2`

is easily a computationally expensive operation. For a K×K filter on an M×N image, `conv2`

costs \(O(MNK^2)\) additions and multiplications, or \(O(N^4)\) supposing M, N, K are similar magnitudes.

For large filters, filtering in the Fourier domain is faster since the computational cost is reduced to \(O(N^2 \log N)\). Using the convolution-multiplication property of the Fourier transform, the convolution is equivalently computed by

```
% Compute y = h*x with periodic boundary extension
k1, k2] = size(h);
[hpad = zeros(size(x));
hpad([end + 1 - floor(k1/2):end, 1:ceil(k1/2)], ...
end + 1 - floor(k2/2):end, 1:ceil(k2/2)]) = h;
[y = real(ifft2(fft2(hpad) .* fft2(x)));
```

The result is equivalent to `conv2padded(x,h)`

except near the boundary, where the above computation uses periodic boundary extension.

Fourier-based filtering can also be done with symmetric boundary extension by reflecting the input in each direction:

```
% Compute y = h*x with symmetric boundary extension
xSym = [x, fliplr(x)]; % Symmetrize horizontally
xSym = [xSym; flipud(xSym)]; % Symmetrize vertically
k1,k2] = size(h);
[hpad = zeros(size(xSym));
hpad([end + 1 - floor(k1/2):end, 1:ceil(k1/2)], ...
end + 1 - floor(k2/2):end, 1:ceil(k2/2)]) = h;
[y = real(ifft2(fft2(hpad) .* fft2(xSym)));
y = y(1:size(y, 1)/2, 1:size(y, 2)/2);
```

*(Note: An even more efficient method is FFT overlap-add or overlap-save filtering. The Signal Processing Toolbox implements FFT overlap-add in one-dimension in fftfilt.)*

## Nonlinear filters

A nonlinear filter is an operation where each filtered pixel \(y_{m,n}\) is a nonlinear function of \(x_{m,n}\) and its neighbors. Here we briefly discuss a few types of nonlinear filters.

### Order statistic filters

If you have the Image Toolbox, order statistic filters can be performed with `ordfilt2`

and `medfilt2`

. An order statistic filter sorts the pixel values over a neighborhood and selects the kth largest value. The min, max, and median filters are special cases.

### Morphological filters

If you have the Image Toolbox, `bwmorph`

implements various morphological operations on binary images, like erosion, dilation, open, close, and skeleton. There are also commands available for morphology on grayscale images: `imerode`

, `imdilate`

, and `imtophat`

, among others.

### Build your own filter

Occasionally we want to use a new filter that Matlab does not have. The code below is a simple generic template for implementing filters.

```
M, N] = size(x);
[y = zeros(size(x));
r = 1; % Adjust for desired window size
for n = 1+r:N-r
for m = 1+r:M-r
% Extract a window of size (2r+1)x(2r+1) around (m,n)
w = x(m + (-r:r),n + (-r:r));
% ... write the filter here ...
y(m, n) = result;
end
end
```

*(Note: A frequent misguided claim is that loops in Matlab are slow and should be avoided. This was once true, back in Matlab 5 and earlier, but loops in modern versions are reasonably fast.)*

For example, the alpha-trimmed mean filter ignores the d/2 lowest and d/2 highest values in the window, and averages the remaining values. The filter is a balance between a median filter and a mean filter. The alpha-trimmed mean filter can be implemented in the template as

```
% The alpha-trimmed mean filter
w = sort(w(:));
y(m, n) = mean(w(1+d/2:end-d/2)); % Compute the result y(m,n)
```

As another example, the bilateral filter is \[ y_{m,n} = \frac{\sum_{j,k} h_{j,k,m,n} x_{m-j,n-k}}{\sum_{j,k} h_{j,k,m,n}} \] where \(h_{j,k,m,n}\) is \[ \mathrm{e}^{-(j^2 + k^2) / (2\sigma_s^2)} \mathrm{e}^{-(x_{m-j,n-k} - x_{m,n})^2 / (2\sigma_d^2)}\]

The bilateral filter can be implemented as

```
% The bilateral filter
k, j] = meshgrid(-r:r, -r:r);
[h = exp( -(j.^2 + k.^2)/(2*sigma_s^2) ) .* ...
exp( -(w - w(r+1,r+1)).^2/(2*sigma_d^2) );
y(m, n) = h(:)'*w(:) / sum(h(:));
```

If you don’t have the Image Toolbox, the template can be used to write substitutes for missing filters, though they will not be as fast as the Image Toolbox implementations.

```
% medfilt2
y(m, n) = median(w(:));
% ordfilt2
w = sort(w(:));
y(m, n) = w(k); % Select the kth largest element
% imdilate
% Define a structure element as a (2r+1)x(2r+1) array
SE = [0,1,0;1,1,1;0,1,0];
y(m, n) = max(w(SE));
```

## Special topics

Up to this point, we have covered those operations that are generally useful in imaging processing. You must look beyond this tutorial for the details on the topic of your interest. A large amount of Matlab code and material is freely available online for filter design, wavelet and multiresolution techniques, PDE-based imaging, morphology, and wherever else researchers have made their code publicly available.