# Image Processing with MATLAB

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.

1. Grayscale. A grayscale image M pixels tall and N pixels wide is represented as a matrix of double datatype of size M×N. Element values (e.g., `MyImage(m,n)`) denote the pixel grayscale intensities in [0,1] with 0=black and 1=white.
2. Truecolor RGB. A truecolor red-green-blue (RGB) image is represented as a three-dimensional M×N×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`.
3. Indexed. Indexed (palette-based) images are represented with an index matrix of size M×N and a colormap matrix of size K×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.
4. Binary. A binary image is represented by an M×N logical matrix where pixel values are 1 (true) or 0 (false).
5. uint8. This type uses less memory and some operations compute faster than with double types. For simplicity, this tutorial does not discuss uint8 further.

Grayscale is usually the preferred 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     INFO        READ      WRITE     ALPHA  DESCRIPTION                            ``
``------------------------------------------------------------------------------------------------``
``bmp       isbmp   imbmpinfo   readbmp   writebmp  0      Windows Bitmap (BMP)``
``gif       isgif   imgifinfo   readgif   writegif  0      Graphics Interchange Format (GIF)``
``jpg jpeg  isjpg   imjpginfo   readjpg   writejpg  0      Joint Photographic Experts Group (JPEG)``
``pbm       ispbm   impnminfo   readpnm   writepnm  0      Portable Bitmap (PBM)``
``pcx       ispcx   impcxinfo   readpcx   writepcx  0      Windows Paintbrush (PCX)``
``pgm       ispgm   impnminfo   readpnm   writepnm  0      Portable Graymap (PGM)``
``png       ispng   impnginfo   readpng   writepng  1      Portable Network Graphics (PNG)``
``pnm       ispnm   impnminfo   readpnm   writepnm  0      Portable Any Map (PNM)``
``ppm       isppm   impnminfo   readpnm   writepnm  0      Portable Pixmap (PPM)``
``...``

When reading images, an unfortunate problem 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:

``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``

Right-click and save 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 image with one-pixel margin``
``uPad = padarray(u,[k,k],'replicate');        % Pad image with k-pixel margin [Image Toolbox]``
``uCrop = u(RowStart:RowEnd,ColStart:ColEnd);  % Crop image``
``uFlip = flipud(u);                           % Flip in the up/down direction``
``uFlip = fliplr(u);                           % Flip left/right``
``uResize = imresize(u,ScaleFactor);           % Interpolate image [Image Toolbox]``
``uRot = rot90(u,k);                           % Rotate by k*90 degrees with integer k ``
``uRot = imrotate(u,Angle);                    % Rotate by Angle degrees [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 {0,1/K,2/K,...,1} ``

``% Simulating noise``
``uNoisy = u + randn(size(u))*sigma;           % Add white Gaussian noise of standard deviation sigma``
``uNoisy = u; uNoisy(rand(size(u)) < p) = round(rand(size(u)));  % Salt and pepper noise``

``% Debugging``
``any(~isfinite(u(:)))                         % Check if any elements are infinite or NaN``
``nnz(u > 0.5)                                 % Count how many elements satisfy 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 L2 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 xm,n of an image, a linear function is evaluated on the pixel and its neighbors to compute a new pixel value ym,n.

### 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 means that filtering results are usually inappropriate for pixels close to the boundary. To work around this, we can pad the input image and use the 'valid' option when calling `conv2`. The following M-function does this.

``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 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(MNK2) additions and multiplications, or O(N4) 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(N2logN). 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 ym,nym,n is a nonlinear function of xm,nxm,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.

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 but 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 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.