Basic ideas of Image Transforms are derived from those showed earlier

Image Transforms Fast Fourier –2-D Discrete Fourier Transform Fast Cosine –2-D Discrete Cosine Transform Radon Transform Slant Walsh, Hadamard, Paley, Karczmarz Haar Chrestenson Reed-Muller

Methods for Digital Image Processing

Spatial Frequency or Fourier Transform Jean Baptiste Joseph Fourier Fourier face in Fourier Transform Domain

Examples of Fourier 2D Image Transform

Fourier 2D Image Transform

Another formula for Two-Dimensional Fourier A cos(x  2  i/N) B cos(y  2  j/M) f x = u = i/N, f y = v =j/M Image is function of x and y Now we need two cosinusoids for each point, one for x and one for y Lines in the figure correspond to real value 1 Now we have waves in two directions and they have frequencies and amplitudes

Fourier Transform of a spot Original imageFourier Transform

Transform Results image spectrum transform

Two Dimensional Fast Fourier in Matlab

Filtering in Frequency Domain … will be covered in a separate lecture on spectral approaches…..

H(u,v) for various values of u and v These are standard trivial functions to compose the image from

<<image..and its spectrum

Image and its spectrum

Let g(u,v) be the kernel Let h(u,v) be the image G(k,l) = DFT[g(u,v)] H(k,l) = DFT[h(u,v)] Then where means multiplication and means convolution. This means that an image can be filtered in the Spatial Domain or the Frequency Domain. Convolution Theorem This is a very important result

Let g(u,v) be the kernel Let h(u,v) be the image G(k,l) = DFT[g(u,v)] H(k,l) = DFT[h(u,v)] Then where means multiplication and means convolution. Convolution Theorem Instead of doing convolution in spatial domain we can do multiplication In frequency domain Convolution in spatial domain Multiplication in spectral domain

v u Image Spectrum Noise and its spectrum Noise filtering

Image v u Spectrum

Image x(u,v) v u Spectrum log(X(k,l)) l k

k l v u Image x(u,v) Image of cow with noise

white noisewhite noise spectrum kernel spectrum (low pass filter) red noisered noise spectrum

Filtering is done in spectral domain. Can be very complicated

Discrete Cosine Transform (DCT) Used in JPEG and MPEGUsed in JPEG and MPEG Another Frequency Transform, with Different Set of Basis FunctionsAnother Frequency Transform, with Different Set of Basis Functions

Discrete Cosine Transform in Matlab absolute Two-dimensional Discrete Cosine Transform trucks Two dimensional spectrum of tracks. Nearly all information in left top corner

“Statistical” Filters Median Filter also eliminates noise preserves edges better than blurring Sorts values in a region and finds the median region size and shape how define the median for color values?

“Statistical” Filters Continued Minimum Filter (Thinning)Minimum Filter (Thinning) Maximum Filter (Growing)Maximum Filter (Growing) “Pixellate” Functions“Pixellate” Functions Now we can do this quickly in spectral domain

ThinningThinning GrowingGrowing thinninggrowing

Pixellate Examples Original image Noise added After pixellate

DCT used in compression and recognition Fringe Pattern DCT DCT Coefficients Zonal Mask (1,1) (1,2) (2,1) (2,2). Feature Vector Artificial Neural Network Can be used for face recognition, tell my story from Japan.

Noise Removal Image with Noise Transform been removed Transforms for Noise Removal Image reconstructed as the noise has been removed

Image Segmentation Recall: Edge Detection f(x,y) Gradient Mask f e (x,y) Now we do this in spectral domain!!

Image Moments 2-D continuous function f(x,y), the moment of order (p+q) is: Central moment of order (p+q) is: Moments were found by convoluti ons

Image Moments (contd.) Normalized central moment of order (p+q) is: A set of seven invariant moments can be derived from  pq Now we do this in spectral domain!! convolutions are now done in spectral domain

Image Textures The USC-SIPI Image Database Grass Sand Brick wall Now we do texture analysis like this in spectral domain!!

Problems There is a lot of Fourier and Cosine Transform software on the web, find one and apply it to remove some kind of noise from robot images from FAB building. Read about Walsh transform and think what kind of advantages it may have over Fourier Read about Haar and Reed-Muller transform and implement them. Experiment

Sources Howard Schultz, Umass Herculano De Biasi Shreekanth Mandayam ECE Department, Rowan University

Image Compression Please visit the website