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Learn about spatial filters, frequency domain filtering, and edge detection techniques in image processing. This course is based on the work of Prof. Shree Nayar and Prof. Srinivasa Narasimhan from prestigious universities in the USA.
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VC 15/16 – TP7Spatial Filters Mestrado em Ciência de Computadores Mestrado Integrado em Engenharia de Redes e Sistemas Informáticos Miguel Tavares Coimbra
Outline • Spatial filters • Frequency domain filtering • Edge detection Acknowledgements: Most of this course is based on the excellent courses offered by Prof. Shree Nayar at Columbia University, USA and by Prof. Srinivasa Narasimhan at CMU, USA. Please acknowledge the original source when reusing these slides for academic purposes. VC 15/16 - TP7 - Spatial Filters
Topic: Spatial filters • Spatial filters • Frequency domain filtering • Edge detection VC 15/16 - TP7 - Spatial Filters
Images are Discrete and Finite Convolution Fourier Transform Inverse Fourier Transform VC 15/16 - TP7 - Spatial Filters
Simple way to process an image. Mask defines the processing function. Corresponds to a multiplication in frequency domain. Spatial Mask Mask Image Convolution – Mask ‘slides’ over the image VC 15/16 - TP7 - Spatial Filters
Each mask position has weight w. The result of the operation for each pixel is given by: Example Mask Image =1*2+2*2+1*2+… =8+0-20 =-12 VC 15/16 - TP7 - Spatial Filters
Definitions • Spatial filters • Use a mask (kernel) over an image region. • Work directly with pixels. • As opposed to: Frequency filters. • Advantages • Simple implementation: convolution with the kernel function. • Different masks offer a large variety of functionalities. VC 15/16 - TP7 - Spatial Filters
For n=2, convolve pixel values with 2D images: 1 1 1 2 1 then (a) use or (b) use 1 1 1 2 2 2 1 1 1 2 2 4 2 2 1 2 1 1 1 Averaging Let’s think about averaging pixel values Which is faster? VC 15/16 - TP7 - Spatial Filters
large Repeated averaging Gaussian smoothing Averaging The convolution kernel VC 15/16 - TP7 - Spatial Filters
pixels Filter size …can be very large 2D Gaussian is separable! Gaussian Smoothing Gaussiankernel (truncate, if necessary) Use two 1DGaussianFilters! VC 15/16 - TP7 - Spatial Filters
Gaussian Smoothing • A Gaussian kernel gives less weight to pixels further from the center of the window • This kernel is an approximation of a Gaussian function: VC 15/16 - TP7 - Spatial Filters
original VC 15/16 - TP7 - Spatial Filters
Mean Filtering • We are degrading the energy of the high spatial frequencies of an image (low-pass filtering). • Makes the image ‘smoother’. • Used in noise reduction. • Can be implemented with spatial masks or in the frequency domain. VC 15/16 - TP7 - Spatial Filters
Mean filter Gaussian filter VC 15/16 - TP7 - Spatial Filters
VC 15/16 - TP7 - Spatial Filters http://www.michaelbach.de/ot/cog_blureffects/index.html
VC 15/16 - TP7 - Spatial Filters http://www.michaelbach.de/ot/cog_blureffects/index.html
Median filtering sort median Median Filter (a) • Smoothing is averaging (a) Blurs edges (b) Sensitive to outliers (b) • Sort values around the pixel • Select middle value (median) • Non-linear (Cannot be implemented with convolution) VC 15/16 - TP7 - Spatial Filters
Gaussian noise Salt and pepper noise 3x3 5x5 7x7 VC 15/16 - TP7 - Spatial Filters
What a computer sees Border Problem How do we apply our mask to this pixel? VC 15/16 - TP7 - Spatial Filters
Border Problem • Ignore • Output image will be smaller than original • Pad with constant values • Can introduce substantial 1st order derivative values • Pad with reflection • Can introduce substantial 2nd order derivative values VC 15/16 - TP7 - Spatial Filters
Topic: Frequency domain filtering • Spatial filters • Frequency domain filtering • Edge detection VC 15/16 - TP7 - Spatial Filters
Image Processing in the Fourier Domain Magnitude of the FT Does not look anything like what we have seen VC 15/16 - TP7 - Spatial Filters
Convolution in the Frequency Domain |F(sx,sy)| f(x,y) h(x,y) |H(sx,sy)| g(x,y) |G(sx,sy)| VC 15/16 - TP7 - Spatial Filters
Low-pass Filtering Lets the low frequencies pass and eliminates the high frequencies. Generates image with overall shading, but not much detail VC 15/16 - TP7 - Spatial Filters
High-pass Filtering Lets through the high frequencies (the detail), but eliminates the low frequencies (the overall shape). It acts like an edge enhancer. VC 15/16 - TP7 - Spatial Filters
Boosting High Frequencies VC 15/16 - TP7 - Spatial Filters
The Ringing Effect An ideal low-pass filter causes ‘rings’ in the spatial domain! http://homepages.inf.ed.ac.uk/rbf/HIPR2/freqfilt.htm VC 15/16 - TP7 - Spatial Filters
Topic: Edge detection • Spatial filters • Frequency domain filtering • Edge detection VC 15/16 - TP7 - Spatial Filters
Convert a 2D image into a set of curves Extracts salient features of the scene More compact than pixels Edge Detection VC 15/16 - TP7 - Spatial Filters
Edges are caused by a variety of factors Origin of Edges surface normal discontinuity depth discontinuity surface color discontinuity illumination discontinuity VC 15/16 - TP7 - Spatial Filters
How can you tell that a pixel is on an edge? VC 15/16 - TP7 - Spatial Filters
Edge Types Step Edges Line Edges Roof Edge VC 15/16 - TP7 - Spatial Filters
Edge Magnitude Edge Orientation High Detection Rate and Good Localization Real Edges Noisy and Discrete! We want an Edge Operator that produces: VC 15/16 - TP7 - Spatial Filters
Gradient • Gradient equation: • Represents direction of most rapid change in intensity • Gradient direction: • The edge strength is givenby the gradient magnitude VC 15/16 - TP7 - Spatial Filters
Ideal edge Unit step function: Image intensity (brightness): Theory of Edge Detection VC 15/16 - TP7 - Spatial Filters
Partial derivatives (gradients): • Squared gradient: Edge Magnitude: (normal of the edge) Edge Orientation: Rotationally symmetric, non-linear operator Theory of Edge Detection VC 15/16 - TP7 - Spatial Filters
Laplacian: Rotationally symmetric, linear operator zero-crossing Theory of Edge Detection VC 15/16 - TP7 - Spatial Filters
Discrete Edge Operators • How can we differentiate a discrete image? Finite difference approximations: Convolution masks : VC 15/16 - TP7 - Spatial Filters
Second order partial derivatives: • Laplacian : Convolution masks : or Discrete Edge Operators (more accurate) VC 15/16 - TP7 - Spatial Filters
The Sobel Operators • Better approximations of the gradients exist • The Sobel operators below are commonly used VC 15/16 - TP7 - Spatial Filters
Comparing Edge Operators Good Localization Noise Sensitive Poor Detection Gradient: Roberts (2 x 2): Sobel (3 x 3): Sobel (5 x 5): Poor Localization Less Noise Sensitive Good Detection VC 15/16 - TP7 - Spatial Filters
Effects of Noise • Consider a single row or column of the image • Plotting intensity as a function of position gives a signal Where is the edge?? VC 15/16 - TP7 - Spatial Filters
Solution: Smooth First Look for peaks in Where is the edge? VC 15/16 - TP7 - Spatial Filters
Derivative Theorem of Convolution …saves us one operation. VC 15/16 - TP7 - Spatial Filters
Laplacian of Gaussian (LoG) Laplacian of Gaussian Laplacian of Gaussian operator Where is the edge? Zero-crossings of bottom graph ! VC 15/16 - TP7 - Spatial Filters
2D Gaussian Edge Operators Derivative of Gaussian (DoG) Laplacian of Gaussian Gaussian Mexican Hat (Sombrero) • is the Laplacian operator: VC 15/16 - TP7 - Spatial Filters
Canny Edge Operator • Smooth image I with 2D Gaussian: • Find local edge normal directions for each pixel • Compute edge magnitudes • Locate edges by finding zero-crossings along the edge normal directions (non-maximum suppression) VC 15/16 - TP7 - Spatial Filters
Non-maximum Suppression • Check if pixel is local maximum along gradient direction • requires checking interpolated pixels p and r VC 15/16 - TP7 - Spatial Filters