Gaussian Filtering Continued

• The multidimensional Gaussian distribution.
• The specific instantiation of this distribution for 2D f(x,y) images.
• The specification of the radially symmetric Gaussian distribution.
• The derivation of the radially symmetric Gaussian distribution from the 2D (x,y) Gaussian distribution.

Blurring and the Point Spread Function

• The Point Spread Function (PSF) is a function which denotes the degradation of a point source of light due to imperfections in an optical imaging system.
• Examples of a point spread function are shown.
• The PSF characterizes the blurring effects due to the imperfections in the imaging system. It is often assumed to be a linear effect which is modeled as the convolution of some unknown blurring function h(x,y) with the image f(x,y). Hence, the output, g(x,y) = f(x,y) * h(x,y) where '*' denotes convolution.

Order Statistic Filters

• Median filter, how to compute the median. Why it is effective for "salt and pepper" noise perturbations of an image.
• Min Filter
• Max Filter
• All of these filters are non-linear. Example proof.

Sharpening Filters in the Spatial Domain

• What do sharpening filters do? - Highlight details in the image normally associated with significantly different neighboring pixel values, i.e., areas where the image derivative is large.
• Sharpening Transform goals :
1. Constant intensity areas should be dark, i.e., low detail areas are visibly suppressed and have low pixel intensities.
2. Details, i.e., ares with significantly different pixel values have high pixel intensities.
• What is the appropriate operator ? -- Ans. Differentiation .
• Definition of the differentiation filter as both a forward difference discrete operator, i.e., filter, and a backward difference operator.
• Relation of the discrete operator to the continous definition of differentiation.
• Generalization of the differentiation operator for 2D functions, i.e., partial differentiation in both x and y. This vector is the gradient vector.
• Discussion of the image gradient, edge detection, and the geometric interpretation of the image gradient.
• Examples of the image gradient for different orientations of a step function. The dual representation of the (x,y) gradient as a phasor or exponential, i.e., radius or gradient magnitude and angle or edge orientation rather than the Cartesian gradient.
• How to display edge images. Normalizations to use to enhance the visualization of edge detection.

The Laplacian Operator

• Derivation of the 1D laplacian and its relation to the curvature of a 1D function.
• Generalization of the Laplacian as a 2D operator.
• The representation of the Laplacian as the divergence of the image gradient.
• The fact that this formalism is N-dimensional generically and its formulation for 2D images.
• The geometric interpretation of the Laplacian.