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- Published: 05 April 2006
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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 :

- Constant intensity areas should be dark, i.e., low detail areas are visibly suppressed and have low pixel intensities.

- 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.