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**Spatial Filtering (Chapter 3)**

CS474/674 - Prof. Bebis

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**Spatial Filtering Methods (or Mask Processing Methods)**

output image

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Spatial Filtering The word “filtering” has been borrowed from the frequency domain. Filters are classified as: Low-pass (i.e., preserve low frequencies) High-pass (i.e., preserve high frequencies) Band-pass (i.e., preserve frequencies within a band) Band-reject (i.e., reject frequencies within a band)

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**Spatial Filtering (cont’d)**

Spatial filtering is defined by: (1) A neighborhood (2) An operation that is performed on the pixels inside the neighborhood output image

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**Spatial Filtering - Neighborhood**

Typically, the neighborhood is rectangular and its size is much smaller than that of f(x,y) - e.g., 3x3 or 5x5

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**Spatial filtering - Operation**

Assume the origin of the mask is the center of the mask. for a 3 x 3 mask: for a K x K mask:

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**Spatial filtering - Operation**

A filtered image is generated as the center of the mask moves to every pixel in the input image. output image

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**Handling Pixels Close to Boundaries**

pad with zeroes 0 0 0 ……………………….0 or 0 0 0 ……………………….0

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**Linear vs Non-Linear Spatial Filtering Methods**

A filtering method is linear when the output is a weighted sum of the input pixels. Methods that do not satisfy the above property are called non-linear. e.g.,

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**Linear Spatial Filtering Methods**

Two main linear spatial filtering methods: Correlation Convolution

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Correlation Output Image w(i,j) f(i,j) g(i,j)

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Correlation (cont’d) Often used in applications where we need to measure the similarity between images or parts of images (e.g., pattern matching).

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Convolution Similar to correlation except that the mask is first flipped both horizontally and vertically. Note: if w(x,y) is symmetric, that is w(x,y)=w(-x,-y), then convolution is equivalent to correlation!

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Example Correlation: Convolution:

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**How do we choose the elements of a mask?**

Typically, by sampling certain functions. Gaussian 1st derivative of Gaussian 2nd derivative

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**Filters Smoothing (i.e., low-pass filters)**

Reduce noise and eliminate small details. The elements of the mask must be positive. Sum of mask elements is 1 (after normalization) Gaussian

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**Filters (cont’d) Sharpening (i.e., high-pass filters)**

Highlight fine detail or enhance detail that has been blurred. The elements of the mask contain both positive and negative weights. Sum of the mask weights is 0 (after normalization) 1st derivative of Gaussian 2nd derivative

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**Smoothing Filters: Averaging (Low-pass filtering)**

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**Smoothing Filters: Averaging (cont’d)**

Mask size determines the degree of smoothing and loss of detail. original 3x3 5x5 7x7 15x15 25x25

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**Smoothing Filters: Averaging (cont’d)**

Example: extract, largest, brightest objects 15 x 15 averaging image thresholding

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**Smoothing filters: Gaussian**

The weights are samples of the Gaussian function σ = 1.4 mask size is a function of σ :

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**Smoothing filters: Gaussian (cont’d)**

σ controls the amount of smoothing As σ increases, more samples must be obtained to represent the Gaussian function accurately. σ = 3

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**Smoothing filters: Gaussian (cont’d)**

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**Averaging vs Gaussian Smoothing**

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**Smoothing Filters: Median Filtering (non-linear)**

Very effective for removing “salt and pepper” noise (i.e., random occurrences of black and white pixels). median filtering averaging

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**Smoothing Filters: Median Filtering (cont’d)**

Replace each pixel by the median in a neighborhood around the pixel.

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**Sharpening Filters (High Pass filtering)**

Useful for emphasizing transitions in image intensity (e.g., edges).

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**Sharpening Filters (cont’d)**

Note that the response of high-pass filtering might be negative. Values must be re-mapped to [0, 255] sharpened images original image

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**Sharpening Filters: Unsharp Masking**

Obtain a sharp image by subtracting a lowpass filtered (i.e., smoothed) image from the original image: - =

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**Sharpening Filters: High Boost**

Image sharpening emphasizes edges but details (i.e., low frequency components) might be lost. High boost filter: amplify input image, then subtract a lowpass image. (A-1) + =

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**Sharpening Filters: High Boost (cont’d)**

If A=1, we get a high pass filter If A>1, part of the original image is added back to the high pass filtered image.

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**Sharpening Filters: High Boost (cont’d)**

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**Sharpening Filters: Derivatives**

Taking the derivative of an image results in sharpening the image. The derivative of an image can be computed using the gradient.

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**Sharpening Filters: Derivatives (cont’d)**

The gradient is a vector which has magnitude and direction: or (approximation)

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**Sharpening Filters: Derivatives (cont’d)**

Magnitude: provides information about edge strength. Direction: perpendicular to the direction of the edge.

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**Sharpening Filters: Gradient Computation**

Approximate gradient using finite differences: sensitive to vertical edges Δx sensitive to horizontal edges

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**Sharpening Filters: Gradient Computation (cont’d)**

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Example

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**Sharpening Filters: Gradient Computation (cont’d)**

We can implement and using masks: (x+1/2,y) good approximation at (x+1/2,y) (x,y+1/2) * * good approximation at (x,y+1/2) Example: approximate gradient at z5

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**Sharpening Filters: Gradient Computation (cont’d)**

A different approximation of the gradient: good approximation (x+1/2,y+1/2) * We can implement and using the following masks:

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**Sharpening Filters: Gradient Computation (cont’d)**

Example: approximate gradient at z5 Other approximations Sobel

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Example

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**Sharpening Filters: Laplacian**

The Laplacian (2nd derivative) is defined as: (dot product) Approximate derivatives:

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**Sharpening Filters: Laplacian (cont’d)**

Laplacian Mask detect zero-crossings

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**Sharpening Filters: Laplacian (cont’d)**

Sobel Laplacian

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