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Image Enhancement in the Frequency Domain
Frequency Domain Filtering
Basic steps for filtering in the frequency domain
Basics of filtering in the frequency domain1. multiply the input image by (-1)x+y to center the
transform to u = M/2 and v = N/2 (if M and N are even numbers, then the shifted coordinates will be integers)
2. compute F(u,v), the DFT of the image from (1)3. multiply F(u,v) by a filter function H(u,v)4. compute the inverse DFT of the result in (3)5. obtain the real part of the result in (4)6. multiply the result in (5) by (-1)x+y to cancel the
multiplication of the input image.
Images Black and white image is a 2D matrix. Intensities represented as pixels. Color images are 3D matrix, RBG.
Linear Filtering About modifying pixels based on neighborhood.
Local methods simplest. Linear means linear combination of neighbors.
Linear methods simplest. Useful to:
• Integrate information over constant regions.• Scale.• Detect changes.
Fourier analysis. Many nice slides taken from image database.
Filtering to reduce noise Noise is what we’re not interested in.
• We’ll discuss simple, low-level noise: Light fluctuations; Sensor noise; Quantization effects; Finite precision
• Not complex: shadows; extraneous objects. A pixel’s neighborhood contains
information about its intensity. Averaging noise reduces its effect.
Additive noise I = S + N. Noise doesn’t depend on
signal. We’ll consider:
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Average Filter• Mask with positive
entries, that sum 1.• Replaces each pixel
with an average of its neighborhood.
• If all weights are equal, it is called a BOX filter.
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Does it reduce noise?
• Intuitively, takes out small variations.
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Example: Smoothing by Averaging
Smoothing as Inference About the Signal
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Nearby points tell more about the signal than distant ones.
Neighborhood for averaging.
Gaussian Averaging Rotationally
symmetric. Weights nearby
pixels more than distant ones.• This makes sense
as probabalistic inference. A Gaussian gives a
good model of a fuzzy blob
exp x2 y2
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An Isotropic Gaussian The picture shows a
smoothing kernel proportional to
(which is a reasonable model of a circularly symmetric fuzzy blob)
Smoothing with a Gaussian
The effects of smoothing Each row shows smoothingwith gaussians of differentwidth; each column showsdifferent realizations of an image of gaussian noise.
Efficient Implementation
Both, the BOX filter and the Gaussian filter are separable:• First convolve each row with a 1D filter• Then convolve each column with a 1D filter.
Smoothing as Inference About the Signal: Non-linear Filters.
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What’s the best neighborhood for inference?
Filtering to reduce noise: Lessons Noise reduction is probabilistic inference. Depends on knowledge of signal and
noise. In practice, simplicity and efficiency
important.
Filtering and Signal Smoothing also smooths signal. Removes detail This is good and bad: - Bad: can’t remove noise w/out blurring
shape. - Good: captures large scale structure
Notch filter
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• this filter is to force the F(0,0) which is the average value of an image (dc component of the spectrum)• the output has prominent edges• in reality the average of the displayed image can’t be zero as it needs to have negative gray levels. the output image needs to scale the gray level
Low pass filter
high pass filter
Add the ½ of filter height to F(0,0) of the high pass filter
Correspondence between filter in spatial and frequency domains
Convolution
Convolution kernel g, represented as matrix.• it’s associative
Result is:
Smoothing Frequency-domain filters: Ideal Lowpass filter
image power circles
Result of ILPF
Example
Butterworth Lowpass Filter: BLPF
Example
Spatial representation of BLPFs
Gaussian Lowpass Filter: GLPF
Example
Example
Example
Example
Sharpening Frequency Domain Filter: Ideal highpass filter
Butterworth highpass filter
Gaussian highpass filter
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Spatial representation of Ideal, Butterworth and Gaussian highpass filters
Example: result of IHPF
Example: result of BHPF
Example: result of GHPF
Laplacian in the Frequency domain
Example: Laplacian filtered image
Example: high-boost filter
Examples
2-D Fourier Transform Properties