Image Enhancement Frequency Domain

Resmi N.G. Reference: Digital Signal Processing Rafael C. Gonzalez Richard E. Woods

Frequency Domain Methods Basics of filtering in frequency domain Basic Filters and Properties

Notch filter Lowpass Filter Highpass Filter

Smoothing Frequency Domain Filters

Ideal Lowpass Filters Butterworth Lowpass Filters Gaussian Lowpass Filters

Sharpening Frequency Domain Filters

Ideal Highpass Filters Butterworth Highpass Filters Gaussian Highpass Filters Enhancement using The Laplacian Unsharp Masking High Boost Filtering High-Frequency Emphasis Filtering

Homomorphic Filtering 3/20/2012

CS04 804B Image Processing - Module2

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Basics of Filtering in Frequency Domain 1. Multiply the input image by (-1)x+y to center the transform. 2. Compute the DFT, F(u,v) of the resulting image. 3. Multiply F(u,v) by a filter function H(u,v) to obtain G (u,v). 4.Compute the inverse DFT of G(u,v) to obtain g*(x,y). 5. Obtain the real part of g*(x,y). 6. Multiply the result by (-1)x+y to obtain g (x,y). 3/20/2012


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Basic Steps for Filtering in Frequency Domain

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Basic Filters and Properties Notch Filter It is a constant function with a hole at the origin. Sets F(0,0) to zero. Lowpass Filter It attenuates high frequencies and passes low frequencies. Highpass Filter It attenuates low frequencies and passes high frequencies. 3/20/2012


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Smoothing Frequency Domain Filters Low Pass Filter (Smoothing Filter) The result in the spatial domain is equivalent to that of a smoothing filter as the blocked high frequencies correspond to sharp intensity changes, i.e. to the finescale details and noise in the spatial domain image.

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High Pass Filter(Sharpening Filter) A highpass filter attenuates the low-frequency components without disturbing the high frequency information in the Fourier Transform. It yields edge enhancement or edge detection in the spatial domain, because edges contain many high frequencies. Areas of constant gray level consist mainly of low frequencies and are therefore suppressed. 3/20/2012


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Band Pass Filter

A bandpass filter attenuates very low and very high frequencies, but retains a middle range band of frequencies. Bandpass filtering can be used to enhance edges (suppressing low frequencies) while reducing the noise(attenuating high frequencies).

Bandpass filter is a combination of both lowpass and highpass filters. These filters attenuate all frequencies below a specific frequency and above a specific frequency, while retaining the frequencies between the two cut-offs.

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Ideal Low Pass Filters Transfer Function 1 if D (u, v) ≤ D0 H (u, v) =  0 if D (u, v) > D0 D0 is a specified non − negative quantity. D(u,v)is the distance from point (u,v)to the origin of the frequency rectangle.   N M + v− D(u, v) =  u − 2 2  

(

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2

) (


)

2

1

2

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Ideal – because all frequencies inside a circle of radius D0

are passed without any attenuation, whereas all frequencies outside the circle are completely attenuated. The point of transition between H(u,v) = 1 and H(u,v) = 0

is called the cut-off frequency.

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Ideal Low pass Filter

Produces “Ringing” effect. Cannot be realized in electronic components. Not very Practical

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Butterworth Low Pass Filters The transfer function of a BLPF of order n, and with cut-

off frequency at a distance D0 from the origin, is defined as 1 H (u, v) =

=

 D (u , v)  1+   D 0  

1   N M u v − + −   2 2   1+   2 D0    

(

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2n

2

) (


)

2

n

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Provides a smooth transition between low and high

frequencies. Butterworth filter of order 1 has neither ringing nor

negative values. BLPF of order 2 has mild ringing and small negative

values. Reduced ringing effect than ILPF.

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Gaussian Low Pass Filters − D 2 ( u ,v )

H (u , v) = e

2σ 2

D(u,v)is the distance from the origin of the Fourier Transform. σ is a measure of the spread of the Gaussian curve. When σ = D0 , − D 2 ( u ,v )

H (u , v) = e

2 D02

where D0 is the cut − off frequency. 3/20/2012


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When D(u , v) = 0, H (u , v) = 1 − D0 2

When D(u , v) = D0 , H (u , v) = e 3/20/2012


2 D0 2

=e

−1

2

= 0.607 22

Gaussian Low Pass Filters Very smooth filter function. Inverse DFT of the Gaussian lowpass filter is Gaussian. No “Ringing” effect.

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Applications of Low Pass Filters In the field of machine perception Character Recognition In printing and publishing industry. Cosmetic processing prior to printing For processing satellite and aerial images.

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Sharpening Frequency Domain Filters Ideal High Pass Filters Transfer Function of high pass filter is given by

H hp (u , v) = 1 − H lp (u , v) H lp (u , v) is thetransfer function of corresponding low pass filter. That is, when low pass filter attenuates frequencies, high

pass filter passes them and vice versa. 3/20/2012


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Opposite of ideal lowpass filter.

0 if D(u, v) ≤ D0 H (u , v) =  1 if D (u, v) > D0 Sets to zero all frequencies inside a circle of radius D0

while all frequencies outside the circle are passed without attenuation. Not physically realizable with electronic components. Produces ringing effect. 3/20/2012


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D0 = 15,30,80 3/20/2012


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Butterworth High Pass Filter H (u , v) =

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1  D0  1+   D u v ( , )  


2n

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D0 = 15,30,80

Represents a transition between the sharpness of IHPF and

the total smoothness of Gaussian filter. 3/20/2012


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Gaussian High Pass Filter − D 2 ( u ,v )

H (u , v) = 1 − e

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2 D0 2


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D0 = 15,30,80

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Enhancement using The Laplacian  d n f ( x)  n ( ) ℑ = ju F (u )  n  dx   d 2 f ( x, y ) d 2 f ( x, y )  2 2 ℑ + = ( ju ) F (u , v) + ( jv) F (u, v)  2 2 dy  dx  = −(u 2 + v 2 ) F (u , v) d 2 f ( x, y ) d 2 f ( x, y ) is the Laplacian of f ( x, y ). + 2 2 dx dy 3/20/2012


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∴ℑ ∇ 2 f ( x, y )  = −(u 2 + v 2 ) F (u , v) Laplacian can be implemented in the frequency domain using the filter , H (u, v) = −(u 2 + v 2 ). 2 2   N M + v− H (u , v) = −  u − ( shifted )  2 2   The Laplacian filtered image in the spatial domain is obtained by computing the inverse Fourier Transform of H (u , v) F (u , v) :

) (

(

)

2 2     N M ( , ) ∇ f ( x, y ) = ℑ  −  u − + v− F u v   2 2     2

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−1

(

) (


)

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Unsharp Masking and High Boost Filtering High pass filters eliminate the zero frequency component

of their Fourier transforms and hence average background intensity reduces to near black. Solution: Add a portion of the image back to the filtered

result. Enhancement using Laplacian adds the entire image back

to the filtered result. 3/20/2012


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Unsharp masking consists of generating a sharp image by

subtracting a blurred version of an image from itself. That is, obtaining a highpass-filtered image by subtracting

from the image a lowpass-filtered version of itself.

f hp ( x, y ) = f ( x, y ) − flp ( x, y ) High-boost filtering generalizes this by multiplying f(x,y)

by a constant A≥1.

f hp ( x, y ) = Af ( x, y ) − flp ( x, y )

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High-boost filtering thus increases the contribution made

by the image to the overall enhanced result. When A=1, high-boost filtering reduces to regular

highpass filtering.

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High Frequency Emphasis Filtering To increase the contribution made by high-frequency

components of an image. Multiply a highpass filter function by a constant and add

an offset so that the zero frequency term is not eliminated by the filter. Filter transfer function is given by

H hfe (u , v) = a + bH hp (u , v) Where a ≥0 and b>a. 3/20/2012


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Module 2 Assignment Explain the following point operations: Contrast Stretching Range Compression Image Clipping Explain Homomorphic Filtering. Explain Convolution and Correlation Theorems.

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Thank You

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Image Enhancement Frequency Domain

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