Image Smoothing using Frequency Domain Filters

IMAGE SMOOTHING USING FREQUENCY DOMAIN FILTERS By, H. Suhaila Afzana C. Surega T. Vaitheeswari 1

CONTENTS  Frequency Domain Filters  Lowpass Filters  Ideal Lowpass Filters  Butterworth Lowpass Filters  Gaussian Lowpass Filters  Lowpass Filters – Comparison  Lowpass Filtering Examples 2

FREQUENCY DOMAIN FILTERS  Smoothing(blurring) is achieved in the frequency domain by high- frequency attenuation; that is, by lowpass filtering.  Here, we consider 3 types of lowpass filters:  Ideal lowpass filters  Butterworth lowpass filters  Gaussian lowpass filters  These three categories cover the range from very sharp(ideal), to very smooth(Gaussian) filtering. 3

FREQUENCY DOMAIN FILTERS  The Butterworth filter has a parameter called the filter order.  For high order values, the Butterworth filter approaches the ideal filter. For low order values, Butterworth filter is more like a Gaussian filter.  Thus, the Butterworth filter may be viewed as providing a transition between two “extremes”. 4

LOWPASS FILTERS  The most basic of filtering operations is called “lowpass”.  A lowpass filter is also called a “blurring” or smoothing filter.  The simplest lowpass filter just calculates the average of a pixel and all of its eight immediate neighbours.  Lowpass is also called as blurring mask. 5

IDEAL LOWPASS FILTERS  A 2-D lowpass filter that passes without attenuation all frequencies within a circle of radius D0 from the origin and “cuts off” all frequencies outside this circle is called an ideal lowpass filter(ILPF); it is specified by the function:       0 0 ),(if0 ),(if1 ),( DvuD DvuD vuH 6

IDEAL LOWPASS FILTERS  D0 is a positive constant and D(u,v) is the distance between a point (u,v) in the frequency domain and the center of the frequency rectangle; that is, 2/122 ])2/()2/[(),( QvPuvuD  7

IDEAL LOWPASS FILTERS  The ideal lowpass filter is radially symmetric about the origin, which means that the filter is completely defined by a radial cross section.  Rotating the cross section by 360° yields the filter in 2-D.  For an ILPF cross section, the point of transition between H(u,v)=1 and H(u,v)=0 is called the cutoff frequency D0.  Simply cut off all high frequency components that are at a specified distance D0 from the origin of the transform, changing the distance changes the behaviour of the filter. 8

IDEAL LOWPASS FILTERS A)Perspective plot of an ideal lowpass filter transfer function B)Filter displayed as an image C)Filter radius cross section 9

IDEAL LOWPASS FILTERS  When the lowpass filter is applied ringing occurs in the image.  The narrower the filter in the frequency domain, the more severe are the blurring and ringing.  The more ringing in the image, the more blurring of the image. 10

IDEAL LOWPASS FILTERS  Above we show an image, it’s Fourier spectrum and a series of ideal low pass filters of radius 5, 15, 30, 80 and 230 superimposed on top of it. 11

IDEAL LOWPASS FILTERS Original image Result of filtering with ideal low pass filter of radius 5 Result of filtering with ideal low pass filter of radius 30 Result of filtering with ideal low pass filter of radius 230 Result of filtering with ideal low pass filter of radius 80 Result of filtering with ideal low pass filter of radius 15 12

BUTTERWORTH LOWPASS FILTERS  The Butterworth lowpass filter is a type of signal processing filter designed to have as flat a frequency response as possible in the passband.  It is also referred to as a maximally flat magnitude filter.  It was first described in 1930 by the British Engineer and physicist Stephen Butterworth. 13

BUTTERWORTH LOWPASS FILTERS  The transfer function of a Butterworth lowpass filter of order n with cutoff frequency at distance D0 from the origin is defined as: n DvuD vuH 2 0 ]/),([1 1 ),(   14

BUTTERWORTH LOWPASS FILTERS A)Perspective plot of an Butterworth lowpass filter transfer function B)Filter displayed as an image C)Filter radius cross section of orders 1 through 4 15

BUTTERWORTH LOWPASS FILTERS Original image Result of filtering with Butterworth filter of order 2 and cutoff radius 5 Result of filtering with Butterworth filter of order 2 and cutoff radius 30 Result of filtering with Butterworth filter of order 2 and cutoff radius 230 Result of filtering with Butterworth filter of order 2 and cutoff radius 80 Result of filtering with Butterworth filter of order 2 and cutoff radius 15 16

BUTTERWORTH LOWPASS FILTERS 17

GAUSSIAN LOWPASS FILTERS  The transfer function of a Gaussian lowpass filter is defined as:  Here, is the standard deviation and is a measure of spread of the Gaussian curve.  If we put =D0 we get, 22 2/),( ),( vuD evuH   2 0 2 2/),( ),( DvuD evuH   18

GAUSSIAN LOWPASS FILTERS A)Perspective plot of a GLPF transfer function B)Filter displayed as an image C)Filter radius cross section for various values of D0 19

GAUSSIAN LOWPASS FILTERS  Main advantage of a Gaussian LPF over a Butterworth LPF is that we are assured that there will be no ringing effects no matter what filter order we choose to work with. 20

GAUSSIAN LOWPASS FILTERS Original image Result of filtering with Gaussian filter with cutoff radius 5 Result of filtering with Gaussian filter with cutoff radius 30 Result of filtering with Gaussian filter with cutoff radius 230 Result of filtering with Gaussian filter with cutoff radius 85 Result of filtering with Gaussian filter with cutoff radius 15 21

LOWPASS FILTERS-COMPARISON Result of filtering with ideal low pass filter of radius 15 Result of filtering with Butterworth filter of order 2 and cutoff radius 15 Result of filtering with Gaussian filter with cutoff radius 15 22

LOWPASS FILTERING EXAMPLES  A low pass Gaussian filter is used to connect broken text 23

LOWPASS FILTERING EXAMPLES  Different lowpass Gaussian filters used to remove blemishes in a photograph 24

Image Smoothing using Frequency Domain Filters

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