laplacian of gaussian

7/24/2019 Laplacian of Gaussian

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M.Phil. (Information Technology) ITEC 804 Computer Viion

Laplacian/Laplacian of Gaussian

Common Names:Laplacian, Laplacian of Gaussian, LoG, Marr Filter

Brief Description

The Laplacian is a 2-D isotropicmeasure of the 2nd spatial derivativeof an image.The Laplacian of an image highlights regions of rapid intensity change and is

therefore often used for edge detection (see ero crossing edge detectors!. TheLaplacian is often applied to an image that has first "een smoothed #ith something

appro$imating a Gaussian smoothing filter in order to reduce its sensitivity tonoise, and hence the t#o variants #ill "e descri"ed together here. The operatornormally ta%es a single graylevel image as input and produces another graylevel

image as output.

How It Works

The LaplacianL(x,y)of an image #ith pi$el intensity valuesI(x,y)is given "y&

This can "e calculated using a convolution filter.

'ince the input image is represented as a set of discrete pi$els, #e have to find adiscrete convolution %ernel that can appro$imate the second derivatives in the

definition of the Laplacian. T#o commonly used small %ernels are sho#n in Figure

Prepared by: Zeeshan Hyder Bhatti
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.

Figure 1T#o commonly used discrete appro$imations to the Laplacian filter.

()ote, #e have defined the Laplacian using a negative pea% "ecause this is morecommon* ho#ever, it is e+ually valid to use the opposite sign convention.!

sing one of these %ernels, the Laplacian can "e calculated using standardconvolution methods.

ecause these %ernels are appro$imating a second derivative measurement on theimage, they are very sensitive to noise. To counter this, the image is often Gaussian

smoothed"efore applying the Laplacian filter. This pre-processing step reduces thehigh fre+uency noise components prior to the differentiation step.

n fact, since the convolution operation is associative, #e can convolve theGaussian smoothing filter #ith the Laplacian filter first of all, and then convolve

this hy"rid filter #ith the image to achieve the re+uired result. Doing things this#ay has t#o advantages&

'ince "oth the Gaussian and the Laplacian %ernels are usually much smaller

than the image, this method usually re+uires far fe#er arithmetic operations.

The LoG (/Laplacian of Gaussian0! %ernel can "e precalculated in advance

so only one convolution needs to "e performed at run-time on the image.

The 2-D LoG function centered on ero and #ith Gaussian standard deviationhas the form&

and is sho#n in Figure 2.

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Figure The 2-D Laplacian of Gaussian (LoG! function. Thexandya$es are

mar%ed in standard deviations ( !.

1 discrete %ernel that appro$imates this function (for a Gaussian .3! is sho#n

in Figure 4.

Figure !Discrete appro$imation to LoG function #ith Gaussian .3

)ote that as the Gaussian is made increasingly narro#, the LoG %ernel "ecomes thesame as the simple Laplacian %ernels sho#n in Figure . This is "ecause smoothing

#ith a very narro# Gaussian ( 5 6.7 pi$els! on a discrete grid has no effect.8ence on a discrete grid, the simple Laplacian can "e seen as a limiting case of the

LoG for narro# Gaussians.



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Gui"elines for #se

The LoG operator calculates the second spatial derivative of an image. This meansthat in areas #here the image has a constant intensity ( i.e.#here the intensity

gradient is ero!, the LoG response #ill "e ero. n the vicinity of a change inintensity, ho#ever, the LoG response #ill "e positive on the dar%er side, and

negative on the lighter side. This means that at a reasona"ly sharp edge "et#eent#o regions of uniform "ut different intensities, the LoG response #ill "e&

ero at a long distance from the edge,

positive 9ust to one side of the edge,

negative 9ust to the other side of the edge,

ero at some point in "et#een, on the edge itself.

Figure 3 illustrates the response of the LoG to a step edge.

Figure $:esponse of -D LoG filter to a step edge. The left hand graph sho#s a -D image, 266 pi$els long, containing a step edge. The right hand graph sho#s the

response of a -D LoG filter #ith Gaussian 4 pi$els.

y itself, the effect of the filter is to highlight edges in an image.

For e$ample,

is a simple image #ith strong edges.



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The image

is the result of applying a LoG filter #ith Gaussian .6. 1 ;


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is the result of com"ining (i.e.su"tracting! the filtered image and the originalimage. )ote that the filtered image had to "e suita"le scaled"efore com"ining in

order to produce a sensi"le enhancement. 1lso, it may "e necessary to translatethefiltered image "y half the #idth of the convolution %ernel in "oth

thexandydirections in order to register the images correctly.

The enhancement has made edges sharper "ut has also increased the effect of

noise. f #e simply filter the image #ith a Laplacian (i.e.use a LoG filter #ith avery narro# Gaussian! #e o"tain

=erforming edge enhancement using this sharpening image yields the noisy result

()ote that unsharp filteringmay produce an e+uivalent result since it can "e

defined "y addingthe negative Laplacian image (or any suita"le edge image! ontothe original.! >onversely, #idening the Gaussian smoothing component of theoperator can reduce some of this noise, "ut, at the same time, the enhancement

effect "ecomes less pronounced.

The fact that the output of the filter passes through ero at edges can "e used to

detect those edges. 'ee the section on ero crossing edge detection.

)ote that since the LoG is an isotropic filter, it is not possi"le to directly e$tractedge orientation information from the LoG output in the same #ay that it is for

other edge detectors such as the:o"erts >rossand 'o"eloperators.

>onvolving #ith a %ernel such as the one sho#n in Figure 4 can very easily

produce output pi$el values that are much larger than any of the input pi$elsvalues, and #hich may "e negative. Therefore it is important to use an image type

(e.g.floating point! that supports negative num"ers and a large range in order toavoid overflo# or saturation. The %ernel can also "e scaled do#n "y a constant

factor in order to reduce the range of output values.

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Common %ariants

t is possi"le to appro$imate the LoG filter #ith a filter that is 9ust the difference of

t#o differently sied Gaussians. 'uch a filter is %no#n as a DoGfilter (short for/Difference of Gaussians0!.

1s an aside it has "een suggested (Marr ?@2! that LoG filters (actually DoG

filters! are important in "iological visual processing.

1n even cruder appro$imation to the LoG ("ut much faster to compute! is the Do

filter (/Difference of o$es0!. This is simply the difference "et#een t#o meanfiltersof different sies. t produces a %ind of s+uared-off appro$imate version of

the LoG.

Interacti&e '(perimentation

Aou can interactively e$periment #ith the Laplacian operator "y clic%ing here.

Aou can interactively e$periment #ith the Laplacian of Gaussian operator "y

clic%ing here.

'(ercises

. Try the effect of LoG filters using different #idth Gaussians on the image

Bhat is the general effect of increasing the Gaussian #idthC )otice

particularly the effect on features of different sies and thic%nesses.

2. >onstruct a LoG filter #here the %ernel sie is much too small for the

chosen Gaussian #idth (i.e.the LoG "ecomes truncated!. Bhat is the effecton the outputC n particular #hat do you notice a"out the LoG output in

different regions each of uniform "ut different intensitiesC

4. Devise a rule to determine ho# "ig an LoG %ernel should "e made in

relation to the of the underlying Gaussian if severe truncation is to "e

avoided.

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Gaussian +moot,ing

Common Names:Gaussian smoothing

Brief Description

The Gaussian smoothing operator is a 2-D convolution operatorthat is used to/"lur0 images and remove detail and noise. n this sense it is similar to the mean

filter, "ut it uses a different %ernelthat represents the shape of a Gaussian (/"ell-shaped0! hump. This %ernel has some special properties #hich are detailed "elo#.

How It Works

The Gaussian distri"ution in -D has the form&

#here is the standard deviation of the distri"ution. Be have also assumed that the

distri"ution has a mean of ero (i.e.it is centered on the linex6!. The distri"utionis illustrated in Figure .

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Figure 1-D Gaussian distri"ution #ith mean 6 and

n 2-D, an isotropic (i.e.circularly symmetric! Gaussian has the form&

This distri"ution is sho#n in Figure 2.

Figure 2-D Gaussian distri"ution #ith mean (6,6! and



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The idea of Gaussian smoothing is to use this 2-D distri"ution as a /point-spread0function, and this is achieved "y convolution. 'ince the image is stored as a

collection of discrete pi$els #e need to produce a discrete appro$imation to theGaussian function "efore #e can perform the convolution. n theory, the Gaussian

distri"ution is non-ero every#here, #hich #ould re+uire an infinitely largeconvolution %ernel, "ut in practice it is effectively ero more than a"out three

standard deviations from the mean, and so #e can truncate the %ernel at this point.Figure 4 sho#s a suita"le integer-valued convolution %ernel that appro$imates a

Gaussian #ith a of .6. t is not o"vious ho# to pic% the values of the mas% to

appro$imate a Gaussian. ne could use the value of the Gaussian at the centre of api$el in the mas%, "ut this is not accurate "ecause the value of the Gaussian varies

non-linearly across the pi$el. Be integrated the value of the Gaussian over the#hole pi$el ("y summing the Gaussian at 6.66 increments!. The integrals are not

integers& #e rescaled the array so that the corners had the value . Finally, the 2;4

is the sum of all the values in the mas%.

Figure !Discrete appro$imation to Gaussian function #ith .6

nce a suita"le %ernel has "een calculated, then the Gaussian smoothing can "eperformed using standard convolution methods. The convolution can in fact "e

performed fairly +uic%ly since the e+uation for the 2-D isotropic Gaussian sho#na"ove is separa"le intoxandycomponents. Thus the 2-D convolution can "e

performed "y first convolving #ith a -D Gaussian in thexdirection, and thenconvolving #ith another -D Gaussian in theydirection. (The Gaussian is in factthe onycompletely circularly symmetric operator #hich can "e decomposed in

such a #ay.! Figure 3 sho#s the -Dxcomponent %ernel that #ould "e used toproduce the full %ernel sho#n in Figure 4 (after scaling "y 2;4, rounding and

truncating one ro# of pi$els around the "oundary "ecause they mostly have thevalue 6. This reduces the ;$; matri$ to the 7$7 sho#n a"ove.!. Theycomponent is

e$actly the same "ut is oriented vertically.

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Figure $ne of the pair of -D convolution %ernels used to calculate the full

%ernel sho#n in Figure 4 more +uic%ly.

1 further #ay to compute a Gaussian smoothing #ith a large standard deviation is

to convolve an image several times #ith a smaller Gaussian. Bhile this iscomputationally comple$, it can have applica"ility if the processing is carried out

using a hard#are pipeline.

The Gaussian filter not only has utility in engineering applications. t is also

attracting attention from computational "iologists "ecause it has "een attri"uted#ith some amount of "iological plausi"ility, e.g.some cells in the visual path#ays

of the "rain often have an appro$imately Gaussian response.

Gui"elines for #se

The effect of Gaussian smoothing is to "lur an image, in a similar fashion to

the mean filter. The degree of smoothing is determined "y the standard deviation ofthe Gaussian. (Larger standard deviation Gaussians, of course, re+uire larger

convolution %ernels in order to "e accurately represented.!

The Gaussian outputs a /#eighted average0 of each pi$el0s neigh"orhood, #ith the

average #eighted more to#ards the value of the central pi$els. This is in contrastto the mean filter0s uniformly #eighted average. ecause of this, a Gaussian

provides gentler smoothing and preserves edges "etter than a similarly sied meanfilter.

ne of the principle 9ustifications for using the Gaussian as a smoothing filter isdue to its !re"uency response. Most convolution-"ased smoothing filters act as

lo#pass fre+uency filters. This means that their effect is to remove high spatialfre+uency components from an image. The fre+uency response of a convolution

filter, i.e. its effect on different spatial fre+uencies, can "e seen "y ta%ing theFourier transformof the filter. Figure 7 sho#s the fre+uency responses of a -D

mean filter #ith #idth 7 and also of a Gaussian filter #ith 4.

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Figure .Fre+uency responses of o$ (i.e.mean! filter (#idth 7 pi$els! and

Gaussian filter ( 4 pi$els!. The spatial fre+uency a$is is mar%ed in cycles per

pi$el, and hence no value a"ove 6.7 has a real meaning.

oth filters attenuate high fre+uencies more than lo# fre+uencies, "ut the meanfilter e$hi"its oscillations in its fre+uency response. The Gaussian on the other

hand sho#s no oscillations. n fact, the shape of the fre+uency response curve isitself (half a! Gaussian. 'o "y choosing an appropriately sied Gaussian filter #e

can "e fairly confident a"out #hat range of spatial fre+uencies are still present inthe image after filtering, #hich is not the case of the mean filter. This has

conse+uences for some edge detection techni+ues, as mentioned in the section onero crossings. (The Gaussian filter also turns out to "e very similar to the optimal

smoothing filter for edge detection under the criteria used to derive the >anny edge

detector.!

Be use

to illustrate the effect of smoothing #ith successively larger and larger Gaussian

filters.

The image

sho#s the effect of filtering #ith a Gaussian of .6 (and %ernel sie 7


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sho#s the effect of filtering #ith a Gaussian of 2.6 (and %ernel sie ?


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#hich has "een corrupted "y salt and pepper noise ( i.e.individual "its have"een flipped #ith pro"a"ility !. The image

sho#s the result of Gaussian smoothing (using the same convolution as a"ove!.>ompare this #ith the original

)otice that much of the noise still e$ists and that, although it has decreased in

magnitude some#hat, it has "een smeared out over a larger spatial region.

ncreasing the standard deviation continues to reduceH"lur the intensity of thenoise, "ut also attenuates high fre+uency detail (e.g.edges! significantly, as sho#nin

This type of noise is "etter reduced using median filtering, conservative

smoothingor >rimmins 'pec%le :emoval.

Interacti&e '(perimentation

Aou can interactively e$periment #ith this operator "y clic%ing here.

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'(ercises

. 'tarting from the Gaussian noise(mean 6, 4! corrupted image

compute "oth mean filterand Gaussian filter smoothing at various scales,

and compare each in terms of noise removal vs loss of detail.

2. 1t ho# many standard deviations from the mean does a Gaussian fall to 7

of its pea% valueC n the "asis of this suggest a suita"le s+uare %ernel sie

for a Gaussian filter #ith s.

4. Istimate the fre+uency response for a Gaussian filter "y Gaussiansmoothing an image, and ta%ing its Fourier transform"oth "efore and

after#ards. >ompare this #ith the fre+uency response of a mean filter.

3. 8o# does the time ta%en to smooth #ith a Gaussian filter compare #ith the

time ta%en to smooth #ith a mean filter!or a #erne o! the same si$e%)otice

that in "oth cases the convolution can "e speeded up considera"ly "ye$ploiting certain features of the %ernel.

)eferences

'* Da&iesMachine Vision& 'heory, gorithms and racticaities, 1cademic =ress,

??6, pp 32 - 33.

)* Gonale an" )* Woo"sDigita Image rocessing, 1ddison-Besley

=u"lishing >ompany, ??2, p ?.

)* Haralick an" L* +,apiroComputer and Robot Vision, 1ddison-Besley

=u"lishing >ompany, ??2, ol. , >hap. ;.

B* HornRobot Vision, MT =ress, ?@E, >hap. @.

D* %ernonMachine Vision, =rentice-8all, ??, pp 7? - E, 23.

Local Information

'pecific information a"out this operator may "e found here.

http://homepages.inf.ed.ac.uk/rbf/HIPR2/noise.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/mean.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/mean.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/mean.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/local/gsmooth.txthttp://homepages.inf.ed.ac.uk/rbf/HIPR2/noise.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/mean.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/mean.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/mean.htmhttp://homepages.inf.ed.ac.uk/rbf/HIPR2/local/gsmooth.txt


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More general advice a"out the local 8=: installation is availa"le in the LocaIn!ormationintroductory section.

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laplacian of gaussian

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