color interpolation algorithmss

7/29/2019 Color Interpolation Algorithmss

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Color Interpolation Algorithms

Gayathri devi Kotapati

Depatment of computer science

K L UniversityVaddeswaram,India

[email protected]

Jaya deepthi Thiyyagura

Depatment of computer science

K L UniversityVaddeswaram,India

[email protected]

AbstractIn this paper we discuss about different colorinterpolation algorithms and compare these algorithms based on

subjective quality measure ,objective quality measure and

computational cost .Interpolation is a method of constructing

new data points with in the range of discrete set of known data

points

I.INTRODUCTIONAll Usage of digital cameras is spreading widely as they are

easy image input devices. The increasing popularity of digital

cameras has provided motivation to improve all elements of

the digital photography signal chain. To lower cost, digital

color cameras typically use a single image detector. Color

imaging with a single detector requires the use of a ColorFilter Array (CFA) which covers the detector array. In this

arrangement each pixel in the detector samples the intensity of

just one of the many-color separations. The recovery of full-

color images from a CFA-based detector requires a method of

calculating values of the other color separations at each pixel.

These methods are commonly referred as color interpolation

or color demosaicing algorithms.

In a single-detector camera, varying intensities of light are

measured at an rectangular grid of image sensors. To constructa color image, a CFA must be placed between the lens and the

sensors. A CFA typically has one color filter element for each

sensor. Obviously, the color interpolation algorithms depend

on the CFA configuration. Many different CFA configurations

have been proposed. One of the most popular is the Bayer

pattern[1], which uses the three additive primary colors, red,

green and blue (RGB), for the filter elements. Since our goal

in this study is to examine color interpolation algorithms, we

don't pay special attention to CFA patterns. Instead we

exclusively work with the Bayer pattern in RGB color space,

as shown in Figure.(1) below. The selection of Bayer patternis mainly due to its popularity.

Figure 1

In Section II, we will describe the color interpolationalgorithms included in this study. Section III gives software

implementation of some of these algorithms in Matlab. We

then compare the performance of these algorithms in Section

IV.

Notice even though our study is specifically targeted at Bayer

pattern with RGB color space, many algorithms studies here

can be extended to different pattern with different color spaceeasily.

In our effort to include as many algorithms as possible, wehave tried the most extensive search within our resource

limitation. But by no means this inclusion is complete and we

apologize if some other work in this area should have been

included.

II.ALGORITHM DESCRIPTIION. In this section we will describe a number of color

interpolation algorithms used in digital cameras as reported in

liberatures. To describe these algorithms, we classify them

into two distinct groups, non-adaptive

algorithms and adaptive algorithms.

As their names suggest, non-adaptive algorithms denote those

algorithms that perform interpolation in a fixed pattern for

every pixel (within a group). While adaptive algorithms imply


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that those algorithms can detect local spatial features present

in the pixel neighborhood, then make effective choices as to

which predictor to use for that neighborhood. In other words,adaptive algorithms pocesses some intelligence and therefore

are more sophisticated in general.

There are many ways to describe these algorithms. Graphical

description, kernel convolution, arithmetic equations or

description via examples are all useful means to explain how

the algorithm works. In our presentation of each algorithm, wewill try to adopt any one of these means or combinations of

those means, whichever makes the description as easily

understandable as possible. As a result, different algorithms

may be subject to different description means.

Before we jump into the detailed description, we want to

mention briefly some other algorithms that are not included

here in detail. This includesMedian Filtering Method[28],

some other linear estimator methods in [29][31][33],Pattern

Classification Method[32], and a new Gradient based

methoddescribed in a newly approved patent[30]. We would

like include them all in the future.

III.NON ADAPTIVE ALGORITHM

A.Nearest Neighbor Replication

In this interpolation method[2][3][4][5], eachinterpolated output pixel is assigned the value of the

nearest pixel in the input image. The nearest neighbor

can be any one of the upper, lower, left and rightpixels.

An example is illustrated below in Figure 2 for a 3x3block in green plane. Here we assume the leftneighboring pixel value is used to fill the missing

ones.

Figure 2

B.Bilinear Interpolation

figure 1

Interpolation of green pixels : the average of theupper, lower, left and right pixel values is assigned as

the G value of the interpolated pixel. For example :

G8 = (G3+G7+G9+G13) / 4

Interpolation of red/blue pixels :o Interpolation of a red/blue pixel at

a green position : the average of two

adjacent pixel values in corresponding color

is assigned to the interpolated pixel. Forexample : B7 = (B6+B8) / 2 ; R7 =

(R2+R12) / 2

o Interpolation of a red/blue pixel ata blue/red position : the average of four

adjacent diagonal pixel values is assigned to

the interpolated pixel. For example : R8 =

(R2+R4+R12+R14) / 4 ; B12 =(B6+B8+B16+B18) / 4

C.Cubic Convolution Interpolation

We will rely on system kernel convolution todescribe this interpolation scheme. In general,

interpolation can be expressed as :

where h is the interpolation kernel and c(x,y) is the data

sample itself at a pixel ( xk,yl ). Apparently, in this formation,

h, the interpolation kernel, is the main part of the interpolation

algorithm.

Cubic convolution is a third-degree interpolationusing 16 adjacent pixels and some typical cubic

convolution kernels are :

o one-variable cubic interpolation :


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and the parameter "a" has been reported as -

1 in [22], 0.5 in [23] and 0.75 in [24].

o two-variable cubic interpolation [25]:

where "b" and "c" are two parameters and(b,c) = (0.33,0.33), (b,c) = (1.5,-0.25) have

been reported in [26].

o Cubic B-Spline interpolation [27] :

Cubic B-spline is the three-degree B-spline.

D.Smooth Hue Transition Interpolation

One of the key objection of the bilinear interpolationis that the hues of adjacent pixels change abuptly and

in an unnatural manner. The Bayer CFA pattern, on

the other hand, can be thought of as consisting of a

luminance channel (the relatively numerous greenpixels) and a chrominance channel (the relative

sparse red and blue pixels). A scheme can be created

to interpolate these channels differently.

For convenience, figure 1 is duplicated here

Interpolation of green pixels : same as in bilinearinterpolation. Also note interpolation of green pixels

has to be done before interpolations of red/blue

pixels.

Interpolation of red/blue pixels :o The idea here is to try to impose a smooth

transition in hue from pixel to pixel.

o To do so, define blue "hue value" as : B/G.And red "hue value" can be analogously

defined.

o Considering the interpolation of blue pixelvalues : there are three different cases of

blue pixel value interpolations.

estimating blue pixel value at thegreen position and the adjacent

blue pixels are on left and right :

e.g. B7 = G7 / 2 * (B6 / G6 + B8 /

G8)


blue pixels are on top and bottom :

e.g. B13 = G13 / 2 * (B8 / G8 +

B18 / G18)

estimating blue pixel value at thered postion : e.g. B12 = G12 / 4 *(B6 / G6 + B8 / G8 + B16 / G16 +

B18 / G18)

o Interpolation of red pixel values can becarried out analogously

E.Smooth Hue Transition Interpolation in Logarithmic

Exposure Space

Figure 1

Interpolation of green pixels : same as in bilinearinterpolation. Also note interpolation of green pixels


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has to be done before interpolations of red/blue

pixels.

Key step : After the green pixel interpolations areperformed, transfer the original pixel data together

with the interpolated green pixel values from linear

exposure space into logarithmic exposure space. Let's

denote these log values ad Glog, Blog and Rlog,

respectively.

Interpolation of red/blue pixels in log space :o In logarithmic space, define blue "hue

value" as : Blog- Glog . And red "hue

value" can be analogously defined.

o Considering the interpolation of blue pixelvalues : there are three different cases of

blue pixel value interpolations.


blue pixels are on left and right :

e.g. B7log = G7log + (B6log - G6log

+ B8log - G8log) / 2


blue pixels are on top and bottom :

e.g. B13log = G13log + (B8log -

G8log + B18log - G18log) / 2

estimating blue pixel value at thered postion : e.g. B12 log = G12log +

(B6log - G6log + B8log - G8log +

B16log - G16log + B18log - G18log) /

4

o Interpolation of red pixel values can becarried out analogously

Finally, transfer the all pixel values back into linearexposure space

.

IV.ADAPTIVE ALGORITHMA.Edge Sensing Interpolation Algorithm I

From our description of non-adaptive algorithms, itcan be seen that most of the color interpolation is

done by averaging neighboring pixels

indiscriminately. This causes an artifact -- the "zipper

effect" in the interpolated image. To combat with this

artifact, it is natural to derive an algorithm that can

detect local spatial features present in the pixel

neighborhood and then makes effective choices as towhich predictor to use that neighborhood. The result

is a reduction or elimination of "zipper-type"

artifacts. And algorithms that involve this kind of

"intelligent" detection and decision process are

referred as adaptive color interpolation algorithms.

Human visual systems are sensitive to edges presentin the images and non-adaptive color interpolation

algorithms often fail around edges since they are not

able to detect "edges". To deal with edges, let's first

look at an edge sensing algorithm[14][3][15].

figure 1

Interpolation of green pixels :o First, define two gradients, one in horizontal

direction, the other in vertical direction, for

each blue/red position. For instance,

consider B8 : define two gradients

as ,where | . | denotes absolute value

o Define some threshold value To The algorithm then can be described as

follows :

o The choice of T depends on the images andcan have defferent optimum values from

different neighborhoods. A particular choice

of T is [4]. In this case,the algorithm becomes :


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Interpolation of red/blue pixels : same as in Smoothhue transition interpolation in logarithmic exposure

space. Other methods may be used as well.

B.Edge Sensing Interpolation Algorithm II

A slightly different edge sensing interpolationalgorithm is described in [17].

Interpolation of green pixels : same as in edgesensing interpolation algorithm I except that

horzitonal and vertical gradients are defined

differently. Refer to figure 6 below, to estimate G5 at

B5, we define gradients as :

figure 6

and the actual algorithm follows as in the edge sensing

interpolation algorithm I.

Interpolation of red/blue pixels : similar to bilinearinterpolation algorithm except that color difference is

interpolated instead of color itself.

C.Linear Interpolation with Laplacian second-order correctionterms I

This algorithm [18] is designed for optimizingperformance of images with horizontal and vertical

edges. To illustrate the algorithm, we'll utilize real

examples to make description a little bit easier to

understand.

Interpolation of green pixels : refer to figure 6,duplicated below for convenience. We'll consider

estimating the green value at a blue pixel.

Interpolation done at a red pixel can be carried out

similarly. Let's find out G5 at B5.

o Define horizontal and vertical gradients as

:

o The algorithm is:

Interpolation of red/blue pixels : refer to figure 7below. Depending on the positions, we have three

cases.

figure 7

o case 1 : find red/blue values at a green pixelwhere nearest neighbors are in the samecolumn. For instance, to estimate R4 at G4,

use predictor R4 = (R1 + R7) / 2 + (G4 - G1

+ G4 - G7) / 4

o case 2 : find red/blue values at a green pixelwhere nearest neightbors are in the same

row. For instance, to estimate R2 at G2, use


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predictor R2 = (R1 + R3) / 2 + (G2 - G1 +

G2 - G3) / 4

o case 3 : find red/blue values ata blue/red pixel. For instance, estimate R5 atB5.

Define two diagonal (negative andpositive) directions as follows :

And the actual algorithm isdescribed as :

D.Linear Interpolation with Laplacian second-order correction

terms II

An extension to previous interpolation method withLaplacian second-order correction terms is described

in [19][20].

Interpolation of green pixels : again, we will borrowthe figure from previous algorithm descriptions. Forconvenience, it's duplicated below. We'll consider

estimating the green value at a blue pixel.Interpolation done at a red pixel can be carried out

similarly. Let's find out G5 at B5.

o Define eleven color correction terms withcorresponding predictors as follows :

CC1 = B5 - B3 + B5 - B7; P1 = ( G4 + G6 ) / 2 +CC1 / 4;

CC2 = B5 - B1 + B5 - B9; P2 = ( G2 + G8 ) / 2 +

CC2 / 4;

CC3 = B5 - B1 + B5 - B7; P3 = ( G2 + G6 ) / 2 +

CC3 / 4;CC4 = B5 - B7 + B5 - B9; P4 = ( G6 + G8 ) / 2 +

CC4 / 4;

CC5 = B5 - B9 + B5 - B3; P5 = ( G4 + G8 ) / 2 +

CC5 / 4;

CC6 = B5 - B3 + B5 - B1; P6 = ( G2 + G4 ) / 2 +

CC6 / 4;CC7 = B5 - B1; P7 = G2 + CC7 / 2;

CC8 = B5 - B7; P8 = G6 + CC8 / 2;

CC9 = B5 - B9; P9 = G8 + CC9 / 2;

CC10 = B5 - B3; P10 = G4 + CC10 / 2;

CC11 = B5 - B1 + B5 - B3 + B5 - B7 + B5 - B9; P11

= ( G2 + G4 + G6 + G8) / 4 + CC11 / 8;

notice CC1 - CC6 are Laplacian second-

order derivative operators and CC7 - CC10are Laplacian first-order derivative

operators.

o The algorithm is :If min{CC1, CC2 } < a given

Threshold (e.g. 69 in 8-bit log

space)G5 = the corresponding

predictor; (e.g. if CC1 < CC2 and

CC1 < Threshold, then G5 = P1)

Else if min{CC3,CC4,CC5,CC6} B>C>D,

then G12 =median{G7,G11,G13,G17} = ( B +

C ) / 2

stripe pattern : a larger pixelneighborhood as described in

figure 3 is needed

Figure 3

and where

M = SUM(G) / 4 , S = SUM(X) / 8

and 'clip' function constrains the

prediction to lie between the values

B and C. If the prediction is greaterthan B, it is set o B; if the

prediction is less than C, it is set to

C

corner pattern : again, a larger pixelneighborhood is needed, as

described in figure 4.

figure 4

and just as in the stripe pattern

case, ,

where M = median{H's, L's} and S= SUM(X) / 4

Interpolation of red/blue pixels : can use any methoddescribed before, for example, bilinear interpolationor smooth hue transition interpolation

G.Pattern Matching Interpolation

People have observed that there's a high degree ofcorrelation between the RGB channels in natural

scenes.


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The idea here is to exploit the correlations betweenthe green, red and blue components, and determine

the missing components from the pixel contexts.

Since green is sampled with the highest sampling

frequency, the reproduction of the green components

is much more accurate than the reproduction of the

red or blue components. For this reason, this

algorithm specifically targets at the interpolations of

chrominance channels.

Interpolation of green pixels : use any methoddescribed before.

Interpolation of red/blue pixels :o estimate red/blue values at green pixels : use

any method described before. For instance,use Bilinear Interpolation, i.e., use the

average of two adjacent color samples.

o estimate blue values at red pixels : First let's define "pattern" p(x,y)

for a red pixel at position (x,y),

refer to figure 5, as a four-

dimensional vector or a green

pattern : P(x,y) = [G(x-1,y);G(x+1,y); G(x,y-1); G(x,y+1)],

where G(x,y) is the green value at

position (x,y).

Next compare the green patternp(x,y) with its four neighboringgreen patterns and measure their

similarities by:

where || . || denotes the l-1 norm ofa vector

Sort these delta values and let =the ith largest value,

therefore

Define some threshold value T

The actual algorithm is as follows :if < T, then we have high

confidence that the image is

smooth around (x,y), so the bluevalue is the average of all four

adjacent diagonal pixel values. else

the desired blue value=

Let's refer to a real example now.Consider figure 5, we want toestimate the blue value at R15 :

figure 5

the procedures are :

1. P15 = [G14; G16; G9;G21]; P8 = [G7; G9; G2;

G14]; P10 = [G9; G11;G4; G16]; P20= [G19;

G21; G14; G26]; P22 =

[G21; G23; G16; G28]

2.3. sort them. Suppose the

result is:

4.o estimate red values at blue pixels can be

carried out analogously


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V.ALGORITHM IMPLEMENTATIONWe choose to implement above mentioned algorithms in

software using matlab

VI. ALGORITHMS COMPARISIONYou In this section we will try to compare different color

interpolation algorithms introduced and implemented in

previous sections. In doing so, we compare these algorithms

from three aspects. First, we will compare them using

subjective measures, we will try to identify the major

difference among these algorithms with visual observations.

Secondly, we will switch to objective quality measure, in our

case, we choose Mean Squared Error (MSE) between the

original test image and the reconstructed image. MSE is

chosen for its simplicity and popularity. Finally, we compare

their computational cost, which is vital for implementation in

practice

SUBJECTIVE QUALITY MEASURE:

Example 1:

Example 2:

Observations :o Nearest Neighbor Replication

Interpolation : significant errors occur

throughout the image, very evident blocking

effect, very bad for sharp edges.

Unacceptable for still imaging systems. Invideo applications, many of these problems

might be averaged away.

o Bilinear Interpolation : much improvedperformance over Nearest Neighbor

Replication Interpolation; some blurring due

to averaging adjacent pixels; introduce theso-called "zipper" effect

o Smooth Hue Algorithms : less blur comparedto Bilinear algorithm as seen in the 'BMW'

image. On the other hand, significant 'hue'

interpolation error is imposed as seen in the

image "flowers". This is due to the division

(or subtraction in Log space) in R/B

channel interpolation where large G


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difference between adjacent pixels will

cause huge R/B error during interpolation.

o Edge Sensing Algorithms : as their namessuggest, indeed sharper images can be seen.Since R/B interpolation uses the same

algorithm as in Smooth Hue Transition

algorithms, significant "hue" error can beseen in the image "flowers". Not very

suitable for colored small objects.

o Linear Interpolation with Laplacian 2ndorder color correction terms I: gives one ofthe best quality among all algorithms.

Especially good for edges.

o Interpolation using Threshold-basedVariable Number of Gradients : appears to

give the best result among all algorithms.Sharp edges as well.

o Pattern Recognition : farily good results, butedges aren't dealt as well as variable number

gradients method or Laplacian color

correction method

Objective Quality Measure

To compare these algorithms objectively, we adoptMean Square Error (MSE) as our quality metrics. The

MSE between the original image Io(x,y) and the

interpolated image Ir(x,y) for each color channel isdefined as :

where

m x n is the size of the image

The MSE values for each algorithm that areimplemented in Section III are depicted below forthree color channels separately. We normalized the

MSE value to the minimum since we are only

interested in the relative performance among thesealgorithms.

o Example I : flowers.tif

o Example II : bmw.bmp

From these plots, we conclude that based onobjective measure using MSE :

o Nearest Neighbor ReplicationInterpolation gives significant larger mean

squared error than other algorithms and

therefore the interpolated image is expectedto show the worst artifacts, which can be

verified in previous subsection.

o In terms of Green (or Luminance) channelinterpolations, Variable Number Gradients

Methodgives the best performance in both

examples.Laplacian Color Correction Imethodalso gives impressive results. And as

far as the rest algorithms are concerned,

their MSE values don't show any major

difference.

o In terms of Red/Blue (or Chrominance)channel interpolations, there's no obvious

superior algorithm that can stand out


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alone. Variable Number Gradients

Methodgives slightly better results.

Computational Cost

Here we will compare the computational costassociated with each of the algorithms implemented.

The way we compute the cost is to count the flopsoccurred in Matlab when processing some input

images. For those non-adaptive algorithms, the cost is

fixed regardless of the content of the input image as

long as the image size is fixed. While for adaptive

algorithms, the computational cost is scene dependent

since those algorithms involve comparisons or

decision making steps where branching can occur.

This variation on computational cost due to image

content is, however, not evident for two examples we

included here.

Since the computational cost is solely determinedfrom the software point of view, it is of course related

to the actual implementation of the algorithm.

Efficient program reduces the cost. In our study no

special effort is denoted to deal with thisimplementation-dependent feature of the cost and

none of the algorithm code is tuned intentionally to

increase its efficiency, nor do we decrease it on

purpose. We assume the behavior of our

implementation observes the average in general.

Figure below shows the comparison for the two

images.

Observations :o Nearest Neighbor Replication

Interpolation doesn't involve any arithmetic

operation and therefore doesn't impose any

computational cost

o Excluding Nearest Neighbor Replicationalgorithm,Bilinear interpolation imposesthe least amount of computation among the

other eight interpolation schemes

o On the other hand, interpolation usingthreshold-based variable number gradients

methodhas a significant computational cost

penalty compared to others, and it's very

impractical for real-time digital videoapplications.

o Linear interpolation with Laplacian second-order color correction terms Ialso imposes

a large computational cost compared to

other algorithms but with a much lesspenalty compared to variable number

gradients method.

o The rest algorithms have similarcomputational cost

SUMMARY

In this study, we described twelve color interpolationalgorithms used for single-detector digital cameras.

We selectively implemented nine of them usingMatlab and finally did comparative performance

analysis from three aspects of the algorithm :

subjective quality of reproduced images, objective

quality (MSE) of reproduced images and

computational cost.

We made some observations for these algorithmsbased on our simulation results. In particular, we note

thatNearest Neighbor Replication

Interpolation andInterpolation using Threshold-

based Variable Number of Gradients represent two

extremes of interpolation algorithms.Nearest

Neighbor Replication Interpolation is extremelycomputationally efficient but with the worst quality

performance. Variable Number Gradients

Methodgives impressive reproduced image quality

but imposes a huge computation penalty.

As far as the future work is concerned, an algorithmthat is both superior in image reproduction and

computationaly efficient is still worth pursuing giventhe fact that none of the existing algorithm satisfies

both criteria.


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