Single-Image Vignetting Correction

Yuanjie Zheng∗

Shanghai Jiaotong UniversityStephen Lin

Microsoft Research AsiaSing Bing Kang

Microsoft Research

Abstract

In this paper, we propose a method for determining thevignetting function given only a single image. Our methodis designed to handle both textured and untextured regionsin order to maximize the use of available information. Toextract vignetting information from an image, we presentadaptations of segmentation techniques that locate imageregions with reliable data for vignetting estimation. Withineach image region, our method capitalizes on frequencycharacteristics and physical properties of vignetting to dis-tinguish it from other sources of intensity variation. Thevignetting data acquired from regions are weighted accord-ing to a presented reliability measure to promote robustnessin estimation. Comprehensive experiments demonstrate theeffectiveness of this technique on a broad range of images.

1. Introduction

Vignetting refers to the phenomenon of brightness atten-uation away from the image center, and is an artifact thatis prevalent in photography. Although not objectionable tothe average viewer at low levels, it can significantly impaircomputer vision algorithms that rely on precise intensitydata to analyze a scene. Applications in which vignettingdistortions can be particularly damaging include photomet-ric methods such as shape from shading, appearance-basedtechniques such as object recognition, and image mosaic-ing.

Several mechanisms may be responsible for vignettingeffects. Some arise from the optical properties of cameralenses, the most prominent of which is off-axis illumina-tion falloff or the cos4 law. This contribution to vignettingresults from foreshortening of the lens when viewed fromincreasing angles from the optical axis [7]. Other sourcesof vignetting are geometric in nature. For example, light ar-riving at oblique angles to the optical axis may be partiallyobstructed by the field stop or lens rim.

∗This work was done while Yuanjie Zheng was a visiting student atMicrosoft Research Asia.

To determine the vignetting effects in an image, themost straightforward approach involves capturing an imagecompletely spanned by a uniform scene region, such thatbrightness variations can solely be attributed to vignetting[12, 1, 6, 14]. In such a calibration image, ratios of inten-sity with respect to the pixel on the optical axis describe thevignetting function. Suitable imaging conditions for this ap-proach, however, can be challenging to produce due to un-even illumination and camera tilt, and the vignetting mea-surements are valid only for images captured by the cameraunder the same camera settings. Moreover, a calibrationimage can be recorded only if the camera is at hand; con-sequently, this approach cannot be used to correct imagescaptured by unknown cameras, such as images downloadedfrom the web.

A vignetting function can alternatively be computedfrom image sequences with overlapping views of an arbi-trary static scene [5, 9, 4]. In this approach, point corre-spondences are first determined in the overlapping imageregions. Since a given scene point has a different positionin each image, its brightness may be differently attenuatedby vignetting. From the aggregate attenuation informationfrom all correspondences, the vignetting function can be ac-curately recovered without assumptions on the scene.

These previous approaches require either a collectionof overlapping images or an image of a calibration scene.However, often in practice only a single image of an arbi-trary scene is available. The previous techniques gain in-formation for vignetting correction from pixels with equalscene radiance but differing attenuations of brightness. Fora single arbitrary input image, this information becomeschallenging to obtain, since it is difficult to identify pixelshaving the same scene radiance while differing appreciablyin vignetting attenuation.

In this paper, we show that it is possible to correct orreduce vignetting given just a single image. To maximizethe use of available information in the image, our tech-nique extracts vignetting information from both texturedand untextured regions. Large image regions appropriatefor vignetting function estimation are identified by pro-posed adaptations to segmentation methods. To counter theadverse effects of vignetting on segmentation, our method

Figure 1. Tilt angles τ and χ in the Kang-Weiss vignetting model.

iteratively re-segments the image with respect to progres-sively refined estimates of the vignetting function. Addi-tionally, spatial variations in segmentation scale are usedin a manner that enhances collection of reliable vignettingdata.

In extracting vignetting information from a given region,we take advantage of physical vignetting characteristics todiminish the influence of textures and other sources of in-tensity variation. With the joint information of disparateimage regions, we describe a method for computing the vi-gnetting function. The effectiveness of this vignetting cor-rection method is supported by experiments on a wide vari-ety of images.

2. Vignetting model

Most methods for vignetting correction use a paramet-ric vignetting model to simplify estimation and minimizethe influence of image noise. Typically used are empiricalmodels such as polynomial functions [4, 12] and hyperboliccosine functions [14]. Models based on physical consider-ations include that of Asada et al. [1], which accounts foroff-axis illumination and light path obstruction, and that ofKang and Weiss [6] which additionally incorporates scene-based tilt effects. Tilt describes intensity variations within ascene region that are caused by differences in distance fromthe camera, i.e., closer points appear brighter due to the in-verse square law of illumination. Although not intrinsic tothe imaging system, the intensity attenuation effects causedby tilt must be accounted for in single-image vignetting es-timation.

Besides having physically meaningful parameters, animportant property of physical models is that their highlystructured and constrained form facilitates estimation incases where data is sparse and/or noisy. In this work, we usean extension of the Kang-Weiss model, originally designedfor a single planar surface of constant albedo, to multiplesurfaces of possibly different color. Additionally, we gen-eralize its linear model of geometric vignetting to a polyno-mial form.

2.1. Kang-Weiss model

We consider an image with zero skew, an aspect ratio of1, and principal point at the image center with image coor-dinates (u, v)=(0, 0). In the Kang-Weiss vignetting model[6], brightness ratios are described in terms of an off-axisillumination factor A, a geometric factor G, and a tilt factorT . For a pixel i at (ui, vi) with distance ri from the imagecenter, the vignetting function ϕ is expressed as

ϕi = AiGiTi = ϑriTi for i = 1 · · ·N, (1)

where

Ai =1

(1 + (ri/f)2)2,

Gi = (1 − α1ri),

ϑri= AiGi,

Ti = cos τ

(1 +

tan τ

f(ui sin χ − vi cos χ)

)3

. (2)

N is the number of pixels in the image, f is the effectivefocal length of the camera, and α1 represents a coefficientin the geometric vignetting factor. The tilt parameters χ, τrespectively describe the rotation angle of a planar scenesurface around an axis parallel to the optical axis, and therotation angle around the x-axis of this rotated plane, asillustrated in Fig. 1.

The model ϕ in Eq. (1) can be decomposed into theglobal vignetting function ϑ of the camera and the naturalattenuation caused by local tilt effects T in the scene. Notethat ϑ is rotationally symmetric; thus, it can be specified as a1D function of the radial distance ri from the image center.

2.2. Extended vignetting model

In an arbitrary input image, numerous regions with dif-ferent local tilt factors may exist. To account for multiplesurfaces in an image, we present an extension of the Kang-Weiss model in which different image regions can have dif-ferent tilt angles. The tilt factor of Eq. (2) is modified to

Ti = cos τsi

(1 +

tan τsi

f(ui sin χsi

− vi cos χsi))3

, (3)

where si indexes the region containing pixel i.We also extend the linear geometric factor to a more gen-

eral polynomial form:

Gi = (1 − α1ri − · · · − αprpi ), (4)

where p represents a polynomial order that can be arbitrarilyset according to a desired precision. This generalized rep-resentation provides a closer fit to the geometric vignettingeffects that we have observed in practice. In contrast to us-ing a polynomial as the overall vignetting model, represent-ing only the geometric component by a polynomial allowsthe overall model to explicitly account for local tilt effectsand global off-axis illumination.

Figure 3. Overview of vignetting function estimation.

Figure 2. Vignetting over multiple regions. Top row: without andwith vignetting for a single uniform region. Bottom row: withoutand with vignetting for multiple regions.

2.3. Vignetting energy function

Let the scene radiance Is of a region s be expressed byits ratio λs to the scene radiance I0 of the center pixel, i.e.,Is = λsI0. Given an image with M regions of differentscene radiance, we formulate the vignetting solution as theminimization of the following energy function:

E =M∑

s=1

Ns∑i=1

wi(λsI0Tiϑri− zi)2, (5)

where i indexes the Ns pixels in region s, zi is the pixelvalue in the vignetted image, and wi is a weight assigned topixel i. In color images, z represents an RGB vector. Forease of explanation, we express z in this paper as a singlecolor channel, and overall energies are averaged from sepa-rate color components.

In this energy function, the parameters to be estimatedare the focal length f in the off-axis component, the α coef-ficients of the geometric factor, the tilt angles τs and χs, thescene radiance of the center pixel I0, and the radiance ra-tio λs of each region. In processing multiple image regionsas illustrated in Fig. 2, minimization of this energy func-tion can intuitively be viewed as simultaneously solving forlocal region parameters Is, τs and χs that give a smoothalignment of vignetting attenuations between regions, while

optimizing the underlying global vignetting parameters f ,α1, · · · , αp.

With the estimated parameters, the vignetting correctedimage is then given by zi/ϑri

. We note that the estimatedlocal tilt factors may contain other effects that can appearsimilar to tilt, such as non-uniform illumination or shad-ing. In the vignetting corrected image, these tilt and tilt-likefactors are all retained so as not to produce an unnatural-looking result. Only the attenuation attributed to the imag-ing system itself (off-axis illumination and geometric fac-tors) is corrected.

In this formulation, the scene is assumed to contain somepiecewise planar Lambertian surfaces that are uniformly il-luminated and occupy significant portions of the image. Al-though typical scenes are considerably more complex thanuniform planar surfaces, we will later describe how vi-gnetting data in an image can be separated from other in-tensity variations such as texture, and how the weights ware set to enable robust use of this energy function.

3. Algorithm overview

The high-level flow of our algorithm is illustrated inFig. 3. In each iteration through the procedure, the image isfirst segmented at a coarse scale, and for each region a relia-bility measure of the region data for vignetting estimation iscomputed. For regions that exhibit greater consistency withphysical vignetting characteristics and with other regions, ahigher reliability weight is assigned. Low weights may indi-cate regions with multiple distinct surfaces, so these regionsare recursively segmented at incrementally finer scales un-til weights of the smaller regions exceed a threshold or re-gions becomes negligible in size. With this segmentationapproach, the segmentation scale varies spatially in a man-ner that facilitates collection of vignetting data.

After spatially adaptive segmentation, regions with highreliability weights are used to estimate the vignetting modelparameters. Since the preceding segmentations may be cor-rupted by the presence of vignetting, the subsequent itera-tion of the procedure re-computes segmentation boundariesfrom an image corrected using the currently estimated vi-gnetting model. Better segmentation results lead to im-

proved vignetting estimates, and these iterations are re-peated until the estimates converge.

The major components of this algorithm are described inthe following sections.

4. Vignetting-based image segmentation

To obtain information for vignetting estimation, pixelshaving the same scene radiance need to be identified in theinput image. Our method addresses this problem with pro-posed adaptations to existing segmentation methods. To fa-cilitate the location of reliable vignetting data, segmentationscales are spatially varied over the image, and the adverseeffects of vignetting on segmentation are progressively re-duced as the vignetting function estimate is refined.

4.1. Spatial variations in scale

Sets of pixels with the same scene radiance provide morevaluable information if they span a broader range of vi-gnetting attenuations. In the context of segmentation, largerregions are therefore preferable. While relatively large re-gions can be obtained with a coarse segmentation scale,many of these regions may be unreliable for vignetting esti-mation since they may contain multiple surfaces or includeareas with non-uniform illumination. In an effort to gainuseful data from an unreliable region, our method recur-sively segments it into smaller regions that potentially con-sist of better data for vignetting estimation. This recursivesegmentation proceeds until regions have a high reliabilityweight or become of negligible size according to a thresh-old of 225 pixels used in our implementation. Regions ofvery small size generally contain insignificant changes invignetting attenuation, and the inclusion of such regionswould bias the optimization process.

In the recursive segmentation procedure, incrementallyfiner scales of segmentation are used. For methods such asmean shift [3] and region competition [13], segmentationscale is essentially controlled by a parameter on variationwithin each feature class, where a feature may simply bepixel intensity or color. With such approaches, a finer parti-tioning of a low-weight region can be obtained by segment-ing the region with a decreased parameter value. In othertechniques such as graph cuts [15] and Blobworld [2], thedegree of segmentation is set according to a given numberof feature classes in an image. There exist various ways toset the number of classes, including user specification, dataclustering, and minimum description length criteria [11].For recursive segmentation, since each region belongs toa certain class, a finer partitioning of the region can be ob-tained by segmenting it with the number of feature classesspecified as two.

With this general adaptation, segmentation scale variesover an image in a manner designed to maximize the qual-

ity of vignetting data. In our implementation, we employgraph cut segmentation [15] with per-pixel feature vectorscomposed of six color/texture attributes. The color compo-nents are the RGB values, and the local texture descriptorsare the polarity, anisotropy and normalized texture contrastdescribed in [2].

4.2. Accounting for vignetting

Two pixels of the same scene radiance may exhibit sig-nificantly different image intensities due to variations in vi-gnetting attenuation. In segmentation, a consequence of thisvignetting is that a homogeneous scene area may be dividedinto separate image regions. Vignetting may also result inheterogeneous image areas being segmented together dueto lower contrasts at greater radial distances. For better sta-bility in vignetting estimation, the effects of vignetting onsegmentation should be minimized.

To address vignetting effects in segmentation, after eachiteration through the procedure in Fig. 3, the estimated vi-gnetting function is accounted for in segmentations duringthe subsequent iteration. Specifically, the vignetting cor-rected image computed with the currently estimated param-eters is used in place of the original input image in deter-mining segmentation boundaries. The corrected image isused only for segmentation purposes, and the colors in theoriginal image are still used for vignetting estimation.

As the segmentations improve from reduced vignettingeffects, the estimated vignetting function also is progres-sively refined. This process is repeated until the differencebetween vignetting functions in consecutive iterations fallsbelow a prescribed threshold, where the difference is mea-sured as

∆ϑ =1k

∑r

||ϑr(t) − ϑr(t − 1)||. (6)

ϑ(t) represents the global vignetting function at iteration t,and radial distances r are sampled at k uniform intervals,where k = 100 in our implementation.

5. Region weighting

To guide the vignetting-based segmentation process andpromote robust vignetting estimation, the reliability of datain each image region is evaluated and used as a regionweight. A region is considered to be reliable if it ex-hibits consistency with physical vignetting characteristicsand conforms to vignetting observed elsewhere in the im-age.

Initially, no vignetting estimates are known, so reliabilityis measured in the first iteration of the algorithm accordingto how closely the region data can be represented by ourphysically-based vignetting model. For a given region, anestimate ϑ′ of the vignetting function is computed similarly

(a) (b) (c) (d) (e)

(f) (g)

Figure 4. Effects of vignetting compensation in segmentation. (a) Original image; (b) Vignetting correction with segmentation that doesnot account for vignetting; (c) Segmentation without accounting for vignetting; (d) Vignetting correction with segmentation that accountsfor vignetting; (e) Segmentation that accounts for vignetting; (f) Estimated vignetting functions after each iteration in comparison to theground truth; (g) Intensity profile before (red) and after (blue) correction, shown for the image row that passes through the image center.

to the technique described in Section 6, and the weight forregion s is computed as

ws =1

Nsexp

{−

Ns∑i=1

||ϑ′ri− zi

λsI0Ti||}

. (7)

Each pixel is assigned the weight of its region.The presence of texture in a region does not preclude it

from having a high weight. In contrast to textures whichtypically exhibit high frequency variations, vignetting is alow frequency phenomenon with a wavelength on the orderof the image width. This difference in frequency charac-teristics allows vignetting effects to be discerned in manytextured regions.

At the end of each iteration, an estimate of the vignettingfunction is determined and used as ϑ′ in the following itera-tion. As the vignetting parameters are progressively refined,computed weights will more closely reflect the quality ofregion data. In cases where the texture or shading in a re-gion coincidentally approximates the characteristics of vi-gnetting, it will be assigned a low weight if it is inconsistentwith the vignetting observed in other parts of the image.

6. Vignetting estimation

For a collection of segmented regions, the many un-known parameters create a complicated solution space. Tosimplify optimization, we use a stepwise method for param-eter initialization prior to estimating the vignetting function.

In the first step, initial values of relative scene radiancesλs are determined for each region without consideration ofvignetting and tilt parameters. For pixels i and j at the sameradius r but from different regions, their vignetting attenua-tion should be equal, so their image values zi and zj shoulddiffer only in scene radiance. Based on this property, rela-tive scene radiance values are initialized by minimizing thefunction

E1 =∑

r

∑ri,rj=r;si �=sj

wiwj

(zi

λsi

− zj

λsj

)2

.

The λs values are solved in the least squares sense by sin-gular value decomposition (SVD) on a system of equations√

wiwj( zi

λsi− zj

λsj) = 0 where 1

λsiand 1

λsjare unknowns.

To expedite minimization of this function, a set of pixels at agiven radius and within the same region may be representedby a single pixel with the average color of the set.

With the initial values of λs, the second step initializesthe parameters f , I0, and α1, ..., αp, where p is the polyno-mial order used in the geometric factor of Eq. 4. Ignoringlocal tilt factors, this is computed with the energy function

E2 =M∑

s=1

Ns∑i=1

wi(λsI0ϑri− zi)2. (8)

This function is iteratively solved by incrementally increas-ing the polynomial order from k = 1 to k = p, and using thepreviously computed polynomial coefficients α1, ..., αk−1

(a)

(b) (c)

(d) (e)

(f) (g)

Figure 5. Recursive segmentation on low-weight regions. (a)Original image; (b) Regions prior to recursive segmentation; (c)Regions after recursive segmentation of one region; (d) Regionweights of (b), where higher intensity indicates a higher weight;(e) Region weights of (c); (f) Vignetting correction result usingregion information of (b); (g) Vignetting correction result usingregion information of (c).

as initializations. In our implementation, we use a polyno-mial order of p = 4.

In the third step, the local tilt parameters τs, χs are esti-mated by optimizing the energy function in Eq. 5 with theother parameters fixed to their initialization values. Afterthis initialization stage, all the parameters are jointly opti-mized in Eq. 5 to finally estimate the vignetting function.The optimizations of Eq. 5 and Eq. 8 are computed usingthe Levenberg-Marquardt algorithm [10].

7. Results

Our algorithm was evaluated on images captured with aCanon G3, Canon EOS 20D, and a Nikon E775. To ob-tain a linear camera response, the single-image radiometriccalibration method of [8] can be applied as a preprocessingstep to our algorithm. Ground truth vignetting functions ofthe cameras at different focal lengths were computed frommultiple images of a distant white surface under approxi-

(a) (b)

(c) (d)

Figure 6. Tilt effects in vignetting estimation. (a) Original imagewith vignetting and tilt; (b) Image corrected for only vignettingusing the proposed method; (c) Tilt image, where brighter areasindicate more distant points on a surface; (d) Estimated attenuationfunction with both vignetting and tilt.

Image (a) (b) (c) (d) (e) (f)Error 1.373 2.988 1.812 2.327 2.592 0.973

Image (g) (h) (i) (j) (k) (l)Error 2.368 3.176 0.823 2.782 2.501 1.473

Table 1. Error(×10−3

)in estimated vignetting function for im-

ages in Fig. 7.

mately uniform illumination. A distant surface was used tominimize tilt effects, but generally a distant surface does notfully cover the image plane. We captured multiple imageswith camera translation such that each image pixel viewsthe surface in at least one view. The image fragments of thewhite surface were joined and blended to obtain an accuratecalibration image.

We first examine the effects of the proposed segmenta-tion adaptations. Accounting for vignetting in segmenta-tion leads to progressive improvements in the estimated vi-gnetting function as exemplified in Fig. 4. The correctionand segmentation results in (b) and (c) without vignettingcompensation are equivalent to those after a single passthrough the overall procedure shown in Fig. 3. With ad-ditional iterations, the enhanced segmentations lead to vi-gnetting estimates that trend towards the ground truth.

The effect of recursive segmentation on a given region isillustrated in Fig. 5. Further segmentation of a low-weightregion can produce sub-regions of higher weight. With thisimprovement in data quality, a more accurate vignettingfunction can be estimated.

While the goal of this work is to estimate and correctfor the global vignetting function of the camera, tilt ef-fects computed in vignetting estimation as shown in Fig. 6could potentially provide some geometric information of

the scene. It should be noted though that estimated tilt val-ues are only accurate for reliable regions with high weights.

Some vignetting correction results of our technique arepresented in Fig. 7, along with the vignetting-based seg-mentation regions and their weights. Errors computed sim-ilarly to Eq. 6 between the estimated vignetting functionsand the ground truth functions are listed in Table 1. Whilesome slight vignetting artifacts may be visible under closeexamination, the correction quality is reasonable especiallyconsidering that only a single arbitrary input image is pro-cessed. For some indoor images, the amount of reliabledata can possibly be low due to greater illumination non-uniformity. Images with poor data quality could potentiallybe identified within our method by examination of regionweights, and indicated to the user.

In Fig. 8, we show the application of our method to im-age mosaicing. Even though vignetting correction was per-formed independently on each image of the sequence, a rea-sonable mosaicing result was still obtained. In cases whereoverlapping images are available, joint consideration of thevignetting data among all images in the sequence wouldlikely lead to better results. In contrast to previous workson image mosaicing [5, 9, 4], our proposed method can alsojointly process data from images containing completely dif-ferent content if they are captured by the same camera underthe same camera setting.

8. Conclusion

In this paper, we introduced a method for vignetting cor-rection using only the information available in a single ar-bitrary image. Adaptations to general segmentation tech-niques are presented for locating regions with reliable vi-gnetting data. Within an image region, the proposed methodtakes advantage of frequency characteristics and physicalproperties of vignetting to distinguish it from other sourcesof intensity variation. Experimental results demonstrate ef-fective vignetting correction on a broad range of images.

Accurate correction results are generally obtained de-spite many regions having non-planar geometry and non-uniform illumination. The detrimental effects of non-planargeometry are reduced when distance variations of surfacepoints from the camera are small in comparison to the dis-tance of the surface itself, since variations in scene radi-ances become negligible. In many instances, the effectsof non-uniform illumination appear similar to tilt, suchthat its effects on image intensity are incorporated intothe estimated tilt factor. Low frequency vignetting effectsalso remain distinct when geometry and illumination ex-hibit texture-like high-frequency variations, such as amongleaves on a tree. As a result, reliable vignetting data oftenexists even in image areas with significant geometry and il-lumination variation.

Directions for future work include joint estimation of

Figure 8. The image mosaic on top exhibits obvious vignettingeffects. Below, the same sequence after vignetting is correctedseparately in each image using our method. No image blendinghas been applied.

camera parameters such as principal point, aspect ratio, andskew, in addition to the vignetting function. In our cur-rent method, these camera parameters are assumed to beknown from prior geometric calibration, but could poten-tially be recovered from vignetting information. Anotherinteresting topic for future investigation is the examinationof data in the RGB channels for region weighting, sincevignetting should attenuate RGB values in a similar way,while other causes of region variation may not affect thechannels equally.

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