«Abstract. Image stitching is used to combine several individual images having some overlap into a composite image. The quality of image stitching is ...»
Seamless Image Stitching in the Gradient Domain
Anat Levin, Assaf Zomet, Shmuel Peleg, and Yair Weiss
School of Computer Science and Engineering
The Hebrew University of Jerusalem
91904, Jerusalem, Israel
alevin,peleg,yweiss @cs.huji.ac.il, firstname.lastname@example.org
Abstract. Image stitching is used to combine several individual images having
some overlap into a composite image. The quality of image stitching is measured
by the similarity of the stitched image to each of the input images, and by the visibility of the seam between the stitched images.
In order to deﬁne and get the best possible stitching, we introduce several formal cost functions for the evaluation of the quality of stitching. In these cost functions, the similarity to the input images and the visibility of the seam are deﬁned in the gradient domain, minimizing the disturbing edges along the seam. A good image stitching will optimize these cost functions, overcoming both photometric inconsistencies and geometric misalignments between the stitched images.
This approach is demonstrated in the generation of panoramic images and in ob- ject blending. Comparisons with existing methods show the beneﬁts of optimiz- ing the measures in the gradient domain.
1 Introduction Image stitching is a common practice in the generation of panoramic images and ap- plications such as object insertion, super resolution  and texture synthesis . An example of image stitching is shown in Figure 1. Two images Á ½,Á¾ capture different portions of the same scene, with an overlap region viewed in both images. The images should be stitched to generate a mosaic image Á. A simple pasting of a left region from Á½ and a right region from Á ¾ produces visible artiﬁcial edges in the seam between the images, due to differences in camera gain, scene illumination or geometrical misalign- ments.
The aim of a stitching algorithm is to produce a visually plausible mosaic with two desirable properties: First, the mosaic should be as similar as possible to the input images, both geometrically and photometrically. Second, the seam between the stitched images should be invisible. While these requirements are widely acceptable for visual examination of a stitching result, their deﬁnition as quality criteria was either limited or implicit in previous approaches.
In this work we present several cost functions for these requirements, and deﬁne the mosaic image as their optimum. The stitching quality in the seam region is measured This research was supported (in part) by the EU under the Presence Initiative through contract IST-2001-39184 BENOGO.
Current Address: Computer Science Department, Columbia University, 500 West 120th Street, New York, NY 10027 Input image ½½
Fig. 1. Image stitching. On the left are the input images. is the overlap region. On top right is a simple pasting of the input images. On the bottom right is the result of the GIST1 algorithm.
in the gradient domain. The mosaic image should contain a minimal amount of seam artifacts, i.e. a seam should not introduce a new edge that does not appear in either Á ½ or Á¾. As image dissimilarity, the gradients of the mosaic image Á are compared with the gradients of Á½ Á¾. This reduces the effects caused by global inconsistencies between the stitched images. We call our framework GIST: Gradient-domain Image STitching.
We demonstrate this approach in panoramic mosaicing and object blending. Analytical and experimental comparisons of our approach to existing methods show the beneﬁts in working in the gradient domain, and in directly minimizing gradient artifacts.
1.1 Related Work There are two main approaches to image stitching in the literature, assuming that the images have already been aligned. Optimal seam algorithms[3, 2, 4] search for a curve in the overlap region on which the differences between Á ½ Á¾ are minimal. Then each image is copied to the corresponding side of the seam. In case the difference between Á½ Á¾ on the curve is zero, no seam gradients are produced in the mosaic image Á.
However, the seam is visible when there is no such curve, for example when there is a global intensity difference between the images. This is illustrated on the ﬁrst row of Figure 2. In addition, optimal seam methods are less appropriate when thin strips are taken from the input images, as in the case of manifold mosaicing .
The second approach minimizes seam artifacts by smoothing the transition between the images. In Feathering  or alpha blending, the mosaic image Á is a weighted combination of the input images Á ½ Á¾. The weighting coefﬁcients (alpha mask) vary as Inp. image ½½ Inp. image ½¾ Feathering Pyr. blending Opt. Seam GIST Fig. 2. Comparing stitching methods with various sources for inconsistencies between the input images. The left side of Á½ is stitched to right side of Á¾. Optimal seam methods produce a seam artifact in case of photometric inconsistencies between the images (ﬁrst row). Feathering and pyramid blending produce double edges in case of horizontal misalignments (second row). In case there is a vertical misalignments (third row), the stitching is less visible with Feathering and GIST.
a function of the distance from the seam. In pyramid blending, different frequency bands are combined with different alpha masks. Lower frequencies are mixed over a wide region, and ﬁne details are mixed in a narrow region. This produces gradual transition in lower frequencies, while reducing edge duplications in textured regions. A related approach was suggested in , where a smooth function was added to the input images to force a consistency between the images in the seam curve. In case there are misalignments between the images, these methods leave artifacts in the mosaic such as double edges, as shown in Figure 2.
In our approach we compute the mosaic image Á by an optimization process that uses image gradients. Computation in the gradient domain was recently used in compression of dynamic range, image editing , image inpainting  and separation of images to layers [12–15]. The closest work to ours was done by Perez et. al. , who suggest to edit images by manipulating their gradients. One application is object insertion, where an object is cut from an image, and inserted to a new background image. The insertion is done by optimizing over the derivatives of the inserted object, with the boundary determined by the background image. In sections 4,5 we compare our approach to .
2 GIST: Image Stitching in the Gradient Domain We describe two approaches to image stitching in the gradient domain. Section 2.1 describes GIST1, where the mosaic image is inferred directly from the derivatives of the input images. Section 2.2 describes GIST2, a two-steps approach to image stitching.
Section 2.3 compares the two approaches to each other, and with other methods.
Ô ´Â½ Â¾ Ï µ is
where Í is a uniform image, and the distance between Â½ Â¾ on :
Ô Ô ´Â½ Â¾ Ïµ Ï ´Õµ Â½ ´Õµ Â¾ ´Õµ Ô (2) Õ¾ with ¡ Ô denoting the Ô -norm.
The dissimilarity Ô between the images is deﬁned by the distance between their derivatives. A dissimilarity in the gradient domain is invariant to the mean intensity of the image. In addition it is less sensitive to smooth global differences between the input images, e.g. due to non-uniformness in the camera photometric response and due to scene shading variations. On the overlap region, the cost function Ô penalizes for derivatives which are inconsistent with any of the input images. In image locations where both Á ½ and Á¾ have low gradients, Ô penalizes for high gradient values in the mosaic image. This property is useful in eliminating false stitching edges.
The choice of norm (parameter Ô) has implications on both the optimization algorithm and the mosaic image. The minimization of Ô (Equation 1) for Ô ½ is convex, and hence efﬁcient optimization algorithms can be used. Section 3 describes a minimization scheme for ¾ by existing algorithms, and a novel fast minimization scheme for ½. The mask image Ï was either a uniform mask (for ½ ) or the Feathering mask (for ¾ ), which is linear with the signed-distance from the seam. The inﬂuence of the choice of Ô on the result image is addressed in the following sections, with the introduction of alternative stitching algorithms in the gradient domain.
2.2 GIST2: Stitching Derivative Images
A simpler approach is to stitch the derivatives of the input images:
In stage (2) above, any stitching algorithm may be used. We have experimented with Feathering, pyramid blending , and optimal seam. For the optimal seam we used ´Ýµ that minimizes the sum of absolute the algorithm in , ﬁnding the curve Ü differences in the input images. Stage (3), the optimization under ½ ¾, is described in Section 3. Unlike the GIST1 algorithm described in the previous section, we found minor differences in the result images when minimizing Ô under ½ and ¾.
Optimal seam Optimal seam on the gradients
Fig. 3. Stitching in the gradient domain. The input images appear in Figure 1, with the overlap region marked by a black rectangle. With the image domain methods (top panels) the stitching is observable. Gradient-domain methods (bottom panels) overcome global inconsistencies.
2.3 Which Method to Use ?
In the previous sections we presented several stitching methods. Since stitching results are tested visually, selecting the most appropriate method may be subject to personal taste. However, a formal analysis of properties of these methods is provided below.
Based on those properties in conjunction with the experiments in Section 4, we recommend using GIST1 under ½.
Theorem 1. Let Á½ Á¾ be two input images for a stitching algorithm, and assume there ´Ýµ, such that for each Õ ¾ ´ ´Ýµ Ýµ, Á ½ ´Õµ Á¾ ´Õµ.
Let Í be a is a curve Ü uniform image. Then the optimal seam solution Á, deﬁned below, is a global minimum of Ô ´Á Á½ Á¾ Í µ deﬁned in Eq.1, for any ¼ Ô ½.
The reader is referred to  for a proof. The theorem implies that GIST1 under ½ is as good as the optimal seam methods when a perfect seam exists. Hence the power of GIST1 under ½ to overcome geometric misalignments similarly to the optimal seam methods. The advantage of GIST1 over optimal seam methods is when there is no perfect seam, for example due to photometric inconsistencies between the input images.
This was validated in the experiments.
We also show an equivalence between GIST1 under ¾ and Feathering of derivatives (GIST2) under ¾ (Note that feathering derivatives is different from Feathering the images).
Theorem 2. Let Á½ Á¾ be two input images for a stitching algorithm, and let Ï be a Feathering mask.
Let, the overlap region of Á ½ Á¾, be the entire image (without loss of generality, as Ï ´Õµ ½ for Õ ¾ ½, and Ï ¼ for Õ ¾ ¾ ). Let Á ×Ø be the minimum
of ¾ ´Á Á½ Á¾ Ï µ deﬁned in Eq. 1. Let be the following ﬁeld:
´ÕµÖÁ½ ´Õµ · ´½ Ï ´ÕµµÖÁ¾ ´ÕµÏ Then Á ×Ø is the image with the closest gradient ﬁeld to under ¾.
The proof can be found in  as well. This provides insight into the difference between GIST1 under ½ and under ¾ : Under ¾, the algorithm tends to mix the derivatives and hence blur the texture in the overlap region. Under ½, the algorithm tends to behave similarly to the optimal seam methods, while reducing photometric inconsistencies.
For an even number of samples, the median is taken to be the average of the two middle samples.
In regions where a single image Á is used, the median is taken on the predictions of Á´Ü Ýµ given its four neighbours and the derivatives of image Á. For example, when the derivatives of image Á are 0, the algorithm performs an iterated median ﬁlter of the neighbouring pixels. In the overlap region of Á½ Á¾, the median is taken over the predictions from both images.
At every iteration, the algorithm performs a coordinate descent and improves the cost function until convergence. As the cost function is bounded by zero, the algorithm always converges.
However, although the cost function is convex, the algorithm does not always converge to the global optimum1. To improve the algorithm convergence and speed, we combined it in a multiresolution scheme using multigrid . In extensive experiments with the multi-resolution extension the algorithm always converged to the global optimum.
4 Experiments We have implemented various versions of GIST and applied them to panoramic mosaicing and object blending.
First, we compared GIST to existing image stitching techniques, which work on the image intensity domain: Feathering , Pyramid Blending , and ’optimal seam’ (Implemented as in ). The experiments (Figure 3) validated the advantage in working in the gradient for overcoming photometric inconsistencies. Second, we compared the results of GIST1 (Section 2.1), GIST2 (Section 2.2) and the method by Perez. et. al. . Results of these comparisons are shown, for example, in Figures 4,5, and analyzed in the following sections.
4.1 Stitching Panoramic Views The natural application for image stitching is the construction of panoramic pictures from multiple input pictures. Geometrical misalignments between input images are caused by lens distortions, by the presence of moving objects, and by motion parallax. Photometric inconsistencies between input images may be caused by a varying gain, by lens vignetting, by illumination changes, etc.