GRAYTONE IMAGE METAMORPHOSIS USING 3D

INTERPOLATION FUNCTION

Marcin Iwanowski

Institute of Control and Industrial Electronics

Warsaw University of Technology

ul.Koszykowa 75, 00-662 Warszawa Poland

Keywords:

Image metamorphosis, Morphing, Morphological image processing.

Abstract:

Image metamorphosis process produces deformation sequence which transforms one input image into another

one. The method described in the paper applies morphological approach to achieve this goal. It is based on

morphological interpolation which makes use of the interpolation functions produced from geodesic distance

functions. The described method allows applying this approach to graytone images via its 3D umbra. It

produces 3D interpolation function. Its thresholding at given level followed by inverse umbra transform allows

obtaining frame of the interpolated sequence.

1 INTRODUCTION

This paper describes a method for graytone image

metamorphosis (also called morphing) by means of

binary morphological interpolation (Meyer, 1996;

Serra, 1996; Beucher, 1998; Iwanowski and Serra,

2000; Iwanowski, 2000). Interpolation between two

images consists in generating a sequence of inter-

mediary images, content of which is transformed

from the content of ﬁrst input image (initial) into the

content of the second one (ﬁnal). A method pro-

posed in the paper is based on the method introduced

in (Meyer, 1996), where a content of the interpolated

image is obtained by a thresholding of the interpola-

tion function, which is computed from the morpho-

logical geodesic distance functions. Using increas-

ing threshold values results in production of succes-

sive images which creates the interpolation sequence

converting one input image into another. The novelty

of the proposed method is an application of binary

interpolation into graytone images. This is achieved

through an umbra transform, which converts 2D gray-

tone image into 3D binary one. The graytone im-

ages which are morphed are ﬁrst converted into their

umbras. To interpolate between umbras the method

based on interpolation function in 3D is applied. Con-

trary to methods that already appeared in the litera-

ture (Meyer, 1996; Iwanowski, 2000), the proposed

one interpolates between 3D binary images - umbras

of input graytone images. The interpolated 3D shape

is transformed back into graytone image. By pro-

ducing interpolated image at increasing levels the in-

terpolation sequence (or: morphing sequence) is ob-

tained.

The methods of the morphological interpola-

tion consisting in creation of the intermediary two-

dimensional images between two given ones (’inter-

frame’ interpolation) are developed since 1994. Two

principal approaches were introduced. The ﬁrst one,

based on the morphological median, was presented

in (Beucher, 1998; Serra, 1996). It is a ﬂexible ap-

proach, which is applicable to any kind of image:

binary, mosaic and graytone. In (Iwanowski and

Serra, 1999) the area of applications of this method

was extended into color images, by using a lexico-

graphic ordering of colors in the comparative color

space. Another approach is represented by the in-

terpolation function method introduced in (Meyer,

1996). This method is based on the function which

describes the relative distance between the objects

and can be applied to binary and mosaic images.

The results of further research (Iwanowski and Serra,

2000) allowed combining the morphological interpo-

lation with afﬁne transform. A different approach to

morphological interpolation was presented in (Soille,

1991), which can be called - contrary to previously

described - an ’intraframe’ interpolation. It deals with

a single incomplete image and it reconstructs the im-

5

Iwanowski M. (2009).

GRAYTONE IMAGE METAMORPHOSIS USING 3D INTERPOLATION FUNCTION.

In Proceedings of the Fourth Inter national Conference on Computer Vision Theory and Applications, pages 5-9

DOI: 10.5220/0001754400050009

Copyright

c

SciTePress

age surface starting from the contour lines. This ap-

proach makes use of the geodesic distance function

obtained by the geodesic propagation.

The morphological approach is automatic in such

a sense that it does not require control points, as clas-

sic morphing methods does. Lack of input parameters

places this method together with well-known cross-

dissolving (Wolberg, 1990). It produces however to-

tally different transformation between images. Cross-

dissolving produces a kind of blending while morpho-

logically interpolated sequence contains the change of

shape of objects on the images.

The paper is organized as follows. Section 2 de-

scribes the classic approach to morphological inter-

polation using the interpolation function. Section 3

presents the proposed approach - the way of apply-

ing this interpolation into graytone images. Section 4

shows some results, and ﬁnally Section 5 concludes

the paper.

2 MORPHOLOGICAL

INTERPOLATION FUNCTION

This section recalls the principles of binary inter-

polation using the distance function (Meyer, 1996;

Iwanowski, 2000).

2.1 Binary Object

Binary image i.e. image of pixel values equal either 0

or 1, is usually deﬁned in one of two ways. According

to the ﬁrst one binary image is a mapping from deﬁ-

nition domain D into {0, 1}. According to the second

one, binary image X is a set of pixels of value 1 (fore-

ground pixels). The complement of this set (X

C

) is

referred to as image background. Image X can consist

of many connected components i.e. subsets of image

pixels such that any two pixels belonging to the same

subset can be connected by a path of pixels of value 1

entirely included in this subset. The single connected

component of binary image will be referred to as ob-

ject. The metamorphosis using interpolation function

allows to morph an object on the initial image another

object on the ﬁnal one.

2.2 The Interpolator

An interpolator provides a transformation which pro-

duces an interpolated object. It is a function of three

principal arguments: two input objects (initial and ﬁ-

nal) and an interpolation level α. An interpolation

level is a real number α such that 0 ≤ α ≤ 1. In

this paper the interpolator is denoted as: Int

Q

P

(α),

where Q represents the initial binary object, P - the

ﬁnal one. Shapes of interpolated objects are turning

from a shape of the object Q to shape of the object

P. For α = 0, the interpolated image is equal to the

initial one (Int

Q

P

(0) = Q); for α = 1 - to the ﬁnal im-

age (Int

Q

P

(1) = P). A sequence of interpolated images

produced for increasing values of α is an interpola-

tion sequence.

2.3 Interpolation Method

The way of deﬁning the interpolator depends on the

mutual relation between input objects. First, the case

of nested objects will be considered where objects lo-

cated on the initial image are included in appropriated

objects on the ﬁnal image. Later on, the general case

of any two images will be described.

Let X and Y be nested objects (X ⊂ Y). The in-

terpolation function proposed in (Meyer, 1996) is de-

ﬁned as:

int

Y

(X)[p] =

d

Y

(X)[p]

d

Y

(X)[p] + d

X

C

(Y

C

)[p]

, (1)

where X

C

and Y

C

stand for the complements of

binary images X and Y respectively. d

A

(B) stands for

the geodesic distance function describing the distance

to B inside A (B ⊂ A). Geodesic distance is deﬁned

as the length of the shortest path connecting given

pixel in Y \ X with the set X. In digital grid various

ways of computing the distance function are in com-

mon use. The simplest way is propagation in either

4- or 8-connectivity in 2D and 6,18 or 26 connectiv-

ity in 3D. This however is not an Euclidean distance.

The latter could be obtained using specialized algo-

rithms (Vincent, 1991).

The interpolator based on the Eq. 1 is deﬁned as:

Int

X

Y

(α) = T

[α]

(int

Y

(X)), (2)

where T

[α]

stands for the thresholding operator at

level α which sets 1 for graylevels below threshold α,

and 0 otherwise.

The case of two input objects which are not nested

(but which have a non-empty intersection) is split into

two interpolations between nested sets.

Let P and Q be the initial and ﬁnal objects - nested

or intersected. A ﬁnal result of the interpolation at

given level α is obtained as an sum of two interpola-

tions of nested objects:

Int

Q

P

(α) = Int

P∩Q

P

(α) ∪ Int

P∩Q

Q

(1− α). (3)

The interpolator in the general case (deﬁned by

the Eq. 3) is based on two interpolators. Each of

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

6

them transforms the input image into the intersection

of both input images. This approach requires two in-

terpolation functions. In order to speed up with the

computations, the single function can be computed.

This case is described in section 3.

The interpolated object at level α = 0.5 is located

in the midway between both input objects. This object

is also referred to as morphological median (Beucher,

1998) deﬁned as:

M = ∪

λ≥0

{((P∩ Q) ⊕ λB) ∩ ((P∪ Q) ⊖ λB)}, (4)

where ⊕ stands for morphological dilation and ⊖

- for erosion.

3 METAMORPHOSIS OF

GRAYTONE IMAGES

3.1 Umbra

Let f : D → V be two-dimensional graytone image

where: D = {0, 1, ..., x

max

− 1} × {0, 1, ..., y

max

− 1}

is the set of pixels coordinates - image deﬁnition do-

main and V = {0, 1, ..., g

max

− 1} is set of possible

graytones. x

max

and y

max

are sizes of graytone image

and g

max

is the highest possible graytone value of this

image.

Graytone image can be transformed into 3D um-

bra, which is 3D binary image - a ’relief’ of a terrain

whose map is the input 2D graytone image. The um-

bra transformation is deﬁned as follows:

U[ f ] =

(p, q) ∈ D × V : f(p) ≥ q

. (5)

The above equation describes the transformation

which converts 2D graytone image into 3D umbra.

The inverse transformation, which transforms umbra

back into graytone image, is denoted as follows:

f = U

−1

[X] ⇔ X = U[ f ]. (6)

3.2 Interpolation Function for Graytone

Images

Let f

A

and f

B

be to input graytone images. In the ﬁrst

step their umbra are computed:

X

A

= U[ f

A

] ; X

B

= U[ f

B

]. (7)

In order to produce the interpolation function,

their sum and intersection have to be computed, re-

spectively:

X

∩

= X

A

∩ X

B

= U[ f

A

∧ f

B

], (8)

X

∪

= X

A

∪ X

B

= U[ f

A

∨ f

B

], (9)

where ∧ and ∨ stand for the point-wise minimum

and maximum of two graytone images, respectively.

These two 3D binary objects are always nested: X

∩

⊂

X

∪

. Owing to this, the geodesic distance functions in

X

∪

\ X

∩

can be computed (for any input images) and

combined together to obtain the interpolation function

according to the Eq. 1.

To simplify the calculations of the interpolations

function, single function can be computed instead of

two as stated in the Eq. 3. Let deﬁne the interpola-

tion function between intersection X

∩

and union X

∪

denoted by Int

X

∩

X

∪

. The following relation holds:

Int

X

∩

X

A

(α) = Int

X

∩

X

∪

(α) ∩ X

A

, (10)

Int

X

∩

X

B

(α) = Int

X

∩

X

∪

(α) ∩ X

B

. (11)

The result of thresholding of the interpolation

function have ﬁnally to be converted from the um-

bra form into graytone image using U

−1

transform

(Eq. 6). The complete interpolator can be thus for-

mulated as follows:

f

α

= U

−1

h

Int

X

∩

X

∪

(α) ∩ X

A

∪ (Int

X

∩

X

∪

(1− α) ∩ X

B

)

i

.

(12)

Having in mind earlier considerations the above

equation can be re-written as follows:

f

α

=

U

−1

h

Int

X

∩

X

∪

(α)

i

∧ f

A

∨

U

−1

h

Int

X

∩

X

∪

(1− α)

i

∧ f

B

,

(13)

where f

α

stands for the morphologically interpo-

lated graytone image at level α.

Similiarily to the binary case, in the graytone one,

the image f

0.5

is referred to morphological median of

graytone images (Beucher, 1998).

By producing the interpolated images at increas-

ing levels, the sequence of frames is computed - the

interpolation sequence.

4 RESULTS

Figure 1 shows the interpolation sequence produced

using the proposed method starting from two in-

put test images containing rectangles of different

graytones. The sequence consists of 6 interpolated

GRAYTONE IMAGE METAMORPHOSIS USING 3D INTERPOLATION FUNCTION

7

Figure 1: Morphological interpolation of test images.

Figure 2: Cross-dissolving of test images.

frames, generated for equidistanced interpolation lev-

els: 0.14;0.18;0.43;0.57;0.72;0.86. The shapes of

objects on the sequence are changing from the ini-

tial to the ﬁnal image. In the same time the gray-

value assigned to every rectangle is varying accord-

ingly. Comparing to the alternative way of producing

the image metamorphosis - cross-dissolving shown in

Fig. 2, the morphological interpolation gives a real

change of shape instead of linear combination of pix-

els as cross-dissolving method generates. The umbras

of graytone images are shown in Fig. 3. Pictures 3(a)

and (b) show input images, picture (c) shows the inter-

section of both, (d) - the union. Picture 3(e) presents

the umbra of interpolated image at level α = 0.5

(which is equivalent to morphological median of im-

(a) (b)

(c) (d)

(e)

Figure 3: Umbra of test images: (a),(b) input images, (c) -

their intersection, (d) - sum, (e) median image.

Figure 4: Cross-section of interpolation function for various

values of ’z’-coordinate: (a) z = 70; (b) z = 110; (c) z = 150;

(d) z = 190.

ages shown in (a) and (b)).

The result of morphological interpolation of

real images are shown in Fig. 5. The ’lena’ image

is morphed into ’baboon’. This time the whole

sequence (including input images) consist of 12

images, which was produced for interpolation levels:

0;0.09;0.18;0.27;0.36;0.45;0.54;0.63;0.72;0.81;

0.9;1. The cross-sections of 3D interpolation func-

tion are shown in Fig. 4. These cross-sections was

produced by cutting the 3D interpolation function

at various levels along the ’z’-coordinate. White

VISAPP 2009 - International Conference on Computer Vision Theory and Applications

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Figure 5: First morphing sequence obtained by morpholog-

ical interpolation.

Figure 6: Second morphing sequence.

regions belongs to the intersection of both umbras

(where interpolation function is not computed), black

regions - to the regions which are above the union

of them (here again the interpolation function is not

computed), colors refer to values of the interpolation

function (between 0 and 1).

Another example of interpolation sequence is

shown in Fig. 6. This sequence counts 16 frames pro-

duced, as in the prvious case, for equidistanced con-

secutive values of interpolation levels.

5 CONCLUSIONS

In the paper a method for transforming one graytone

image into another was proposed. The method is

based on the morphological interpolation using the

interpolation function. 2D graytone image image is

transformed into 3D binary image (its umbra) and in-

terpolated morphologically. The interpolated 3D ob-

ject is transformed back into graytone image. The

method can be applied to produce image metamor-

phosis for visual special effects. It is fully automatic,

the only parameter is the number of frames of the out-

put morphing sequence. Comparing with the cross-

dissolving the morphological interpolation produces

a real change of shapes of objects instead of blending

produced by the cross-dissolving. Further research

plans includes the application of the proposed ap-

proach to color images.

REFERENCES

Beucher, S. (1998). Interpolation of sets, of partitions and

of functions. In Mathematical morphology and its ap-

plication to image and signal processing. Kluwer.

Iwanowski, M. (2000). Application of mathematical mor-

phology to interpolation of digital images. Warsaw

University of Technology, School of Mines of Paris.

Iwanowski, M. and Serra, J. (1999). Morphological inter-

polation and color images. In Proc.of: 10-th Interna-

tional Conference on Image Analysis and Processing

ICIAP99. IEEE.

Iwanowski, M. and Serra, J. (2000). Morphological-afﬁne

object deformation. In Mathematical morphology

and its application to image and signal processing.

Kluwer.

Meyer, F. (1996). Morphological interpolation method for

mosaic images. In Mathematical morphology and its

application to image and signal processing. Kluwer.

Serra, J. (1996). Hausdorff distance and interpolations. In

Mathematical morphology and its application to im-

age and signal processing. Kluwer.

Soille, P. (1991). Spatial distributions from contour lines:

an efﬁcient methodology based on distance transfor-

mations. J. of Visual Communication and Image Rep-

resentation, 2.

Vincent, L. (1991). Exact euclidean distance function by

chain propagations. In Proc. IEEE Computer Vision

and Pattern Recognition. IEEE.

Wolberg, G. (1990). Digital Image Warping. IEEE Com-

puter Society Press.

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