Multidimensional Dataflow Graphs - Signal Processing Systems - page 1134

Fig. 7 Border processing by
pixel mirroring. Each
rectangle corresponds to a
pixel. Dashed arrows
represent the pixel mirroring

Mirror
axes

x=0

y=1
x=0

x=1

y=0 y=0

￿ o ( x , y , t ) is the pixel at the coordinates ( x , y ) within the output image at time t ,

￿ i ( x , y , t ) is the pixel at the coordinates ( x , y ) within the input image at time t .

Equation ( 2 ) defines the border processing as explained later on. Consider first the
case where i ( x , y , t )= i ( x , y , t ) and assume for a moment that β 1 = β 2 and α 1 = α 2 ,

then obviously each output pixel o ( x , y , t ) is computed from ( 2 · α 1 + 1 ) · ( 2 · β 1 + 1 )

pixels centered at the input pixel i Hence, only a local subset of

the input image is required to compute an output pixel. In more detail, the filter
represents a sliding window algorithm, where a window samples the input image
i

( 2 · x , 2 · y , t ) .

is
generated. For the example given in Eq. ( 1 ) , the factor of two causes that the sliding
window moves by two pixels in horizontal and vertical direction.
While this operation is straightforward when the complete window is situated
within the input image i

( x , y , t ) , and for each window position, a corresponding output pixel o ( x , y , t )

( x , y , t ) , computation of output pixel o ( x = 0 , y = 0 , t ) for

instance depends on the input pixels i .Pixels

with negative coordinates are situated outside of the input image and are thus not
defined. Hence a border processing is required as expressed in Eq. ( 2 ) . One might
for instance set all pixels situated outside the image to a constant value. In this case,
Eq. ( 2 ) gets

( x , y , t ) , − α ≤ x ≤ α ,− β ≤ y ≤ β

i

( x , y , t ) if 0 ≤ x < w i , 0 ≤ y < h i

i

( , , )=

x y t

ι = const otherwise

Alternatively, often a mirroring of the pixels at the image border is performed as
expressed by the following equation and graphically depicted in Fig. 2 (Fig. 7 ) :

i

( , , )=

x y t

⎛ ⎧

⎨

⎧
⎨

⎞

x if 0 ≤ x < w i y if 0 ≤ y < y i

⎝ ⎠

i , , t

− x − 1 if x < 0 − y − 1 if y < 0

⎩ ⎩

2 · w i − 1 − x if x ≥ w i 2 · h i − 1 − y if y ≥ h i