IntroductionThe integral image repr

Introduction
The integral image representation was introduced [10, 5] with the purpose of evaluating sums of
image values over axis aligned rectangular regions in constant time. Crow [5] applied it in graphics
(under the name of summed area table) for eﬃciently computing mipmaps [26] by ﬁltering the images
with spatially varying rectangular ﬁlters. Lewis [17] used integral images for accelerating template
matching. Viola and Jones [24] popularized it in the computer vision community with its application
in their real-time object detection framework. More recently, Bay et al. [1] took advantage of integral
images for the purpose of rapidly computing approximations of SIFT features with their Speeded Up
Robust Features (SURF).
Block-matching aims at identifying corresponding blocks (or patches) between two images. A
patch is a rectangular portion of an image. For each pixel of the ﬁrst image (which we shall call
from now on reference image) we need to compute the matching cost (denoted also patch distance)
between the patch centered at this reference pixel, and all the patches of the second image within
a given range around the reference pixel. For each patch in the reference image, the patch in the
1
http://dx.doi.org/10.5201/ipol.2014.57
Gabriele Facciolo, Nicolas Limare, Enric Meinhardt, Integral Images for Block Matching, Image Processing On Line, 4 (2014), pp. 344–
369. http://dx.doi.org/10.5201/ipol.2014.57Integral Images for Block Matching
second image yielding the minimum matching cost is called “the corresponding block”. A popular
matching cost is computed as the sum of squared diﬀerences (SSD) of pixel values within the patch.
The correspondences and matching costs provide valuable information about the similarity between the patches, which can be applied to image denoising [3], video compression [15], motion and
depth estimation [21] among others. One of the ﬁrst uses of integral image applied to block matching
appears in the work by Faugeras et al. [8]. In that work the authors describe an optimization to spare
redundant computations, that is much in the spirit of integral images. Later, Veksler [23] applied
the integral images to stereo block matching for eﬃciently computing matching costs with multiple
window sizes, and Wang et al. [25] used integral images for accelerating the patch comparison in the
NL-means denoising algorithm [3].
The naive evaluation of the matching cost for a set of M relative oﬀsets, using patches of size r
2
,
entails O(Mr
2
) operations for each patch in the reference image. Therefore, for an image with N
pixels, the naive computation of all the matching costs requires O(MNr
2
) operations. Note that the
patch size aﬀects directly the computational cost. Many techniques have been proposed to reduce
this computational load by avoiding unnecessary comparisons, for instance exploiting the regularity
of the oﬀset ﬁelds [20]. Covering all these techniques is well beyond the scope of this work. Here
we are going to describe how the integral image can be applied to compute the matching costs with
O(MN) operations. Thus, with a cost only proportional to the number of oﬀsets, regardless of the
window size. This performance gain comes at the expense of some ﬂexibility, a tradeoﬀ that will be
discussed in detail.
We start by describing the integral image computation and its evaluation in Section 2. In Section 3
we explain how to apply the integral image for block matching and comment some issues. Sections 4
and 5 conclude by highlighting the beneﬁts and limitations of the integral image applied to block
matching.
2 Algorithm for Computing Integral Images
For a discrete image i its integral image ii at the pixel (x, y) is deﬁned as the sum of the pixel values
of i above and to the left of (x, y), namely
ii(x, y) =
X
p≤x ,q≤y
i(p, q). (1)
Given the integral image, the sum of all the pixels within a rectangle aligned with the image axes
can be evaluated with four array references, regardless of the size of the rectangle. This is illustrated
in Figure 1, where the sum over the rectangle [x0, x] × [y0, y] is computed as L4 + L1 − (L2 + L3),
with L1 = ii(x0 − 1, y0 − 1), L4 = ii(x, y), L3 = ii(x0 − 1, y) and L2 = ii(x, y0 − 1).
In [24] the authors describe an algorithm for computing the integral image in one-pass over the
image by using the recurrence relation
s(x, y) = s(x, y − 1) + i(x, y)
ii(x, y) = ii(x − 1, y) + s(x, y),
where s(x, y) is the column-wise cumulated sum, and s(x, −1) = ii(−1, y) = 0. Algorithm 1 details
the implementation of this recurrence, where the image s is replaced with a scalar accumulator
storing the current row-wise sum. A performance comparison of this method with two other slower
algorithm variants is provided in Appendix A. Algorithm 2 details how to compute the sum of a
rectangular region of the image by evaluating the integral image.
345Gabriele Facciolo, Nicolas Limare, Enric Meinhardt
Figure 1: Integral image and evaluation over a rectangle. The diagram illustrates the computation of the integral image: the value of ii(x, y) is the sum of the pixels in the gray region, where
the upper left corner of the image is the origin of the image’s coordinate system. The diagram on
the right shows how to compute the sum of the samples of the discrete image on a rectangle with
integer bounds, A = [x0, x] × [y0, y] using the integral image ii. It is enough to obtain the values of
the integral image at L1 = ii(x0 − 1, y0 − 1), L4 = ii(x, y), L3 = ii(x0 − 1, y) and L2 = ii(x, y0 − 1),
and to compute L4 + L1 − (L2 + L3).
Algorithm 1: Computing Integral Image.
input: An image of size nx × ny: i
output: The integral image of size nx × ny: ii
1 ii(0, 0) ← i(0, 0);
2 for x ← 1 to nx − 1 do
3 ii(x, 0) ← ii(x − 1, 0) + i(x, 0);
4 for y ← 1 to ny − 1 do
5 s ← i(0, y); /* scalar accumulator */
6 ii(0, y) ← ii(0, y − 1) + s;
7 for x ← 1 to nx − 1 do
8 s ← s + i(x, y);
9 ii(x, y) ← ii(x, y − 1) + s;
Algorithm 2: Summing the pixel values in a rectangle of an image using the Integral Image.
input: The integral image for an image i: ii. Position, width (w) and height (h) of a
rectangle contained in the image: [x0, x0 + w − 1] × [y0, y0 + h − 1].
output: Sum of the pixel values of i in the rectangle of size w × h : S
1 S ← ii(x0 + w − 1, y0 + h − 1);
2 if x0 > 0, then S ← S − ii(x0 − 1, y0 − 1 + h);
3 if y0 > 0, then S ← S − ii(x0 − 1 + w, y0 − 1);
4 if x0 > 0 and y0 > 0, then S ← S + ii(x0 − 1, y0 − 1);
2.1 Interpretation
The simplest interpretation of the integral image comes from considering a 1D discrete signal l of
length N. The sum over any interval of the signal l (for instance from x0 to x1) rewrites as
Xx1
j=x0
l(j) =
Xx1
j=0
l(j) −
xX0−1
j=0
l(j) = ll(x1) − ll(x0 − 1), (2)
346Integral Images for Block Matching
0 5 1 2 3 4
x
0 5 1 2 3 4
x
Figure 2: Interpolating an integral image. The left image depicts a row of pixels and the right
one its associated integral line. Note, on the left image, that the samples are interpolated with nearest
neighbor and that the signal is displaced half-pixel to the right. With this convention the integral
image at an integer position coincides with the integral of the piecewise constant interpolation of the
samples (for instance the integral image at 2 accumulates the areas of the ﬁrst and second pixels).
This also implies that linearly interpolating the integral image yields the integral of the piecewise
constant interpolation of the samples.
where ll(x) =
Px
i=0
l(x) is the “integral line”. Thus given ll, any sum over an interval of l can be
computed with two lookups of ll.
It is possible to compute sums over non integer intervals by interpolating the integral line [5].
Indeed, Figure 2 illustrates a 1D discrete signal which we interpret as a piecewise constant function
and the corresponding integral line. Linearly interpolating the integral line permits to integrate the
underlying piecewise constant function over any (non integer) interval.
In two dimensions the integral image ii at any position (x, y) is the sum of all the pixels in the
rectangular region above and to the left of (x, y) as illustrated in Figure 1. The orientation of the
summed rectangle determines the “privileged orientation” for evaluating the integral image. This
implies that the sums over any rectangle with sides parallel to the image axes can be evaluated with
four lookups of ii. But for rectangles non parallel to the axes the evaluation of the integral image has
a cost proportional to its perimeter. For 45° oriented rectangles, Lienhart and Maydt [18] proposed a
rotated integral image. The rotated integral image is computed traversing the image in the diagonal
directions. The sums over rotated rectangles are then computed, like the integral image, with four
table lookups.
Regarding the sum of values of a signal f as a convolution with a box ﬁlter g entails that any
invertible linear operator can be applied to f if its inverse is applied to g (for commodity we switch
to continuous notation). Concretely with the derivative/integral pair we have (in 1D)
(f ∗ g)(x) =
Z
f(t)g(x − t)dt =
Z
F
′
(t)g(x − t)dt =
Z
F (t)g
′
(x − t)dt, (3)
where F (t) =
R
t
−∞ f(t
′
)dt
′
. Observing that the derivative of the box ﬁlter is the sum of two delta
functions, the computation of the sum is reduced to the evaluation F at the two points. This insight
was used by the authors of [22] to eﬃciently ﬁlter signals with a piecewise constant ﬁlter g, by
converting g into a sequence of deltas at its discontinuities. The same computation extends to two
(and more) dimensions by simply taking partial derivatives and integrals on each dimension:
(f ∗ g)(x, y) =
Z Z
F (t, s)(
d
dy
d
dx
g(x − t, y − s))dtds, with F (t, s) =
Z s
−∞
Z t
−∞
f(t
′
, s
′
)dt
′
ds
′
.
347Gabriele Facciolo, Nicolas Limare, Enric Meinhardt
-0.5 0.5
box
box * box
box * box * box
-1 0 1
-1.5 -0.5 0.5 1.5
1
-1
1
-2
1
-3
3
-1

Introduction
The integral image representation was introduced [10, 5] with the purpose of evaluating sums of
image values over axis aligned rectangular regions in constant time. Crow [5] applied it in graphics
(under the name of summed area table) for eﬃciently computing mipmaps [26] by ﬁltering the images
with spatially varying rectangular ﬁlters. Lewis [17] used integral images for accelerating template
matching. Viola and Jones [24] popularized it in the computer vision community with its application
in their real-time object detection framework. More recently, Bay et al. [1] took advantage of integral
images for the purpose of rapidly computing approximations of SIFT features with their Speeded Up
Robust Features (SURF).
Block-matching aims at identifying corresponding blocks (or patches) between two images. A
patch is a rectangular portion of an image. For each pixel of the ﬁrst image (which we shall call
from now on reference image) we need to compute the matching cost (denoted also patch distance)
between the patch centered at this reference pixel, and all the patches of the second image within
a given range around the reference pixel. For each patch in the reference image, the patch in the
1
http://dx.doi.org/10.5201/ipol.2014.57
Gabriele Facciolo, Nicolas Limare, Enric Meinhardt, Integral Images for Block Matching, Image Processing On Line, 4 (2014), pp. 344–
369. http://dx.doi.org/10.5201/ipol.2014.57Integral Images for Block Matching
second image yielding the minimum matching cost is called “the corresponding block”. A popular
matching cost is computed as the sum of squared diﬀerences (SSD) of pixel values within the patch.
The correspondences and matching costs provide valuable information about the similarity between the patches, which can be applied to image denoising [3], video compression [15], motion and
depth estimation [21] among others. One of the ﬁrst uses of integral image applied to block matching
appears in the work by Faugeras et al. [8]. In that work the authors describe an optimization to spare
redundant computations, that is much in the spirit of integral images. Later, Veksler [23] applied
the integral images to stereo block matching for eﬃciently computing matching costs with multiple
window sizes, and Wang et al. [25] used integral images for accelerating the patch comparison in the
NL-means denoising algorithm [3].
The naive evaluation of the matching cost for a set of M relative oﬀsets, using patches of size r
2
,
entails O(Mr
2
) operations for each patch in the reference image. Therefore, for an image with N
pixels, the naive computation of all the matching costs requires O(MNr
2
) operations. Note that the
patch size aﬀects directly the computational cost. Many techniques have been proposed to reduce
this computational load by avoiding unnecessary comparisons, for instance exploiting the regularity
of the oﬀset ﬁelds [20]. Covering all these techniques is well beyond the scope of this work. Here
we are going to describe how the integral image can be applied to compute the matching costs with
O(MN) operations. Thus, with a cost only proportional to the number of oﬀsets, regardless of the
window size. This performance gain comes at the expense of some ﬂexibility, a tradeoﬀ that will be
discussed in detail.
We start by describing the integral image computation and its evaluation in Section 2. In Section 3
we explain how to apply the integral image for block matching and comment some issues. Sections 4
and 5 conclude by highlighting the beneﬁts and limitations of the integral image applied to block
matching.
2 Algorithm for Computing Integral Images
For a discrete image i its integral image ii at the pixel (x, y) is deﬁned as the sum of the pixel values
of i above and to the left of (x, y), namely
ii(x, y) =
X
p≤x ,q≤y
i(p, q). (1)
Given the integral image, the sum of all the pixels within a rectangle aligned with the image axes
can be evaluated with four array references, regardless of the size of the rectangle. This is illustrated
in Figure 1, where the sum over the rectangle [x0, x] × [y0, y] is computed as L4 + L1 − (L2 + L3),
with L1 = ii(x0 − 1, y0 − 1), L4 = ii(x, y), L3 = ii(x0 − 1, y) and L2 = ii(x, y0 − 1).
In [24] the authors describe an algorithm for computing the integral image in one-pass over the
image by using the recurrence relation
s(x, y) = s(x, y − 1) + i(x, y)
ii(x, y) = ii(x − 1, y) + s(x, y),
where s(x, y) is the column-wise cumulated sum, and s(x, −1) = ii(−1, y) = 0. Algorithm 1 details
the implementation of this recurrence, where the image s is replaced with a scalar accumulator
storing the current row-wise sum. A performance comparison of this method with two other slower
algorithm variants is provided in Appendix A. Algorithm 2 details how to compute the sum of a
rectangular region of the image by evaluating the integral image.
345Gabriele Facciolo, Nicolas Limare, Enric Meinhardt
Figure 1: Integral image and evaluation over a rectangle. The diagram illustrates the computation of the integral image: the value of ii(x, y) is the sum of the pixels in the gray region, where
the upper left corner of the image is the origin of the image’s coordinate system. The diagram on
the right shows how to compute the sum of the samples of the discrete image on a rectangle with
integer bounds, A = [x0, x] × [y0, y] using the integral image ii. It is enough to obtain the values of
the integral image at L1 = ii(x0 − 1, y0 − 1), L4 = ii(x, y), L3 = ii(x0 − 1, y) and L2 = ii(x, y0 − 1),
and to compute L4 + L1 − (L2 + L3).
Algorithm 1: Computing Integral Image.
input: An image of size nx × ny: i
output: The integral image of size nx × ny: ii
1 ii(0, 0) ← i(0, 0);
2 for x ← 1 to nx − 1 do
3 ii(x, 0) ← ii(x − 1, 0) + i(x, 0);
4 for y ← 1 to ny − 1 do
5 s ← i(0, y); /* scalar accumulator */
6 ii(0, y) ← ii(0, y − 1) + s;
7 for x ← 1 to nx − 1 do
8 s ← s + i(x, y);
9 ii(x, y) ← ii(x, y − 1) + s;
Algorithm 2: Summing the pixel values in a rectangle of an image using the Integral Image.
input: The integral image for an image i: ii. Position, width (w) and height (h) of a
rectangle contained in the image: [x0, x0 + w − 1] × [y0, y0 + h − 1].
output: Sum of the pixel values of i in the rectangle of size w × h : S
1 S ← ii(x0 + w − 1, y0 + h − 1);
2 if x0 > 0, then S ← S − ii(x0 − 1, y0 − 1 + h);
3 if y0 > 0, then S ← S − ii(x0 − 1 + w, y0 − 1);
4 if x0 > 0 and y0 > 0, then S ← S + ii(x0 − 1, y0 − 1);
2.1 Interpretation
The simplest interpretation of the integral image comes from considering a 1D discrete signal l of
length N. The sum over any interval of the signal l (for instance from x0 to x1) rewrites as
Xx1
j=x0
l(j) =
Xx1
j=0
l(j) −
xX0−1
j=0
l(j) = ll(x1) − ll(x0 − 1), (2)
346Integral Images for Block Matching
0 5 1 2 3 4
x
0 5 1 2 3 4
x
Figure 2: Interpolating an integral image. The left image depicts a row of pixels and the right
one its associated integral line. Note, on the left image, that the samples are interpolated with nearest
neighbor and that the signal is displaced half-pixel to the right. With this convention the integral
image at an integer position coincides with the integral of the piecewise constant interpolation of the
samples (for instance the integral image at 2 accumulates the areas of the ﬁrst and second pixels).
This also implies that linearly interpolating the integral image yields the integral of the piecewise
constant interpolation of the samples.
where ll(x) =
Px
i=0
l(x) is the “integral line”. Thus given ll, any sum over an interval of l can be
computed with two lookups of ll.
It is possible to compute sums over non integer intervals by interpolating the integral line [5].
Indeed, Figure 2 illustrates a 1D discrete signal which we interpret as a piecewise constant function
and the corresponding integral line. Linearly interpolating the integral line permits to integrate the
underlying piecewise constant function over any (non integer) interval.
In two dimensions the integral image ii at any position (x, y) is the sum of all the pixels in the
rectangular region above and to the left of (x, y) as illustrated in Figure 1. The orientation of the
summed rectangle determines the “privileged orientation” for evaluating the integral image. This
implies that the sums over any rectangle with sides parallel to the image axes can be evaluated with
four lookups of ii. But for rectangles non parallel to the axes the evaluation of the integral image has
a cost proportional to its perimeter. For 45° oriented rectangles, Lienhart and Maydt [18] proposed a
rotated integral image. The rotated integral image is computed traversing the image in the diagonal
directions. The sums over rotated rectangles are then computed, like the integral image, with four
table lookups.
Regarding the sum of values of a signal f as a convolution with a box ﬁlter g entails that any
invertible linear operator can be applied to f if its inverse is applied to g (for commodity we switch
to continuous notation). Concretely with the derivative/integral pair we have (in 1D)
(f ∗ g)(x) =
Z
f(t)g(x − t)dt =
Z
F
′
(t)g(x − t)dt =
Z
F (t)g
′
(x − t)dt, (3)
where F (t) =
R
t
−∞ f(t
′
)dt
′
. Observing that the derivative of the box ﬁlter is the sum of two delta
functions, the computation of the sum is reduced to the evaluation F at the two points. This insight
was used by the authors of [22] to eﬃciently ﬁlter signals with a piecewise constant ﬁlter g, by
converting g into a sequence of deltas at its discontinuities. The same computation extends to two
(and more) dimensions by simply taking partial derivatives and integrals on each dimension:
(f ∗ g)(x, y) =
Z Z
F (t, s)(
d
dy
d
dx
g(x − t, y − s))dtds, with F (t, s) =
Z s
−∞
Z t
−∞
f(t
′
, s
′
)dt
′
ds
′
.
347Gabriele Facciolo, Nicolas Limare, Enric Meinhardt
-0.5 0.5
 box
 box * box
 box * box * box
-1 0 1
-1.5 -0.5 0.5 1.5
1
-1
1
-2
1
-3
3
-1

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PendahuluanRepresentasi gambar integral diperkenalkan [10, 5] dengan tujuan untuk mengevaluasi jumlahnilai-nilai gambar atas sumbu sejajar wilayah persegi panjang dalam waktu yang konstan. Crow [5] menerapkannya dalam grafis(di bawah nama daerah disimpulkan tabel) untuk eﬃciently komputasi mipmaps [26] dengan ﬁltering gambardengan spasial berbagai ﬁlters persegi panjang. Lewis [17] digunakan integral gambar untuk mempercepat templatepencocokan. Viola dan Jones [24] dipopulerkan di komunitas visi komputer dengan aplikasimereka objek real-time deteksi Framework. Baru-baru ini, Bay et al. [1] mengambil keuntungan dari integralgambar untuk cepat komputasi pendekatan menyaring fitur dengan mereka dipercepat UpFitur kuat (SURF).Pencocokan blok bertujuan mengidentifikasi sesuai blok (atau patch) antara dua gambar. Apatch adalah persegi panjang sebagian dari gambar. Untuk setiap pixel dari gambar posisi (yang akan kita sebutSekarang gambar referensi) kita perlu menghitung biaya pencocokan (dilambangkan juga patch jarak)antara patch berpusat di pixel referensi, dan semua patch gambar kedua dalamberbagai diberikan di sekitar pixel referensi. Untuk setiap patch pada gambar referensi, patch di1http://DX.Doi.org/10.5201/IPOL.2014.57Nicolas Limare, Enric Meinhardt Gabriele Facciolo, Integral gambar untuk blok pencocokan, gambar pengolahan baris, 4 (2014), pp. 344-369. http://dx.doi.org/10.5201/ipol.2014.57Integral gambar untuk blok pencocokangambar kedua yang menghasilkan pencocokan biaya minimum disebut "sesuai blok". Populerpencocokan biaya dihitung sebagai jumlah kuadrat diﬀerences (SSD) dari nilai-nilai piksel dalam patch.Korespondensi dengan pencocokan biaya memberikan informasi berharga tentang kesamaan antara tambalan, yang dapat diterapkan untuk gambar denoising [3], kompresi video [15], gerak danEstimasi kedalaman [21] antara lain. Salah satu penggunaan posisi gambar integral yang diterapkan untuk memblokir pencocokanmuncul dalam karya oleh Faugeras et al. [8]. Dalam karya itu penulis menggambarkan optimasi untuk cadanganperhitungan yang berlebihan, yang banyak dalam semangat integral gambar. Kemudian, diterapkan Veksler [23]gambar integral untuk blok stereo yang cocok untuk komputasi pencocokan biaya dengan beberapa eﬃcientlyukuran jendela, dan Wang et al. [25] digunakan integral gambar untuk mempercepat perbandingan patch diNL-berarti denoising algoritma [3].Evaluasi naif biaya cocok untuk satu set M oﬀsets relatif, menggunakan patch ukuran r2,memerlukan O (Mr2) operasi untuk setiap patch pada gambar referensi. Oleh karena itu, untuk gambar dengan Npiksel, perhitungan naif semua biaya pencocokan memerlukan O (MNr2) operasi. Perhatikan bahwapatch ukuran aﬀects langsung biaya komputasi. Banyak teknik telah diusulkan untuk mengurangibeban ini komputasi dengan menghindari tidak perlu perbandingan, misalnya mengeksploitasi keteraturandari bidang oﬀset [20]. Meliputi semua teknik ini adalah baik di luar lingkup pekerjaan ini. Sinikami akan menjelaskan bagaimana gambar integral dapat diterapkan untuk menghitung biaya yang cocok denganOperasi O(MN). Dengan demikian, dengan biaya hanya sebanding dengan jumlah oﬀsets, terlepas dariukuran jendela. Ini kinerja keuntungan datang dengan mengorbankan beberapa ﬂexibility, tradeoﬀ yang akandibahas secara rinci.Kita mulai dengan menggambarkan perhitungan integral gambar dan evaluasi yang bagian 2. Dalam bagian 3kami menjelaskan bagaimana menerapkan gambar integral untuk blok pencocokan dan komentar beberapa masalah. Bagian 4dan 5 menyimpulkan dengan menyoroti beneﬁts dan keterbatasan gambar integral diterapkan untuk blokpencocokan.2 algoritma untuk komputasi Integral gambarDiskrit gambar saya yang integral gambar ii di pixel (x, y) adalah deﬁned sebagai jumlah dari nilai-nilai pikselsaya di atas dan ke kiri (x, y), yaituII (x, y) =Xp≤x, q≤yi (p, q). (1)Mengingat citra integral, jumlah dari semua piksel dalam persegi panjang yang sejajar dengan sumbu gambardapat dievaluasi dengan empat referensi array, terlepas dari ukuran dari persegi panjang. Ini diilustrasikandalam gambar 1, dimana jumlah atas persegi [x 0, x] × [y0, y] dihitung sebagai L4 + L1 − (L2 + L3),dengan L1 = ii (x 0 − 1, y0 − 1), L4 = ii (x, y), L3 = ii (x 0 − 1, y) dan L2 = ii (x, y0 − 1).Dalam [24] penulis menggambarkan algoritma untuk komputasi integral gambar di atas satu-passgambar dengan menggunakan hubungan kekambuhans (x, y) = s (x, y − 1) + i (x, y)II (x, y) = ii (x − 1, y) + s (x, y)mana s (x, y) adalah column-wise dijaga sum, dan s (x, –1) = ii (–1, y) = 0. Algoritma 1 rincianpelaksanaan ini kambuh, dimana gambar s diganti dengan skalar akumulatormenyimpan jumlah row-wise saat ini. Perbandingan kinerja metode ini dengan dua lambat lainnyaalgoritma varian disediakan di Lampiran A. algoritma 2 rincian bagaimana untuk menghitung jumlahsegi empat gambar dengan mengevaluasi integral gambar.345Gabriele Facciolo, Nicolas Limare, Enric MeinhardtGambar 1: Integral gambar dan evaluasi atas sebuah persegi panjang. Diagram menggambarkan perhitungan integral gambar: nilai ii (x, y) adalah jumlah pixel di wilayah abu-abu, manasudut kiri atas gambar adalah asal-usul sistem kordinat gambar. Diagram padakanan menunjukkan bagaimana untuk menghitung jumlah sampel gambar diskrit pada persegi panjang denganbatas-batas integer, = [x 0, x] × [y0, y] menggunakan integral gambar ii. Itu sudah cukup untuk mendapatkan nilai-nilaigambar integral di L1 = ii (x 0 − 1, y0 − 1), L4 = ii (x, y), L3 = ii (x 0 − 1, y) dan L2 = ii (x, y0 − 1),dan untuk menghitung L4 + L1 − (L2 + L3).Algoritma 1: Komputasi Integral gambar.masukan: gambar ukuran nx × ny: sayaoutput: gambar integral ukuran nx × ny: ii1 ii (0, 0) ← aku (0, 0);2 untuk x ← 1 untuk nx − 1 Apakah3 ii (x, 0) ← ii (x − 1, 0) + i (x, 0);4 untuk y ← 1 ke ny − 1 melakukan5 s ← aku (0, y); / * skalar akumulator * /6 ii (0, y) ← ii (0, y − 1) + s;7 untuk x ← 1 untuk nx − 1 melakukan8 s ← s + saya (x, y);9 ii (x, y) ← ii (x, y − 1) + s;Algoritma 2: Menjumlahkan nilai-nilai piksel dalam persegi panjang dari suatu gambar menggunakan citra Integral.masukan: integral gambar untuk gambar i: ii. Posisi, lebar (w) dan tinggi (h) daripersegi panjang yang terkandung dalam gambar: [x 0, x 0 + w − 1] × [y0, y0 + h − 1].output: jumlah dari nilai-nilai piksel i dari persegi ukuran w × h: S1 S ← ii (x 0 + w − 1, y0 + h − 1);2 x 0 > 0, maka S ← S − ii (x 0 − 1, y0 − 1 + h);3 jika y0 > 0, kemudian S ← S − ii (x 0 − 1 + w, y0 − 1);4 jika x 0 > 0 dan y0 > 0, maka S ← S + ii (x 0 − 1, y0 − 1);2.1 interpretasiPenafsiran sederhana gambar integral berasal dari mengingat 1D l sinyal diskritpanjang N. Jumlah atas setiap interval sinyal l (misalnya dari 0 x-x 1) penulisan ulang sebagaiXx1j = x 0l(j) =Xx1j = 0l(j) −xX0−1j = 0l(j) = ll(x1) − ll (x 0 − 1), (2).346Integral gambar untuk blok pencocokan0 5 1 2 3 4x0 5 1 2 3 4xGambar 2: Interpolasi gambar integral. Kiri gambar menggambarkan deretan piksel dan haksatu lini integral terkait. Catatan, gambar di sebelah kiri, yang sampel adalah sebuah sisipan dengan terdekattetangga dan sinyal yang setengah-pixel pengungsi ke kanan. Dengan Konvensi ini integralgambar pada posisi integer bertepatan dengan integral dari piecewise konstan interpolasi darisampel (misalnya gambar integral di 2 menumpuk bidang posisi dan kedua piksel).Hal ini juga menyiratkan bahwa linear interpolasi gambar integral menghasilkan integral dari piecewisekonstan interpolasi sampel.mana ll(x) =PxAku = 0l (x) adalah "integral line". Dengan demikian mengingat ll, jumlah berapapun selama interval l dapatdihitung dengan dua pencarian dari ll.Dimungkinkan untuk menghitung jumlah selama interval bebas bulat oleh interpolasi garis integral [5].Memang, gambar 2 menggambarkan sinyal diskrit 1D yang kita menafsirkan sebagai konstan piecewise fungsidan garis integral yang sesuai. Linear interpolasi garis integral memungkinkan mengintegrasikanmendasari piecewise konstan fungsi atas setiap interval (bebas bulat).Dalam dua dimensi gambar integral ii di posisi apapun (x, y) adalah jumlah dari semua piksel diwilayah persegi panjang di atas dan ke kiri (x, y) sebagai diilustrasikan pada gambar 1. Orientasidisimpulkan persegi panjang menentukan "orientasi istimewa" untuk mengevaluasi integral gambar. Inimenyiratkan bahwa penjumlahan atas setiap persegi panjang dengan sisi sejajar sumbu gambar dapat dievaluasi denganempat lookup II. Tetapi untuk persegi bebas sejajar sumbu evaluasi integral gambardengan biaya yang sebanding pinggirannya. Untuk 45° berorientasi persegi panjang, Lienhart dan Maydt [18] mengusulkangambar integral diputar. Gambar integral diputar dihitung melintasi gambar di diagonalarah. Penjumlahan atas diputar persegi kemudian dihitung, seperti gambar integral, dengan empatTabel pencarian.Mengenai jumlah dari nilai-nilai f sinyal sebagai lilitan dengan kotak ﬁlter g mensyaratkan bahwa setiapinvertible linear operator dapat diterapkan untuk f jika invers yang diterapkan untuk g (untuk komoditi kita beralihuntuk terus-menerus notasi). Konkret dengan pasangan turunan integral kami memiliki (dalam 1D)(f ∗ g)(x) =Zf (t) g (x − t) dt =ZF′(t) g (x − t) dt =ZG F (t)′(x − t) dt, (3)dimana F (t) =Rt−∞ f (t′) dt′. Mengamati bahwa turunan dari ﬁlter kotak adalah jumlah dari dua deltafungsi, perhitungan jumlah berkurang dengan evaluasi F di dua titik. Ini pemahamandigunakan oleh para penulis [22] untuk eﬃciently ﬁlter sinyal dengan konstan piecewise ﬁlter g, olehmengkonversi g ke dalam urutan Delta di diskontinuitas nya. Perhitungan sama meluas ke dua(dan lebih) dimensi dengan hanya mengambil turunan parsial dan integral pada masing-masing dimensi:(f ∗ g)(x, y) =Z ZF (t, s))dDYdDXmelihat DTD g (t x −, y − s)), dengan F (t, s) =Z s−∞Z t−∞f (t′, s′) dt′DS′.347Gabriele Facciolo, Nicolas Limare, Enric Meinhardt0.5 0,5 kotak kotak * kotak kotak * kotak * kotak-1 0 11.5 0.5 0.5 1.51-11-21-33-1

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