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Neural	
  Art	
  -­‐-­‐	
  電腦作畫	
  
	
  by	
  Mark	
  Chang	
  
A	
  Neural	
  Algorithm	
  of	
  Ar7s7c	
  Style	
  
•  作者:	
  
– Leon	
  A.	
  Gatys.	
  	
  
– Alexander	
  S.	
  Ecker.	
  	
  
– MaAhias	
  Bethge	
  	
  
•  所屬單位:	
  
– Werner	
  Reichardt	
  Centre	
  for	
  Integra7ve	
  Neuroscience	
  
and	
  Ins7tute	
  of	
  Theore7cal	
  Physics,	
  University	
  of	
  
Tubingen,	
  Germany.	
  	
  	
  
– Bernstein	
  Center	
  for	
  Computa7onal	
  Neuroscience,	
  
Tubingen,	
  Germany.	
  
如何作畫?	
  
大腦	
  畫家	
  
景物	
   畫風	
   畫作	
  
電腦	
   類神經網路	
  
大綱	
  
•  人類視覺	
  
•  電腦視覺	
  
•  電腦作畫	
  
•  作品展示	
  
人類視覺	
  
•  神經元	
  
•  視覺傳遞途徑	
  
•  錯覺	
  
神經元	
  
•  Neuron	
   •  Ac7on	
  Poten7al	
  
Dendrite	
  
Axon	
  
Cell	
  Body	
  
Time	
  
Voltage	
  
Threshold	
  
視覺傳遞途徑	
  
Re7na	
  
Visual	
  Area	
  V1	
  
Visual	
  Area	
  V4	
  
Inferior	
  
Temporal	
  Gyrus	
  
(IT)	
  
視覺傳遞途徑	
  
Visual	
  Area	
  V1	
  
Inferior	
  
Temporal	
  
Gyrus	
  (IT)	
  
Recep7ve	
  Fields	
  
Visual	
  Area	
  V4	
  
錯覺	
  
電腦視覺	
  
•  Neural	
  Networks	
  	
  
•  Convolu7onal	
  Neural	
  Networks	
  
•  VGG	
  19	
  
Neural	
  Networks	
  	
  
n
W1
W2
x1
x2
b
 Wb
y
nin = w1x1 + w2x2 + wb
nout =
1
1 + e nin
Sigmoid	
  
Rec7fied	
  Linear	
  
nout =
⇢
nin if nin > 0
0 otherwise
Neural	
  Networks	
  	
  
x
y
n11
n12
n21
n22
b b
z1
z2
Input	
  	
  
Layer
Hidden	
  
Layer
Output	
  
Layer
W12,y
W12,x
W11,y
W11,b
W12,b
W11,x
 W21,11
W22,12
W21,12
W22,11
W21,b
W22,b
Convolu7onal	
  Neural	
  Networks	
  
•  Convolu7onal	
  Layer
depth	
  
width	
  width	
  	
  depth	
  
weights	
  weights	
  
height	
  
shared	
  weight	
  
Convolu7onal	
  Neural	
  Networks	
  
•  Stride
 •  Padding
Stride	
  =	
  1	
  
Stride	
  =	
  2	
  
Padding	
  =	
  0	
  
Padding	
  =	
  1	
  
Convolu7onal	
  Neural	
  Networks	
  
•  Pooling	
  Layer	
  
1
 3
 2
 4
5
 7
 6
 8
0
 0
 4
 4
6
 6
 0
 0
4
 5
3
 2
no	
  overlap	
  
no	
  padding	
   no	
  weights	
  
depth	
  =	
  1	
  
7
 8
6
 4
Maximum	
  
Pooling	
  
Average	
  
Pooling	
  
Convolu7onal	
  Neural	
  Networks	
  
Convolu7onal	
  
Layer	
  
Convolu7onal	
  
Layer	
  	
   Pooling	
  
Layer	
  	
  
Pooling	
  
Layer	
  	
  
Recep7ve	
  Fields	
  
Recep7ve	
  Fields	
  
Input	
  
Layer	
  
Convolu7onal	
  Neural	
  Networks	
  
Input	
  Layer	
  
Convolu7onal	
  
Layer	
  with	
  
Recep7ve	
  Fields:	
  
Max-­‐pooling	
  
Layer	
  with	
  
Width	
  =3,	
  Height	
  =	
  3	
  
Filter	
  Responses	
  
Filter	
  Responses	
  
Input	
  Image	
  
VGG	
  19	
  
•  Karen	
  Simonyan	
  &	
  Andrew	
  Zisserman.	
  Very	
  Deep	
  
Convolu7onal	
  Networks	
  for	
  Large-­‐scale	
  Image	
  
Recogni7on.	
  
•  ImageNet	
  Challenge	
  2014	
  
•  19	
  (+5)	
  layers	
  
– 16	
  Convolu7onal	
  layers	
  (width=3,	
  height=3)	
  
– 5	
  Max-­‐pooling	
  layers	
  (width=2,	
  height=2)	
  
– 3	
  Fully-­‐connected	
  layers	
  
VGG	
  19	
  
depth=64	
  
conv1_1	
  
conv1_2	
  
maxpool
depth=128	
  
conv2_1	
  
conv2_2
maxpool
depth=256	
  
conv3_1	
  
conv3_2	
  
conv3_3	
  
conv3_4
depth=512	
  
conv4_1	
  
conv4_2	
  
conv4_3	
  
conv4_4
depth=512	
  
conv5_1	
  
conv5_2	
  
conv5_3	
  
conv5_4
maxpool
 maxpool
 maxpool
size=4096	
  
FC1	
  
FC2	
  
size=1000	
  
sogmax
電腦作畫	
  
•  內容產生	
  
•  畫風產生	
  
•  作品產生	
  
內容產生	
  
大腦	
  畫家	
  景物	
  
畫布	
  
計算	
  
兩者	
  
差異	
  
神經反應	
  
補上線條和顏色	
  
內容產生	
  
計算	
  
兩者	
  
差異	
  
Filter	
  	
  
Responses	
  VGG19	
  
修
正
差
異	
  
相片	
  
畫布	
  
修正後	
  
Width*Height	
  
Depth	
  
內容產生	
  
Layer	
  l’s	
  Filter	
  l	
  
Responses:	
  
ent(Pl
, Cl
) =
1
2
X
i,j
(Cl
i,j Pl
i,j)2
Layer	
  l’s	
  Filter	
  	
  
Responses:	
  Lcontent(P l
, Cl
) =
1
2
X
i,j
(Cl
i,j P l
i,j)2
Input	
  
Photo:	
  Lcontent(p, c, l) =
1
2
X
i,j
(Cl
i,j Pl
i,j)2
Lcontent(p, x, l) =
1
2
X
i,j
(Xl
i,j Pl
i,j)2
@Lcontent(p, x, l)
@Xl
i,j
= Xl
i,j Pl
i,j
Xl
Xl
i,j
Input	
  
Canvas:	
  x
Width*Height	
  (j)	
  
Depth	
  (i)	
  
Width*Height	
  (j)	
  
Depth	
  (i)	
  
內容產生	
  
•  Backward	
  Propaga7on	
  
Layer	
  l’s	
  Filter	
  l	
  
Responses:	
  Xl
Input	
  
Canvas:	
  
x
VGG19	
  
@Lcontent
@x
=
@Lcontent
@Xl
@Xl
x
x x ⌘
@Lcontent
@x
Update	
  
Canvas	
  
Learning	
  Rate	
  
內容產生	
  
內容產生	
  
VGG19	
  
conv1_2	
   conv2_2	
   conv3_4	
   conv4_4	
   conv5_2	
  conv5_1	
  
畫風產生	
  
•  ”Style”	
  is	
  posi7on-­‐independent	
  
style	
  
extrac7on	
  
畫風產生	
  
VGG19	
  畫作	
  
G	
  
G	
  
Filter	
  Responses	
   Gram	
  Matrix	
  
Width*Height	
  
Depth	
  
Depth	
  
Depth	
  
Posi7on-­‐	
  
dependent	
  
Posi7on-­‐	
  
independent	
  
畫風產生	
  
1.
 .5
.5
.5
1.
1.	
   .5
 .25
 1.
.5
 .25
 .5
.25
 .25
1.
 .5
 1.
Width*Height	
  
Depth	
  
k1	
   k2	
  
k1	
  
k2	
  
Depth	
  
Depth	
  
Layer	
  l’s	
  Filter	
  Responses	
  
Gram	
  Matrix	
  
Fl
1
Fl
2
Fl
3
Fl
4
Fl
1
Fl
2
Fl
3
Fl
4
G	
  
	
  
	
  
	
  
Gl
i,j = Fl
i · Fl
j
Gl
4,1 = Fl
4 · Fl
1
= 1 ⇥ 1 + 0 ⇥ 0.5 + 0 ⇥ 0 + ...
= 1
畫風產生	
  
VGG19	
  
Filter	
  
	
  Responses	
  
Gram	
  	
  
Matrix	
  
計算	
  
兩者	
  
差異	
  
G	
  
G	
  
畫風	
  
畫布	
  
修
正
差
異	
  
修正後	
  
畫風產生	
  
Lstyle(a, x, l) =
1
2
X
i,j
(Xl
i,j Al
i,j)2
@Lstyle(a, x, l)
@Fl
i,j
= ((Fl
)T
(Xl
Al
))j,i
Layer	
  l’s	
  	
  
Filter	
  Responses	
  
Layer	
  l’s	
  	
  
Gram	
  Matrix	
  
Layer	
  l’s	
  	
  
Gram	
  Matrix	
  
Fl
i,j
Al
i,j Xl
i,j
Input	
  
Artwork:	
  
Input	
  
Canvas:	
  a x
畫風產生	
  
畫風產生	
  
VGG19	
  
Conv1_1	
   Conv1_1	
  
Conv2_1	
  
Conv1_1	
  
Conv2_1	
  
Conv3_1	
  
	
  
Conv1_1	
  
Conv2_1	
  
Conv3_1	
  
Conv4_1	
  
Conv1_1	
  
Conv2_1	
  
Conv3_1	
  
Conv4_1	
  
Conv5_1	
  
作品產生	
  
Ltotal = ↵Lcontent + Lstyle
a
ent(p, c, l) =
1
2
X
i,j
(Cl
i,j Pl
i,j)2
x
x x ⌘
@Ltotal
@x
x
Filter	
  Responses	
  VGG19	
  
Lcontent(p, x)
Lstyle(a, x)
Gram	
  Matrix	
  
作品產生	
  
VGG19	
   VGG19	
  
Lcontent(p, x) Lstyle(a, x)
Conv1_1	
  
Conv2_1	
  
Conv3_1	
  
Conv4_1	
  
Conv5_1	
  
Conv4_2	
  
Ltotal = ↵Lcontent + Lstyle
作品產生	
  
作品展示	
  
•  內容 v.s.	
  畫風	
  
•  不同起始狀態	
  
•  不同VGG	
  Layers	
  
•  素描、水彩	
  
•  詩中有畫、畫中有詩	
  
內容 v.s.	
  畫風	
  
0.15	
   0.05	
  
0.02	
   0.007	
  
↵
不同起始狀態	
  
noise	
   0.9	
  *noise	
  +	
  0.1*photo	
   photo	
  
不同VGG	
  Layers	
  
Conv1_1	
  
Conv2_1	
  
Conv1_1	
  
Conv2_1	
  
Conv3_1	
  
Conv1_1	
  
Conv2_1	
  
Conv3_1	
  
Conv4_1	
  
Conv1_1	
  
Conv2_1	
  
Conv3_1	
  
Conv4_1	
  
Conv5_1	
  
↵
= 0.002
素描、水彩	
  
詩中有畫、畫中有詩	
  
延伸閱讀	
  
•  A	
  Neural	
  Algorithm	
  of	
  Ar7s7c	
  Style	
  
–  hAp://arxiv.org/abs/1508.06576	
  
•  Texture	
  Synthesis	
  Using	
  Convolu7onal	
  Neural	
  Networks	
  
–  hAp://arxiv.org/abs/1505.07376	
  
•  Convolu7onal	
  Neural	
  Network	
  
–  hAp://cs231n.github.io/convolu7onal-­‐networks/	
  
•  Neural	
  Network	
  Back	
  Propaga7on	
  
–  hAp://cpmarkchang.logdown.com/posts/277349-­‐neural-­‐
network-­‐backward-­‐propaga7on	
  
•  電腦賦詩:	
  
–  hAp://www.slideshare.net/ckmarkohchang/computa7onal-­‐
poetry	
  
	
  
程式碼
•  Python	
  Tensorflow	
  
– hAps://github.com/ckmarkoh/neuralart_tensorflow	
  
•  Python	
  Theano	
  
– hAps://github.com/woonketwong/ar7fy	
  
•  Python	
  Theano	
  (ipython	
  notebook)	
  
– hAps://github.com/Lasagne/Recipes/blob/master/
examples/styletransfer/Art%20Style
%20Transfer.ipynb	
  
•  Python	
  deeppy	
  
– hAps://github.com/andersbll/neural_ar7s7c_style	
  
圖片來源	
  
•  hAp://
www.taipei-­‐101.com.tw/
upload/news/
201502/2015021711505431
705145.JPG	
  	
  
	
  
•  hAps://github.com/
andersbll/
neural_ar7s7c_style/blob/
master/images/
starry_night.jpg?raw=true	
  
特別感謝	
  
•  	
  臺大資工imlab	
  

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NeuralArt 電腦作畫

  • 1. Neural  Art  -­‐-­‐  電腦作畫    by  Mark  Chang  
  • 2. A  Neural  Algorithm  of  Ar7s7c  Style   •  作者:   – Leon  A.  Gatys.     – Alexander  S.  Ecker.     – MaAhias  Bethge     •  所屬單位:   – Werner  Reichardt  Centre  for  Integra7ve  Neuroscience   and  Ins7tute  of  Theore7cal  Physics,  University  of   Tubingen,  Germany.       – Bernstein  Center  for  Computa7onal  Neuroscience,   Tubingen,  Germany.  
  • 3. 如何作畫?   大腦  畫家   景物   畫風   畫作   電腦   類神經網路  
  • 4. 大綱   •  人類視覺   •  電腦視覺   •  電腦作畫   •  作品展示  
  • 5. 人類視覺   •  神經元   •  視覺傳遞途徑   •  錯覺  
  • 6. 神經元   •  Neuron   •  Ac7on  Poten7al   Dendrite   Axon   Cell  Body   Time   Voltage   Threshold  
  • 7. 視覺傳遞途徑   Re7na   Visual  Area  V1   Visual  Area  V4   Inferior   Temporal  Gyrus   (IT)  
  • 8. 視覺傳遞途徑   Visual  Area  V1   Inferior   Temporal   Gyrus  (IT)   Recep7ve  Fields   Visual  Area  V4  
  • 10. 電腦視覺   •  Neural  Networks     •  Convolu7onal  Neural  Networks   •  VGG  19  
  • 11. Neural  Networks     n W1 W2 x1 x2 b Wb y nin = w1x1 + w2x2 + wb nout = 1 1 + e nin Sigmoid   Rec7fied  Linear   nout = ⇢ nin if nin > 0 0 otherwise
  • 12. Neural  Networks     x y n11 n12 n21 n22 b b z1 z2 Input     Layer Hidden   Layer Output   Layer W12,y W12,x W11,y W11,b W12,b W11,x W21,11 W22,12 W21,12 W22,11 W21,b W22,b
  • 13. Convolu7onal  Neural  Networks   •  Convolu7onal  Layer depth   width  width    depth   weights  weights   height   shared  weight  
  • 14. Convolu7onal  Neural  Networks   •  Stride •  Padding Stride  =  1   Stride  =  2   Padding  =  0   Padding  =  1  
  • 15. Convolu7onal  Neural  Networks   •  Pooling  Layer   1 3 2 4 5 7 6 8 0 0 4 4 6 6 0 0 4 5 3 2 no  overlap   no  padding   no  weights   depth  =  1   7 8 6 4 Maximum   Pooling   Average   Pooling  
  • 16. Convolu7onal  Neural  Networks   Convolu7onal   Layer   Convolu7onal   Layer     Pooling   Layer     Pooling   Layer     Recep7ve  Fields   Recep7ve  Fields   Input   Layer  
  • 17. Convolu7onal  Neural  Networks   Input  Layer   Convolu7onal   Layer  with   Recep7ve  Fields:   Max-­‐pooling   Layer  with   Width  =3,  Height  =  3   Filter  Responses   Filter  Responses   Input  Image  
  • 18. VGG  19   •  Karen  Simonyan  &  Andrew  Zisserman.  Very  Deep   Convolu7onal  Networks  for  Large-­‐scale  Image   Recogni7on.   •  ImageNet  Challenge  2014   •  19  (+5)  layers   – 16  Convolu7onal  layers  (width=3,  height=3)   – 5  Max-­‐pooling  layers  (width=2,  height=2)   – 3  Fully-­‐connected  layers  
  • 19. VGG  19   depth=64   conv1_1   conv1_2   maxpool depth=128   conv2_1   conv2_2 maxpool depth=256   conv3_1   conv3_2   conv3_3   conv3_4 depth=512   conv4_1   conv4_2   conv4_3   conv4_4 depth=512   conv5_1   conv5_2   conv5_3   conv5_4 maxpool maxpool maxpool size=4096   FC1   FC2   size=1000   sogmax
  • 20. 電腦作畫   •  內容產生   •  畫風產生   •  作品產生  
  • 21. 內容產生   大腦  畫家  景物   畫布   計算   兩者   差異   神經反應   補上線條和顏色  
  • 22. 內容產生   計算   兩者   差異   Filter     Responses  VGG19   修 正 差 異   相片   畫布   修正後   Width*Height   Depth  
  • 23. 內容產生   Layer  l’s  Filter  l   Responses:   ent(Pl , Cl ) = 1 2 X i,j (Cl i,j Pl i,j)2 Layer  l’s  Filter     Responses:  Lcontent(P l , Cl ) = 1 2 X i,j (Cl i,j P l i,j)2 Input   Photo:  Lcontent(p, c, l) = 1 2 X i,j (Cl i,j Pl i,j)2 Lcontent(p, x, l) = 1 2 X i,j (Xl i,j Pl i,j)2 @Lcontent(p, x, l) @Xl i,j = Xl i,j Pl i,j Xl Xl i,j Input   Canvas:  x Width*Height  (j)   Depth  (i)   Width*Height  (j)   Depth  (i)  
  • 24. 內容產生   •  Backward  Propaga7on   Layer  l’s  Filter  l   Responses:  Xl Input   Canvas:   x VGG19   @Lcontent @x = @Lcontent @Xl @Xl x x x ⌘ @Lcontent @x Update   Canvas   Learning  Rate  
  • 26. 內容產生   VGG19   conv1_2   conv2_2   conv3_4   conv4_4   conv5_2  conv5_1  
  • 27. 畫風產生   •  ”Style”  is  posi7on-­‐independent   style   extrac7on  
  • 28. 畫風產生   VGG19  畫作   G   G   Filter  Responses   Gram  Matrix   Width*Height   Depth   Depth   Depth   Posi7on-­‐   dependent   Posi7on-­‐   independent  
  • 29. 畫風產生   1. .5 .5 .5 1. 1.   .5 .25 1. .5 .25 .5 .25 .25 1. .5 1. Width*Height   Depth   k1   k2   k1   k2   Depth   Depth   Layer  l’s  Filter  Responses   Gram  Matrix   Fl 1 Fl 2 Fl 3 Fl 4 Fl 1 Fl 2 Fl 3 Fl 4 G         Gl i,j = Fl i · Fl j Gl 4,1 = Fl 4 · Fl 1 = 1 ⇥ 1 + 0 ⇥ 0.5 + 0 ⇥ 0 + ... = 1
  • 30. 畫風產生   VGG19   Filter    Responses   Gram     Matrix   計算   兩者   差異   G   G   畫風   畫布   修 正 差 異   修正後  
  • 31. 畫風產生   Lstyle(a, x, l) = 1 2 X i,j (Xl i,j Al i,j)2 @Lstyle(a, x, l) @Fl i,j = ((Fl )T (Xl Al ))j,i Layer  l’s     Filter  Responses   Layer  l’s     Gram  Matrix   Layer  l’s     Gram  Matrix   Fl i,j Al i,j Xl i,j Input   Artwork:   Input   Canvas:  a x
  • 33. 畫風產生   VGG19   Conv1_1   Conv1_1   Conv2_1   Conv1_1   Conv2_1   Conv3_1     Conv1_1   Conv2_1   Conv3_1   Conv4_1   Conv1_1   Conv2_1   Conv3_1   Conv4_1   Conv5_1  
  • 34. 作品產生   Ltotal = ↵Lcontent + Lstyle a ent(p, c, l) = 1 2 X i,j (Cl i,j Pl i,j)2 x x x ⌘ @Ltotal @x x Filter  Responses  VGG19   Lcontent(p, x) Lstyle(a, x) Gram  Matrix  
  • 35. 作品產生   VGG19   VGG19   Lcontent(p, x) Lstyle(a, x) Conv1_1   Conv2_1   Conv3_1   Conv4_1   Conv5_1   Conv4_2   Ltotal = ↵Lcontent + Lstyle
  • 37. 作品展示   •  內容 v.s.  畫風   •  不同起始狀態   •  不同VGG  Layers   •  素描、水彩   •  詩中有畫、畫中有詩  
  • 38. 內容 v.s.  畫風   0.15   0.05   0.02   0.007   ↵
  • 39. 不同起始狀態   noise   0.9  *noise  +  0.1*photo   photo  
  • 40. 不同VGG  Layers   Conv1_1   Conv2_1   Conv1_1   Conv2_1   Conv3_1   Conv1_1   Conv2_1   Conv3_1   Conv4_1   Conv1_1   Conv2_1   Conv3_1   Conv4_1   Conv5_1   ↵ = 0.002
  • 43. 延伸閱讀   •  A  Neural  Algorithm  of  Ar7s7c  Style   –  hAp://arxiv.org/abs/1508.06576   •  Texture  Synthesis  Using  Convolu7onal  Neural  Networks   –  hAp://arxiv.org/abs/1505.07376   •  Convolu7onal  Neural  Network   –  hAp://cs231n.github.io/convolu7onal-­‐networks/   •  Neural  Network  Back  Propaga7on   –  hAp://cpmarkchang.logdown.com/posts/277349-­‐neural-­‐ network-­‐backward-­‐propaga7on   •  電腦賦詩:   –  hAp://www.slideshare.net/ckmarkohchang/computa7onal-­‐ poetry    
  • 44. 程式碼 •  Python  Tensorflow   – hAps://github.com/ckmarkoh/neuralart_tensorflow   •  Python  Theano   – hAps://github.com/woonketwong/ar7fy   •  Python  Theano  (ipython  notebook)   – hAps://github.com/Lasagne/Recipes/blob/master/ examples/styletransfer/Art%20Style %20Transfer.ipynb   •  Python  deeppy   – hAps://github.com/andersbll/neural_ar7s7c_style  
  • 45. 圖片來源   •  hAp:// www.taipei-­‐101.com.tw/ upload/news/ 201502/2015021711505431 705145.JPG       •  hAps://github.com/ andersbll/ neural_ar7s7c_style/blob/ master/images/ starry_night.jpg?raw=true  
  • 46. 特別感謝   •   臺大資工imlab  

Hinweis der Redaktion

  1. n_{in} = w_{1} x_{1} + w_{2}x_{2}+w_{b} n_{out} = \frac{1}{1+e^{-n_{in}}}
  2. L_{content}(\textbf{p},\textbf{x}, l) = \frac{1}{2} \sum_{i,j}(X^{l}_{i,j}-P^{l}_{i,j})^2 \dfrac{\partial L_{content}(\textbf{p},\textbf{x},l) }{\partial X^{l}_{i,j}} = X^{l}_{i,j}-P^{l}_{i,j} X_^{l}
  3. \textbf{x} \leftarrow \textbf{x} - \eta \dfrac{\partial L_{content}}{\partial \textbf{x}} \textbf{x} \leftarrow \textbf{x} - \eta \dfrac{\partial L_{content}}{\partial \textbf{x}}
  4. G^{l}_{i,j} = F^{l}_{i} \cdot F^{l}_{j} & G^{l}_{4,1} = F^{l}_{4} \cdot F^{l}_{1} \\ & = 1 \times 1 + 0 \times 0.5 + 0\times 0 + ...\\ &= 1
  5. A^{l}_{i,j} L_{style}(\textbf{a},\textbf{x}, l) = \frac{1}{2} \sum_{i,j}(X^{l}_{i,j}-A^{l}_{i,j})^2 \dfrac{\partial L_{style}(\textbf{a},\textbf{x},l) }{\partial F^{l}_{i,j}} = ( (F^{l})^{T} (X^{l}-A^{l}) )_{j,i}
  6. L_{style} L_{total} = \alpha L_{content} + (1-\alpha) L_{style} \dfrac{\partial L_{total}}{\partial \textbf{x}} \textbf{x} \rightarrow \textbf{x} - \eta \dfrac{\partial L_{total}}{\partial \textbf{x}} L_{style}(\textbf{a},\textbf{x}) L_{content}(\textbf{p},\textbf{x})