,

DeePattern: Layout Pattern Generation with TransformingConvolutional Auto-Encoder

Haoyu Yang1, Piyush Pathak2, Frank Gennari2, Ya-Chieh Lai2 and Bei Yu1

1Department of Computer Science and Engineering, The Chinese University of Hong Kong2Cadence Design Systems, Inc.

,

Backgrounds

Moore’s Law to Extreme Scaling

1940 1950 1960 1970 1980 1990 2000 2010 2020

10,000,000,000

110

1001,000

10,000100,000

1,000,00010,000,000

100,000,0001,000,000,000

Intel Microprocessors

Invention of the Transistor

10 1 0.1 0.0145nm<latexit sha1_base64="DCiRJ1BVzI7mhdLE5r9XfdF1SuA=">AAAB7HicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLUlsLDHxgAQuZG/Zgw37cdndMyEXfoONhcbY+oPs/DcucIWCL5nk5b2ZzMyLU86M9f1vr7S1vbO7V96vHBweHZ9UT886RmWa0JAornQvxoZyJmlomeW0l2qKRcxpN57eLfzuE9WGKfloZymNBB5LljCCrZPCukSiPqzW/Ia/BNokQUFqUKA9rH4NRopkgkpLODamH/ipjXKsLSOcziuDzNAUkyke076jEgtqonx57BxdOWWEEqVdSYuW6u+JHAtjZiJ2nQLbiVn3FuJ/Xj+zyW2UM5lmlkqyWpRkHFmFFp+jEdOUWD5zBBPN3K2ITLDGxLp8Ki6EYP3lTdJpNgK/ETw0a61mEUcZLuASriGAG2jBPbQhBAIMnuEV3jzpvXjv3seqteQVM+fwB97nD6q2jd0=</latexit><latexit sha1_base64="DCiRJ1BVzI7mhdLE5r9XfdF1SuA=">AAAB7HicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLUlsLDHxgAQuZG/Zgw37cdndMyEXfoONhcbY+oPs/DcucIWCL5nk5b2ZzMyLU86M9f1vr7S1vbO7V96vHBweHZ9UT886RmWa0JAornQvxoZyJmlomeW0l2qKRcxpN57eLfzuE9WGKfloZymNBB5LljCCrZPCukSiPqzW/Ia/BNokQUFqUKA9rH4NRopkgkpLODamH/ipjXKsLSOcziuDzNAUkyke076jEgtqonx57BxdOWWEEqVdSYuW6u+JHAtjZiJ2nQLbiVn3FuJ/Xj+zyW2UM5lmlkqyWpRkHFmFFp+jEdOUWD5zBBPN3K2ITLDGxLp8Ki6EYP3lTdJpNgK/ETw0a61mEUcZLuASriGAG2jBPbQhBAIMnuEV3jzpvXjv3seqteQVM+fwB97nD6q2jd0=</latexit><latexit sha1_base64="DCiRJ1BVzI7mhdLE5r9XfdF1SuA=">AAAB7HicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLUlsLDHxgAQuZG/Zgw37cdndMyEXfoONhcbY+oPs/DcucIWCL5nk5b2ZzMyLU86M9f1vr7S1vbO7V96vHBweHZ9UT886RmWa0JAornQvxoZyJmlomeW0l2qKRcxpN57eLfzuE9WGKfloZymNBB5LljCCrZPCukSiPqzW/Ia/BNokQUFqUKA9rH4NRopkgkpLODamH/ipjXKsLSOcziuDzNAUkyke076jEgtqonx57BxdOWWEEqVdSYuW6u+JHAtjZiJ2nQLbiVn3FuJ/Xj+zyW2UM5lmlkqyWpRkHFmFFp+jEdOUWD5zBBPN3K2ITLDGxLp8Ki6EYP3lTdJpNgK/ETw0a61mEUcZLuASriGAG2jBPbQhBAIMnuEV3jzpvXjv3seqteQVM+fwB97nD6q2jd0=</latexit><latexit sha1_base64="DCiRJ1BVzI7mhdLE5r9XfdF1SuA=">AAAB7HicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLUlsLDHxgAQuZG/Zgw37cdndMyEXfoONhcbY+oPs/DcucIWCL5nk5b2ZzMyLU86M9f1vr7S1vbO7V96vHBweHZ9UT886RmWa0JAornQvxoZyJmlomeW0l2qKRcxpN57eLfzuE9WGKfloZymNBB5LljCCrZPCukSiPqzW/Ia/BNokQUFqUKA9rH4NRopkgkpLODamH/ipjXKsLSOcziuDzNAUkyke076jEgtqonx57BxdOWWEEqVdSYuW6u+JHAtjZiJ2nQLbiVn3FuJ/Xj+zyW2UM5lmlkqyWpRkHFmFFp+jEdOUWD5zBBPN3K2ITLDGxLp8Ki6EYP3lTdJpNgK/ETw0a61mEUcZLuASriGAG2jBPbQhBAIMnuEV3jzpvXjv3seqteQVM+fwB97nD6q2jd0=</latexit>

nm<latexit sha1_base64="DCiRJ1BVzI7mhdLE5r9XfdF1SuA=">AAAB7HicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLUlsLDHxgAQuZG/Zgw37cdndMyEXfoONhcbY+oPs/DcucIWCL5nk5b2ZzMyLU86M9f1vr7S1vbO7V96vHBweHZ9UT886RmWa0JAornQvxoZyJmlomeW0l2qKRcxpN57eLfzuE9WGKfloZymNBB5LljCCrZPCukSiPqzW/Ia/BNokQUFqUKA9rH4NRopkgkpLODamH/ipjXKsLSOcziuDzNAUkyke076jEgtqonx57BxdOWWEEqVdSYuW6u+JHAtjZiJ2nQLbiVn3FuJ/Xj+zyW2UM5lmlkqyWpRkHFmFFp+jEdOUWD5zBBPN3K2ITLDGxLp8Ki6EYP3lTdJpNgK/ETw0a61mEUcZLuASriGAG2jBPbQhBAIMnuEV3jzpvXjv3seqteQVM+fwB97nD6q2jd0=</latexit><latexit sha1_base64="DCiRJ1BVzI7mhdLE5r9XfdF1SuA=">AAAB7HicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLUlsLDHxgAQuZG/Zgw37cdndMyEXfoONhcbY+oPs/DcucIWCL5nk5b2ZzMyLU86M9f1vr7S1vbO7V96vHBweHZ9UT886RmWa0JAornQvxoZyJmlomeW0l2qKRcxpN57eLfzuE9WGKfloZymNBB5LljCCrZPCukSiPqzW/Ia/BNokQUFqUKA9rH4NRopkgkpLODamH/ipjXKsLSOcziuDzNAUkyke076jEgtqonx57BxdOWWEEqVdSYuW6u+JHAtjZiJ2nQLbiVn3FuJ/Xj+zyW2UM5lmlkqyWpRkHFmFFp+jEdOUWD5zBBPN3K2ITLDGxLp8Ki6EYP3lTdJpNgK/ETw0a61mEUcZLuASriGAG2jBPbQhBAIMnuEV3jzpvXjv3seqteQVM+fwB97nD6q2jd0=</latexit><latexit sha1_base64="DCiRJ1BVzI7mhdLE5r9XfdF1SuA=">AAAB7HicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLUlsLDHxgAQuZG/Zgw37cdndMyEXfoONhcbY+oPs/DcucIWCL5nk5b2ZzMyLU86M9f1vr7S1vbO7V96vHBweHZ9UT886RmWa0JAornQvxoZyJmlomeW0l2qKRcxpN57eLfzuE9WGKfloZymNBB5LljCCrZPCukSiPqzW/Ia/BNokQUFqUKA9rH4NRopkgkpLODamH/ipjXKsLSOcziuDzNAUkyke076jEgtqonx57BxdOWWEEqVdSYuW6u+JHAtjZiJ2nQLbiVn3FuJ/Xj+zyW2UM5lmlkqyWpRkHFmFFp+jEdOUWD5zBBPN3K2ITLDGxLp8Ki6EYP3lTdJpNgK/ETw0a61mEUcZLuASriGAG2jBPbQhBAIMnuEV3jzpvXjv3seqteQVM+fwB97nD6q2jd0=</latexit><latexit sha1_base64="DCiRJ1BVzI7mhdLE5r9XfdF1SuA=">AAAB7HicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLUlsLDHxgAQuZG/Zgw37cdndMyEXfoONhcbY+oPs/DcucIWCL5nk5b2ZzMyLU86M9f1vr7S1vbO7V96vHBweHZ9UT886RmWa0JAornQvxoZyJmlomeW0l2qKRcxpN57eLfzuE9WGKfloZymNBB5LljCCrZPCukSiPqzW/Ia/BNokQUFqUKA9rH4NRopkgkpLODamH/ipjXKsLSOcziuDzNAUkyke076jEgtqonx57BxdOWWEEqVdSYuW6u+JHAtjZiJ2nQLbiVn3FuJ/Xj+zyW2UM5lmlkqyWpRkHFmFFp+jEdOUWD5zBBPN3K2ITLDGxLp8Ki6EYP3lTdJpNgK/ETw0a61mEUcZLuASriGAG2jBPbQhBAIMnuEV3jzpvXjv3seqteQVM+fwB97nD6q2jd0=</latexit>

Year

Num

ber o

f Tra

nsis

tors

per I

nteg

rate

d C

ircui

t

Moore’s LawProcess Technology (µm

<latexit sha1_base64="1XOQUpYJ4Navgj2rMp+JfrNOobQ=">AAAB7nicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLElsLDERMIEL2VsW2LC7d9mdMyEXfoSNhcbY+nvs/DcucIWCL5nk5b2ZzMyLEiks+v63V9ja3tndK+6XDg6Pjk/Kp2cdG6eG8TaLZWweI2q5FJq3UaDkj4nhVEWSd6Pp7cLvPnFjRawfcJbwUNGxFiPBKDqpW+2rlKjqoFzxa/4SZJMEOalAjtag/NUfxixVXCOT1Npe4CcYZtSgYJLPS/3U8oSyKR3znqOaKm7DbHnunFw5ZUhGsXGlkSzV3xMZVdbOVOQ6FcWJXfcW4n9eL8XRTZgJnaTINVstGqWSYEwWv5OhMJyhnDlCmRHuVsIm1FCGLqGSCyFYf3mTdOq1wK8F9/VKs5HHUYQLuIRrCKABTbiDFrSBwRSe4RXevMR78d69j1VrwctnzuEPvM8fNUyOxg==</latexit><latexit sha1_base64="1XOQUpYJ4Navgj2rMp+JfrNOobQ=">AAAB7nicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLElsLDERMIEL2VsW2LC7d9mdMyEXfoSNhcbY+nvs/DcucIWCL5nk5b2ZzMyLEiks+v63V9ja3tndK+6XDg6Pjk/Kp2cdG6eG8TaLZWweI2q5FJq3UaDkj4nhVEWSd6Pp7cLvPnFjRawfcJbwUNGxFiPBKDqpW+2rlKjqoFzxa/4SZJMEOalAjtag/NUfxixVXCOT1Npe4CcYZtSgYJLPS/3U8oSyKR3znqOaKm7DbHnunFw5ZUhGsXGlkSzV3xMZVdbOVOQ6FcWJXfcW4n9eL8XRTZgJnaTINVstGqWSYEwWv5OhMJyhnDlCmRHuVsIm1FCGLqGSCyFYf3mTdOq1wK8F9/VKs5HHUYQLuIRrCKABTbiDFrSBwRSe4RXevMR78d69j1VrwctnzuEPvM8fNUyOxg==</latexit><latexit sha1_base64="1XOQUpYJ4Navgj2rMp+JfrNOobQ=">AAAB7nicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLElsLDERMIEL2VsW2LC7d9mdMyEXfoSNhcbY+nvs/DcucIWCL5nk5b2ZzMyLEiks+v63V9ja3tndK+6XDg6Pjk/Kp2cdG6eG8TaLZWweI2q5FJq3UaDkj4nhVEWSd6Pp7cLvPnFjRawfcJbwUNGxFiPBKDqpW+2rlKjqoFzxa/4SZJMEOalAjtag/NUfxixVXCOT1Npe4CcYZtSgYJLPS/3U8oSyKR3znqOaKm7DbHnunFw5ZUhGsXGlkSzV3xMZVdbOVOQ6FcWJXfcW4n9eL8XRTZgJnaTINVstGqWSYEwWv5OhMJyhnDlCmRHuVsIm1FCGLqGSCyFYf3mTdOq1wK8F9/VKs5HHUYQLuIRrCKABTbiDFrSBwRSe4RXevMR78d69j1VrwctnzuEPvM8fNUyOxg==</latexit><latexit sha1_base64="1XOQUpYJ4Navgj2rMp+JfrNOobQ=">AAAB7nicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLElsLDERMIEL2VsW2LC7d9mdMyEXfoSNhcbY+nvs/DcucIWCL5nk5b2ZzMyLEiks+v63V9ja3tndK+6XDg6Pjk/Kp2cdG6eG8TaLZWweI2q5FJq3UaDkj4nhVEWSd6Pp7cLvPnFjRawfcJbwUNGxFiPBKDqpW+2rlKjqoFzxa/4SZJMEOalAjtag/NUfxixVXCOT1Npe4CcYZtSgYJLPS/3U8oSyKR3znqOaKm7DbHnunFw5ZUhGsXGlkSzV3xMZVdbOVOQ6FcWJXfcW4n9eL8XRTZgJnaTINVstGqWSYEwWv5OhMJyhnDlCmRHuVsIm1FCGLqGSCyFYf3mTdOq1wK8F9/VKs5HHUYQLuIRrCKABTbiDFrSBwRSe4RXevMR78d69j1VrwctnzuEPvM8fNUyOxg==</latexit>

µm<latexit sha1_base64="1XOQUpYJ4Navgj2rMp+JfrNOobQ=">AAAB7nicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLElsLDERMIEL2VsW2LC7d9mdMyEXfoSNhcbY+nvs/DcucIWCL5nk5b2ZzMyLEiks+v63V9ja3tndK+6XDg6Pjk/Kp2cdG6eG8TaLZWweI2q5FJq3UaDkj4nhVEWSd6Pp7cLvPnFjRawfcJbwUNGxFiPBKDqpW+2rlKjqoFzxa/4SZJMEOalAjtag/NUfxixVXCOT1Npe4CcYZtSgYJLPS/3U8oSyKR3znqOaKm7DbHnunFw5ZUhGsXGlkSzV3xMZVdbOVOQ6FcWJXfcW4n9eL8XRTZgJnaTINVstGqWSYEwWv5OhMJyhnDlCmRHuVsIm1FCGLqGSCyFYf3mTdOq1wK8F9/VKs5HHUYQLuIRrCKABTbiDFrSBwRSe4RXevMR78d69j1VrwctnzuEPvM8fNUyOxg==</latexit><latexit sha1_base64="1XOQUpYJ4Navgj2rMp+JfrNOobQ=">AAAB7nicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLElsLDERMIEL2VsW2LC7d9mdMyEXfoSNhcbY+nvs/DcucIWCL5nk5b2ZzMyLEiks+v63V9ja3tndK+6XDg6Pjk/Kp2cdG6eG8TaLZWweI2q5FJq3UaDkj4nhVEWSd6Pp7cLvPnFjRawfcJbwUNGxFiPBKDqpW+2rlKjqoFzxa/4SZJMEOalAjtag/NUfxixVXCOT1Npe4CcYZtSgYJLPS/3U8oSyKR3znqOaKm7DbHnunFw5ZUhGsXGlkSzV3xMZVdbOVOQ6FcWJXfcW4n9eL8XRTZgJnaTINVstGqWSYEwWv5OhMJyhnDlCmRHuVsIm1FCGLqGSCyFYf3mTdOq1wK8F9/VKs5HHUYQLuIRrCKABTbiDFrSBwRSe4RXevMR78d69j1VrwctnzuEPvM8fNUyOxg==</latexit><latexit sha1_base64="1XOQUpYJ4Navgj2rMp+JfrNOobQ=">AAAB7nicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLElsLDERMIEL2VsW2LC7d9mdMyEXfoSNhcbY+nvs/DcucIWCL5nk5b2ZzMyLEiks+v63V9ja3tndK+6XDg6Pjk/Kp2cdG6eG8TaLZWweI2q5FJq3UaDkj4nhVEWSd6Pp7cLvPnFjRawfcJbwUNGxFiPBKDqpW+2rlKjqoFzxa/4SZJMEOalAjtag/NUfxixVXCOT1Npe4CcYZtSgYJLPS/3U8oSyKR3znqOaKm7DbHnunFw5ZUhGsXGlkSzV3xMZVdbOVOQ6FcWJXfcW4n9eL8XRTZgJnaTINVstGqWSYEwWv5OhMJyhnDlCmRHuVsIm1FCGLqGSCyFYf3mTdOq1wK8F9/VKs5HHUYQLuIRrCKABTbiDFrSBwRSe4RXevMR78d69j1VrwctnzuEPvM8fNUyOxg==</latexit><latexit sha1_base64="1XOQUpYJ4Navgj2rMp+JfrNOobQ=">AAAB7nicbVA9TwJBEJ3DL8Qv1NJmI5hYkTsaLElsLDERMIEL2VsW2LC7d9mdMyEXfoSNhcbY+nvs/DcucIWCL5nk5b2ZzMyLEiks+v63V9ja3tndK+6XDg6Pjk/Kp2cdG6eG8TaLZWweI2q5FJq3UaDkj4nhVEWSd6Pp7cLvPnFjRawfcJbwUNGxFiPBKDqpW+2rlKjqoFzxa/4SZJMEOalAjtag/NUfxixVXCOT1Npe4CcYZtSgYJLPS/3U8oSyKR3znqOaKm7DbHnunFw5ZUhGsXGlkSzV3xMZVdbOVOQ6FcWJXfcW4n9eL8XRTZgJnaTINVstGqWSYEwWv5OhMJyhnDlCmRHuVsIm1FCGLqGSCyFYf3mTdOq1wK8F9/VKs5HHUYQLuIRrCKABTbiDFrSBwRSe4RXevMR78d69j1VrwctnzuEPvM8fNUyOxg==</latexit> )

40048086

286386

486 Pentium

Pentium IIPentium 4

Core 2 Duo Core i7

Doubles every 2.1 yrs

EUV Brings Challenges in DFM

I Hotspot detection and fix

I Previous researches show the significance of a diverse and balanced training data set.[Yang+,SPIE’17]

I Hotspot pattern library covering the design space required by machine learning and pattern matching solutions.I Lithographic simulation challenge due to complicated computational lithography model under EUV nodes.

[Levinson+,SPIE’18]

I Early technology node development

I Due to long logic to layout cycle, test layout patterns are not usually available.I OPC convergence problem.I Patterns are required to massage Design rule, OPC recipe, ...

Related Works on Pattern Generation

I Transferring from previous technology node. (not applicable for large technology node gap) [Zhuang+,ICSICT’16]

I Randomly placing patterns according to certain constraints. (limited diversity)

I Generative machine learning models. (violating design rules) [Alec+,ICLR’16]

Preliminaries

Pattern Generation Challenges

I 7nm EUV metal layer unidirectional on-track shapes.

I Pitch, denoted as p, measures the distance between two adjacent tracks that contain shapes.

I T2T, denoted as t, measures the line-end-to-line-end distance between two adjacent shapes in a track.

I Wire length l and width w measure the shape size along and against the design track.

Pitch T2T Width Length

Evaluation of Pattern Library

All shape edges in a fixed-size window are aligned with x-axis and y -axis. If we extend all horizontal and vertical edgesinfinitely into scan lines, more non-overlapping scan lines always come with more complex patterns. We hence define thecomplexity of a layout pattern as follows.

I Pattern Complexity. The complexity of a pattern in x and y directions (denoted as cx and cy) are defined as thenumber of scan lines subtracted by one along x-axis and y -axis, respectively.

We also introduce the concept of pattern diversity (denoted as H) to measure how are the pattern complexities distributedin a given library. A larger H implies the library contains patterns that are more evenly distributed, as in the followingdefinition.

I Pattern Diversity. The diversity of a pattern library is given by the Shannon Entropy of the pattern complexitysampled from the library,

H = −∑i

∑j

P(cxi , cyj) logP(cxi , cyj),

where P(cxi , cyj) is the probability of a pattern sampled from the library has complexities of cxi and cyj in x and ydirections respectively.

Problem (Pattern Generation)

Given a set of layout design rules, the objective of pattern generation is to generate a pattern library such that thepattern diversity and the number of unique DRC-clean patterns in the library is maximized.

Methods

Squish Representation Example

y3<latexit sha1_base64="XRi+s556iR1h9SkAkBRteBpskdY=">AAAB7HicbZDNTgIxFIVv8Q/xD3XpphE0rsgMLnRJ4sYlJg6QwIR0SgcaOp1J2zGZTHgGNy40xq0P5M63scAsFDxJky/n3pvee4JEcG0c5xuVNja3tnfKu5W9/YPDo+rxSUfHqaLMo7GIVS8gmgkumWe4EayXKEaiQLBuML2b17tPTGkey0eTJcyPyFjykFNirOXVs+F1fVitOQ1nIbwObgE1KNQeVr8Go5imEZOGCqJ133US4+dEGU4Fm1UGqWYJoVMyZn2LkkRM+/li2Rm+sM4Ih7GyTxq8cH9P5CTSOosC2xkRM9Grtbn5X62fmvDWz7lMUsMkXX4UpgKbGM8vxyOuGDUis0Co4nZXTCdEEWpsPhUbgrt68jp0mg3X8kOz1ros4ijDGZzDFbhwAy24hzZ4QIHDM7zCG5LoBb2jj2VrCRUzp/BH6PMHv8GN4Q==</latexit><latexit sha1_base64="XRi+s556iR1h9SkAkBRteBpskdY=">AAAB7HicbZDNTgIxFIVv8Q/xD3XpphE0rsgMLnRJ4sYlJg6QwIR0SgcaOp1J2zGZTHgGNy40xq0P5M63scAsFDxJky/n3pvee4JEcG0c5xuVNja3tnfKu5W9/YPDo+rxSUfHqaLMo7GIVS8gmgkumWe4EayXKEaiQLBuML2b17tPTGkey0eTJcyPyFjykFNirOXVs+F1fVitOQ1nIbwObgE1KNQeVr8Go5imEZOGCqJ133US4+dEGU4Fm1UGqWYJoVMyZn2LkkRM+/li2Rm+sM4Ih7GyTxq8cH9P5CTSOosC2xkRM9Grtbn5X62fmvDWz7lMUsMkXX4UpgKbGM8vxyOuGDUis0Co4nZXTCdEEWpsPhUbgrt68jp0mg3X8kOz1ros4ijDGZzDFbhwAy24hzZ4QIHDM7zCG5LoBb2jj2VrCRUzp/BH6PMHv8GN4Q==</latexit><latexit sha1_base64="XRi+s556iR1h9SkAkBRteBpskdY=">AAAB7HicbZDNTgIxFIVv8Q/xD3XpphE0rsgMLnRJ4sYlJg6QwIR0SgcaOp1J2zGZTHgGNy40xq0P5M63scAsFDxJky/n3pvee4JEcG0c5xuVNja3tnfKu5W9/YPDo+rxSUfHqaLMo7GIVS8gmgkumWe4EayXKEaiQLBuML2b17tPTGkey0eTJcyPyFjykFNirOXVs+F1fVitOQ1nIbwObgE1KNQeVr8Go5imEZOGCqJ133US4+dEGU4Fm1UGqWYJoVMyZn2LkkRM+/li2Rm+sM4Ih7GyTxq8cH9P5CTSOosC2xkRM9Grtbn5X62fmvDWz7lMUsMkXX4UpgKbGM8vxyOuGDUis0Co4nZXTCdEEWpsPhUbgrt68jp0mg3X8kOz1ros4ijDGZzDFbhwAy24hzZ4QIHDM7zCG5LoBb2jj2VrCRUzp/BH6PMHv8GN4Q==</latexit><latexit sha1_base64="XRi+s556iR1h9SkAkBRteBpskdY=">AAAB7HicbZDNTgIxFIVv8Q/xD3XpphE0rsgMLnRJ4sYlJg6QwIR0SgcaOp1J2zGZTHgGNy40xq0P5M63scAsFDxJky/n3pvee4JEcG0c5xuVNja3tnfKu5W9/YPDo+rxSUfHqaLMo7GIVS8gmgkumWe4EayXKEaiQLBuML2b17tPTGkey0eTJcyPyFjykFNirOXVs+F1fVitOQ1nIbwObgE1KNQeVr8Go5imEZOGCqJ133US4+dEGU4Fm1UGqWYJoVMyZn2LkkRM+/li2Rm+sM4Ih7GyTxq8cH9P5CTSOosC2xkRM9Grtbn5X62fmvDWz7lMUsMkXX4UpgKbGM8vxyOuGDUis0Co4nZXTCdEEWpsPhUbgrt68jp0mg3X8kOz1ros4ijDGZzDFbhwAy24hzZ4QIHDM7zCG5LoBb2jj2VrCRUzp/BH6PMHv8GN4Q==</latexit>

y2<latexit sha1_base64="gpWsNLdq3u5WliluCVruaPqRmhU=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXLWzYbpvdqQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6ZSGHTdb2djc2t7Z7e0V94/ODw6rpyctk2SacZ9lshEd0NquBSK+yhQ8m6qOY1DyTvh5G5e7zxxbUSiHnGa8iCmIyUiwShay69NB43aoFJ16+5CZB28AqpQqDWofPWHCctirpBJakzPc1MMcqpRMMln5X5meErZhI54z6KiMTdBvlh2Ri6tMyRRou1TSBbu74mcxsZM49B2xhTHZrU2N/+r9TKMboNcqDRDrtjyoyiTBBMyv5wMheYM5dQCZVrYXQkbU00Z2nzKNgRv9eR1aDfqnuWHRrV5VcRRgnO4gGvw4AaacA8t8IGBgGd4hTdHOS/Ou/OxbN1wipkz+CPn8we+PI3g</latexit><latexit sha1_base64="gpWsNLdq3u5WliluCVruaPqRmhU=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXLWzYbpvdqQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6ZSGHTdb2djc2t7Z7e0V94/ODw6rpyctk2SacZ9lshEd0NquBSK+yhQ8m6qOY1DyTvh5G5e7zxxbUSiHnGa8iCmIyUiwShay69NB43aoFJ16+5CZB28AqpQqDWofPWHCctirpBJakzPc1MMcqpRMMln5X5meErZhI54z6KiMTdBvlh2Ri6tMyRRou1TSBbu74mcxsZM49B2xhTHZrU2N/+r9TKMboNcqDRDrtjyoyiTBBMyv5wMheYM5dQCZVrYXQkbU00Z2nzKNgRv9eR1aDfqnuWHRrV5VcRRgnO4gGvw4AaacA8t8IGBgGd4hTdHOS/Ou/OxbN1wipkz+CPn8we+PI3g</latexit><latexit sha1_base64="gpWsNLdq3u5WliluCVruaPqRmhU=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXLWzYbpvdqQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6ZSGHTdb2djc2t7Z7e0V94/ODw6rpyctk2SacZ9lshEd0NquBSK+yhQ8m6qOY1DyTvh5G5e7zxxbUSiHnGa8iCmIyUiwShay69NB43aoFJ16+5CZB28AqpQqDWofPWHCctirpBJakzPc1MMcqpRMMln5X5meErZhI54z6KiMTdBvlh2Ri6tMyRRou1TSBbu74mcxsZM49B2xhTHZrU2N/+r9TKMboNcqDRDrtjyoyiTBBMyv5wMheYM5dQCZVrYXQkbU00Z2nzKNgRv9eR1aDfqnuWHRrV5VcRRgnO4gGvw4AaacA8t8IGBgGd4hTdHOS/Ou/OxbN1wipkz+CPn8we+PI3g</latexit><latexit sha1_base64="gpWsNLdq3u5WliluCVruaPqRmhU=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXLWzYbpvdqQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6ZSGHTdb2djc2t7Z7e0V94/ODw6rpyctk2SacZ9lshEd0NquBSK+yhQ8m6qOY1DyTvh5G5e7zxxbUSiHnGa8iCmIyUiwShay69NB43aoFJ16+5CZB28AqpQqDWofPWHCctirpBJakzPc1MMcqpRMMln5X5meErZhI54z6KiMTdBvlh2Ri6tMyRRou1TSBbu74mcxsZM49B2xhTHZrU2N/+r9TKMboNcqDRDrtjyoyiTBBMyv5wMheYM5dQCZVrYXQkbU00Z2nzKNgRv9eR1aDfqnuWHRrV5VcRRgnO4gGvw4AaacA8t8IGBgGd4hTdHOS/Ou/OxbN1wipkz+CPn8we+PI3g</latexit>

y1<latexit sha1_base64="qV8nJ/elz9m1YWQxAR/RgqJqwvw=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXKWzYbpvdrQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6aCa+O6387G5tb2zm5pr7x/cHh0XDk5beskUwx9lohEdUOqUXCJvuFGYDdVSONQYCec3M3rnSdUmify0UxTDGI6kjzijBpr+bXpwKsNKlW37i5E1sEroAqFWoPKV3+YsCxGaZigWvc8NzVBTpXhTOCs3M80ppRN6Ah7FiWNUQf5YtkZubTOkESJsk8asnB/T+Q01noah7YzpmasV2tz879aLzPRbZBzmWYGJVt+FGWCmITMLydDrpAZMbVAmeJ2V8LGVFFmbD5lG4K3evI6tBt1z/JDo9q8KuIowTlcwDV4cANNuIcW+MCAwzO8wpsjnRfn3flYtm44xcwZ/JHz+QO8t43f</latexit><latexit sha1_base64="qV8nJ/elz9m1YWQxAR/RgqJqwvw=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXKWzYbpvdrQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6aCa+O6387G5tb2zm5pr7x/cHh0XDk5beskUwx9lohEdUOqUXCJvuFGYDdVSONQYCec3M3rnSdUmify0UxTDGI6kjzijBpr+bXpwKsNKlW37i5E1sEroAqFWoPKV3+YsCxGaZigWvc8NzVBTpXhTOCs3M80ppRN6Ah7FiWNUQf5YtkZubTOkESJsk8asnB/T+Q01noah7YzpmasV2tz879aLzPRbZBzmWYGJVt+FGWCmITMLydDrpAZMbVAmeJ2V8LGVFFmbD5lG4K3evI6tBt1z/JDo9q8KuIowTlcwDV4cANNuIcW+MCAwzO8wpsjnRfn3flYtm44xcwZ/JHz+QO8t43f</latexit><latexit sha1_base64="qV8nJ/elz9m1YWQxAR/RgqJqwvw=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXKWzYbpvdrQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6aCa+O6387G5tb2zm5pr7x/cHh0XDk5beskUwx9lohEdUOqUXCJvuFGYDdVSONQYCec3M3rnSdUmify0UxTDGI6kjzijBpr+bXpwKsNKlW37i5E1sEroAqFWoPKV3+YsCxGaZigWvc8NzVBTpXhTOCs3M80ppRN6Ah7FiWNUQf5YtkZubTOkESJsk8asnB/T+Q01noah7YzpmasV2tz879aLzPRbZBzmWYGJVt+FGWCmITMLydDrpAZMbVAmeJ2V8LGVFFmbD5lG4K3evI6tBt1z/JDo9q8KuIowTlcwDV4cANNuIcW+MCAwzO8wpsjnRfn3flYtm44xcwZ/JHz+QO8t43f</latexit><latexit sha1_base64="qV8nJ/elz9m1YWQxAR/RgqJqwvw=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXKWzYbpvdrQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6aCa+O6387G5tb2zm5pr7x/cHh0XDk5beskUwx9lohEdUOqUXCJvuFGYDdVSONQYCec3M3rnSdUmify0UxTDGI6kjzijBpr+bXpwKsNKlW37i5E1sEroAqFWoPKV3+YsCxGaZigWvc8NzVBTpXhTOCs3M80ppRN6Ah7FiWNUQf5YtkZubTOkESJsk8asnB/T+Q01noah7YzpmasV2tz879aLzPRbZBzmWYGJVt+FGWCmITMLydDrpAZMbVAmeJ2V8LGVFFmbD5lG4K3evI6tBt1z/JDo9q8KuIowTlcwDV4cANNuIcW+MCAwzO8wpsjnRfn3flYtm44xcwZ/JHz+QO8t43f</latexit>

y0<latexit sha1_base64="HeEijwzJo/arHqcPnonN5QrgcwY=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXLWzYbpvdqQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6ZSGHTdb2djc2t7Z7e0V94/ODw6rpyctk2SacZ9lshEd0NquBSK+yhQ8m6qOY1DyTvh5G5e7zxxbUSiHnGa8iCmIyUiwShay69NB25tUKm6dXchsg5eAVUo1BpUvvrDhGUxV8gkNabnuSkGOdUomOSzcj8zPKVsQke8Z1HRmJsgXyw7I5fWGZIo0fYpJAv390ROY2OmcWg7Y4pjs1qbm//VehlGt0EuVJohV2z5UZRJggmZX06GQnOGcmqBMi3sroSNqaYMbT5lG4K3evI6tBt1z/JDo9q8KuIowTlcwDV4cANNuIcW+MBAwDO8wpujnBfn3flYtm44xcwZ/JHz+QO7Mo3e</latexit><latexit sha1_base64="HeEijwzJo/arHqcPnonN5QrgcwY=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXLWzYbpvdqQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6ZSGHTdb2djc2t7Z7e0V94/ODw6rpyctk2SacZ9lshEd0NquBSK+yhQ8m6qOY1DyTvh5G5e7zxxbUSiHnGa8iCmIyUiwShay69NB25tUKm6dXchsg5eAVUo1BpUvvrDhGUxV8gkNabnuSkGOdUomOSzcj8zPKVsQke8Z1HRmJsgXyw7I5fWGZIo0fYpJAv390ROY2OmcWg7Y4pjs1qbm//VehlGt0EuVJohV2z5UZRJggmZX06GQnOGcmqBMi3sroSNqaYMbT5lG4K3evI6tBt1z/JDo9q8KuIowTlcwDV4cANNuIcW+MBAwDO8wpujnBfn3flYtm44xcwZ/JHz+QO7Mo3e</latexit><latexit sha1_base64="HeEijwzJo/arHqcPnonN5QrgcwY=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXLWzYbpvdqQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6ZSGHTdb2djc2t7Z7e0V94/ODw6rpyctk2SacZ9lshEd0NquBSK+yhQ8m6qOY1DyTvh5G5e7zxxbUSiHnGa8iCmIyUiwShay69NB25tUKm6dXchsg5eAVUo1BpUvvrDhGUxV8gkNabnuSkGOdUomOSzcj8zPKVsQke8Z1HRmJsgXyw7I5fWGZIo0fYpJAv390ROY2OmcWg7Y4pjs1qbm//VehlGt0EuVJohV2z5UZRJggmZX06GQnOGcmqBMi3sroSNqaYMbT5lG4K3evI6tBt1z/JDo9q8KuIowTlcwDV4cANNuIcW+MBAwDO8wpujnBfn3flYtm44xcwZ/JHz+QO7Mo3e</latexit><latexit sha1_base64="HeEijwzJo/arHqcPnonN5QrgcwY=">AAAB7HicbZBNT8JAEIanfiJ+oR69bASNJ9Jy0SOJF4+YWCCBhmyXLWzYbpvdqQlp+A1ePGiMV3+QN/+NC/Sg4Jts8uSdmezMG6ZSGHTdb2djc2t7Z7e0V94/ODw6rpyctk2SacZ9lshEd0NquBSK+yhQ8m6qOY1DyTvh5G5e7zxxbUSiHnGa8iCmIyUiwShay69NB25tUKm6dXchsg5eAVUo1BpUvvrDhGUxV8gkNabnuSkGOdUomOSzcj8zPKVsQke8Z1HRmJsgXyw7I5fWGZIo0fYpJAv390ROY2OmcWg7Y4pjs1qbm//VehlGt0EuVJohV2z5UZRJggmZX06GQnOGcmqBMi3sroSNqaYMbT5lG4K3evI6tBt1z/JDo9q8KuIowTlcwDV4cANNuIcW+MBAwDO8wpujnBfn3flYtm44xcwZ/JHz+QO7Mo3e</latexit>

x0<latexit sha1_base64="ityVHXWuLPr4YhpOt0ymukyJwW0=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6305hY3Nre6e4W9rbPzg8Kh+ftHSSKYY+S0SiOiHVKLhE33AjsJMqpHEosB2Ob+f19iMqzRP5YCYpBjEdSh5xRo21/OpT3632yxW35i5E1sHLoQK5mv3yV2+QsCxGaZigWnc9NzXBlCrDmcBZqZdpTCkb0yF2LUoaow6mi2Vn5MI6AxIlyj5pyML9PTGlsdaTOLSdMTUjvVqbm//VupmJboIpl2lmULLlR1EmiEnI/HIy4AqZERMLlCludyVsRBVlxuZTsiF4qyevQ6te8yzf1yuNyzyOIpzBOVyBB9fQgDtogg8MODzDK7w50nlx3p2PZWvByWdO4Y+czx+5q43d</latexit><latexit sha1_base64="ityVHXWuLPr4YhpOt0ymukyJwW0=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6305hY3Nre6e4W9rbPzg8Kh+ftHSSKYY+S0SiOiHVKLhE33AjsJMqpHEosB2Ob+f19iMqzRP5YCYpBjEdSh5xRo21/OpT3632yxW35i5E1sHLoQK5mv3yV2+QsCxGaZigWnc9NzXBlCrDmcBZqZdpTCkb0yF2LUoaow6mi2Vn5MI6AxIlyj5pyML9PTGlsdaTOLSdMTUjvVqbm//VupmJboIpl2lmULLlR1EmiEnI/HIy4AqZERMLlCludyVsRBVlxuZTsiF4qyevQ6te8yzf1yuNyzyOIpzBOVyBB9fQgDtogg8MODzDK7w50nlx3p2PZWvByWdO4Y+czx+5q43d</latexit><latexit sha1_base64="ityVHXWuLPr4YhpOt0ymukyJwW0=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6305hY3Nre6e4W9rbPzg8Kh+ftHSSKYY+S0SiOiHVKLhE33AjsJMqpHEosB2Ob+f19iMqzRP5YCYpBjEdSh5xRo21/OpT3632yxW35i5E1sHLoQK5mv3yV2+QsCxGaZigWnc9NzXBlCrDmcBZqZdpTCkb0yF2LUoaow6mi2Vn5MI6AxIlyj5pyML9PTGlsdaTOLSdMTUjvVqbm//VupmJboIpl2lmULLlR1EmiEnI/HIy4AqZERMLlCludyVsRBVlxuZTsiF4qyevQ6te8yzf1yuNyzyOIpzBOVyBB9fQgDtogg8MODzDK7w50nlx3p2PZWvByWdO4Y+czx+5q43d</latexit><latexit sha1_base64="ityVHXWuLPr4YhpOt0ymukyJwW0=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6305hY3Nre6e4W9rbPzg8Kh+ftHSSKYY+S0SiOiHVKLhE33AjsJMqpHEosB2Ob+f19iMqzRP5YCYpBjEdSh5xRo21/OpT3632yxW35i5E1sHLoQK5mv3yV2+QsCxGaZigWnc9NzXBlCrDmcBZqZdpTCkb0yF2LUoaow6mi2Vn5MI6AxIlyj5pyML9PTGlsdaTOLSdMTUjvVqbm//VupmJboIpl2lmULLlR1EmiEnI/HIy4AqZERMLlCludyVsRBVlxuZTsiF4qyevQ6te8yzf1yuNyzyOIpzBOVyBB9fQgDtogg8MODzDK7w50nlx3p2PZWvByWdO4Y+czx+5q43d</latexit>

x1<latexit sha1_base64="D2q8JmKRu75heN8ZgB0SL9eBWCg=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6305hY3Nre6e4W9rbPzg8Kh+ftHSSKYY+S0SiOiHVKLhE33AjsJMqpHEosB2Ob+f19iMqzRP5YCYpBjEdSh5xRo21/OpT36v2yxW35i5E1sHLoQK5mv3yV2+QsCxGaZigWnc9NzXBlCrDmcBZqZdpTCkb0yF2LUoaow6mi2Vn5MI6AxIlyj5pyML9PTGlsdaTOLSdMTUjvVqbm//VupmJboIpl2lmULLlR1EmiEnI/HIy4AqZERMLlCludyVsRBVlxuZTsiF4qyevQ6te8yzf1yuNyzyOIpzBOVyBB9fQgDtogg8MODzDK7w50nlx3p2PZWvByWdO4Y+czx+7MI3e</latexit><latexit sha1_base64="D2q8JmKRu75heN8ZgB0SL9eBWCg=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6305hY3Nre6e4W9rbPzg8Kh+ftHSSKYY+S0SiOiHVKLhE33AjsJMqpHEosB2Ob+f19iMqzRP5YCYpBjEdSh5xRo21/OpT36v2yxW35i5E1sHLoQK5mv3yV2+QsCxGaZigWnc9NzXBlCrDmcBZqZdpTCkb0yF2LUoaow6mi2Vn5MI6AxIlyj5pyML9PTGlsdaTOLSdMTUjvVqbm//VupmJboIpl2lmULLlR1EmiEnI/HIy4AqZERMLlCludyVsRBVlxuZTsiF4qyevQ6te8yzf1yuNyzyOIpzBOVyBB9fQgDtogg8MODzDK7w50nlx3p2PZWvByWdO4Y+czx+7MI3e</latexit><latexit sha1_base64="D2q8JmKRu75heN8ZgB0SL9eBWCg=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6305hY3Nre6e4W9rbPzg8Kh+ftHSSKYY+S0SiOiHVKLhE33AjsJMqpHEosB2Ob+f19iMqzRP5YCYpBjEdSh5xRo21/OpT36v2yxW35i5E1sHLoQK5mv3yV2+QsCxGaZigWnc9NzXBlCrDmcBZqZdpTCkb0yF2LUoaow6mi2Vn5MI6AxIlyj5pyML9PTGlsdaTOLSdMTUjvVqbm//VupmJboIpl2lmULLlR1EmiEnI/HIy4AqZERMLlCludyVsRBVlxuZTsiF4qyevQ6te8yzf1yuNyzyOIpzBOVyBB9fQgDtogg8MODzDK7w50nlx3p2PZWvByWdO4Y+czx+7MI3e</latexit><latexit sha1_base64="D2q8JmKRu75heN8ZgB0SL9eBWCg=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6305hY3Nre6e4W9rbPzg8Kh+ftHSSKYY+S0SiOiHVKLhE33AjsJMqpHEosB2Ob+f19iMqzRP5YCYpBjEdSh5xRo21/OpT36v2yxW35i5E1sHLoQK5mv3yV2+QsCxGaZigWnc9NzXBlCrDmcBZqZdpTCkb0yF2LUoaow6mi2Vn5MI6AxIlyj5pyML9PTGlsdaTOLSdMTUjvVqbm//VupmJboIpl2lmULLlR1EmiEnI/HIy4AqZERMLlCludyVsRBVlxuZTsiF4qyevQ6te8yzf1yuNyzyOIpzBOVyBB9fQgDtogg8MODzDK7w50nlx3p2PZWvByWdO4Y+czx+7MI3e</latexit>

x2<latexit sha1_base64="U6BRC5RERfZQDXwj/WAGhSbfSRk=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtC2zY27vszhnJhd9gY6Extv4gO/+NC1yh4Jts8uSdmezMGyZSGHTdb6ewsbm1vVPcLe3tHxwelY9PWiZONeM+i2WsOyE1XArFfRQoeSfRnEah5O1wcjuvtx+5NiJWDzhNeBDRkRJDwShay68+9evVfrni1tyFyDp4OVQgV7Nf/uoNYpZGXCGT1Jiu5yYYZFSjYJLPSr3U8ISyCR3xrkVFI26CbLHsjFxYZ0CGsbZPIVm4vycyGhkzjULbGVEcm9Xa3Pyv1k1xeBNkQiUpcsWWHw1TSTAm88vJQGjOUE4tUKaF3ZWwMdWUoc2nZEPwVk9eh1a95lm+r1cal3kcRTiDc7gCD66hAXfQBB8YCHiGV3hzlPPivDsfy9aCk8+cwh85nz+8tY3f</latexit><latexit sha1_base64="U6BRC5RERfZQDXwj/WAGhSbfSRk=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtC2zY27vszhnJhd9gY6Extv4gO/+NC1yh4Jts8uSdmezMGyZSGHTdb6ewsbm1vVPcLe3tHxwelY9PWiZONeM+i2WsOyE1XArFfRQoeSfRnEah5O1wcjuvtx+5NiJWDzhNeBDRkRJDwShay68+9evVfrni1tyFyDp4OVQgV7Nf/uoNYpZGXCGT1Jiu5yYYZFSjYJLPSr3U8ISyCR3xrkVFI26CbLHsjFxYZ0CGsbZPIVm4vycyGhkzjULbGVEcm9Xa3Pyv1k1xeBNkQiUpcsWWHw1TSTAm88vJQGjOUE4tUKaF3ZWwMdWUoc2nZEPwVk9eh1a95lm+r1cal3kcRTiDc7gCD66hAXfQBB8YCHiGV3hzlPPivDsfy9aCk8+cwh85nz+8tY3f</latexit><latexit sha1_base64="U6BRC5RERfZQDXwj/WAGhSbfSRk=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtC2zY27vszhnJhd9gY6Extv4gO/+NC1yh4Jts8uSdmezMGyZSGHTdb6ewsbm1vVPcLe3tHxwelY9PWiZONeM+i2WsOyE1XArFfRQoeSfRnEah5O1wcjuvtx+5NiJWDzhNeBDRkRJDwShay68+9evVfrni1tyFyDp4OVQgV7Nf/uoNYpZGXCGT1Jiu5yYYZFSjYJLPSr3U8ISyCR3xrkVFI26CbLHsjFxYZ0CGsbZPIVm4vycyGhkzjULbGVEcm9Xa3Pyv1k1xeBNkQiUpcsWWHw1TSTAm88vJQGjOUE4tUKaF3ZWwMdWUoc2nZEPwVk9eh1a95lm+r1cal3kcRTiDc7gCD66hAXfQBB8YCHiGV3hzlPPivDsfy9aCk8+cwh85nz+8tY3f</latexit><latexit sha1_base64="U6BRC5RERfZQDXwj/WAGhSbfSRk=">AAAB7HicbZA9TwJBEIbn8AvxC7W02QgaK3JHoyWJjSUmHpDAhewtC2zY27vszhnJhd9gY6Extv4gO/+NC1yh4Jts8uSdmezMGyZSGHTdb6ewsbm1vVPcLe3tHxwelY9PWiZONeM+i2WsOyE1XArFfRQoeSfRnEah5O1wcjuvtx+5NiJWDzhNeBDRkRJDwShay68+9evVfrni1tyFyDp4OVQgV7Nf/uoNYpZGXCGT1Jiu5yYYZFSjYJLPSr3U8ISyCR3xrkVFI26CbLHsjFxYZ0CGsbZPIVm4vycyGhkzjULbGVEcm9Xa3Pyv1k1xeBNkQiUpcsWWHw1TSTAm88vJQGjOUE4tUKaF3ZWwMdWUoc2nZEPwVk9eh1a95lm+r1cal3kcRTiDc7gCD66hAXfQBB8YCHiGV3hzlPPivDsfy9aCk8+cwh85nz+8tY3f</latexit>

x3<latexit sha1_base64="Bllpzvelf0oncLixa8U/0BgAU5U=">AAAB7HicbZA9TwJBEIbn/ET8Qi1tNoLGitxhoSWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6387a+sbm1nZhp7i7t39wWDo6buokUwx9lohEtUOqUXCJvuFGYDtVSONQYCsc3c7qrUdUmifywYxTDGI6kDzijBpr+ZWn3lWlVyq7VXcusgpeDmXI1eiVvrr9hGUxSsME1brjuakJJlQZzgROi91MY0rZiA6wY1HSGHUwmS87JefW6ZMoUfZJQ+bu74kJjbUex6HtjKkZ6uXazPyv1slMdBNMuEwzg5ItPooyQUxCZpeTPlfIjBhboExxuythQ6ooMzafog3BWz55FZq1qmf5vlauX+RxFOAUzuASPLiGOtxBA3xgwOEZXuHNkc6L8+58LFrXnHzmBP7I+fwBvjqN4A==</latexit><latexit sha1_base64="Bllpzvelf0oncLixa8U/0BgAU5U=">AAAB7HicbZA9TwJBEIbn/ET8Qi1tNoLGitxhoSWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6387a+sbm1nZhp7i7t39wWDo6buokUwx9lohEtUOqUXCJvuFGYDtVSONQYCsc3c7qrUdUmifywYxTDGI6kDzijBpr+ZWn3lWlVyq7VXcusgpeDmXI1eiVvrr9hGUxSsME1brjuakJJlQZzgROi91MY0rZiA6wY1HSGHUwmS87JefW6ZMoUfZJQ+bu74kJjbUex6HtjKkZ6uXazPyv1slMdBNMuEwzg5ItPooyQUxCZpeTPlfIjBhboExxuythQ6ooMzafog3BWz55FZq1qmf5vlauX+RxFOAUzuASPLiGOtxBA3xgwOEZXuHNkc6L8+58LFrXnHzmBP7I+fwBvjqN4A==</latexit><latexit sha1_base64="Bllpzvelf0oncLixa8U/0BgAU5U=">AAAB7HicbZA9TwJBEIbn/ET8Qi1tNoLGitxhoSWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6387a+sbm1nZhp7i7t39wWDo6buokUwx9lohEtUOqUXCJvuFGYDtVSONQYCsc3c7qrUdUmifywYxTDGI6kDzijBpr+ZWn3lWlVyq7VXcusgpeDmXI1eiVvrr9hGUxSsME1brjuakJJlQZzgROi91MY0rZiA6wY1HSGHUwmS87JefW6ZMoUfZJQ+bu74kJjbUex6HtjKkZ6uXazPyv1slMdBNMuEwzg5ItPooyQUxCZpeTPlfIjBhboExxuythQ6ooMzafog3BWz55FZq1qmf5vlauX+RxFOAUzuASPLiGOtxBA3xgwOEZXuHNkc6L8+58LFrXnHzmBP7I+fwBvjqN4A==</latexit><latexit sha1_base64="Bllpzvelf0oncLixa8U/0BgAU5U=">AAAB7HicbZA9TwJBEIbn/ET8Qi1tNoLGitxhoSWJjSUmHpDAhewtc7Bhb++yu2ckhN9gY6Extv4gO/+NC1yh4Jts8uSdmezMG6aCa+O6387a+sbm1nZhp7i7t39wWDo6buokUwx9lohEtUOqUXCJvuFGYDtVSONQYCsc3c7qrUdUmifywYxTDGI6kDzijBpr+ZWn3lWlVyq7VXcusgpeDmXI1eiVvrr9hGUxSsME1brjuakJJlQZzgROi91MY0rZiA6wY1HSGHUwmS87JefW6ZMoUfZJQ+bu74kJjbUex6HtjKkZ6uXazPyv1slMdBNMuEwzg5ItPooyQUxCZpeTPlfIjBhboExxuythQ6ooMzafog3BWz55FZq1qmf5vlauX+RxFOAUzuASPLiGOtxBA3xgwOEZXuHNkc6L8+58LFrXnHzmBP7I+fwBvjqN4A==</latexit>

x4<latexit sha1_base64="yc1Q/d6h4IMWrSe3O3pd0EhIxZk=">AAAB7HicbZBNSwMxEIZn/az1q+rRS7BVPJXdIuix4MVjBbcttEvJprNtaDa7JFmxlP4GLx4U8eoP8ua/MW33oK0vBB7emSEzb5gKro3rfjtr6xubW9uFneLu3v7BYenouKmTTDH0WSIS1Q6pRsEl+oYbge1UIY1Dga1wdDurtx5RaZ7IBzNOMYjpQPKIM2qs5VeeeleVXqnsVt25yCp4OZQhV6NX+ur2E5bFKA0TVOuO56YmmFBlOBM4LXYzjSllIzrAjkVJY9TBZL7slJxbp0+iRNknDZm7vycmNNZ6HIe2M6ZmqJdrM/O/Wicz0U0w4TLNDEq2+CjKBDEJmV1O+lwhM2JsgTLF7a6EDamizNh8ijYEb/nkVWjWqp7l+1q5fpHHUYBTOINL8OAa6nAHDfCBAYdneIU3RzovzrvzsWhdc/KZE/gj5/MHv7+N4Q==</latexit><latexit sha1_base64="yc1Q/d6h4IMWrSe3O3pd0EhIxZk=">AAAB7HicbZBNSwMxEIZn/az1q+rRS7BVPJXdIuix4MVjBbcttEvJprNtaDa7JFmxlP4GLx4U8eoP8ua/MW33oK0vBB7emSEzb5gKro3rfjtr6xubW9uFneLu3v7BYenouKmTTDH0WSIS1Q6pRsEl+oYbge1UIY1Dga1wdDurtx5RaZ7IBzNOMYjpQPKIM2qs5VeeeleVXqnsVt25yCp4OZQhV6NX+ur2E5bFKA0TVOuO56YmmFBlOBM4LXYzjSllIzrAjkVJY9TBZL7slJxbp0+iRNknDZm7vycmNNZ6HIe2M6ZmqJdrM/O/Wicz0U0w4TLNDEq2+CjKBDEJmV1O+lwhM2JsgTLF7a6EDamizNh8ijYEb/nkVWjWqp7l+1q5fpHHUYBTOINL8OAa6nAHDfCBAYdneIU3RzovzrvzsWhdc/KZE/gj5/MHv7+N4Q==</latexit><latexit sha1_base64="yc1Q/d6h4IMWrSe3O3pd0EhIxZk=">AAAB7HicbZBNSwMxEIZn/az1q+rRS7BVPJXdIuix4MVjBbcttEvJprNtaDa7JFmxlP4GLx4U8eoP8ua/MW33oK0vBB7emSEzb5gKro3rfjtr6xubW9uFneLu3v7BYenouKmTTDH0WSIS1Q6pRsEl+oYbge1UIY1Dga1wdDurtx5RaZ7IBzNOMYjpQPKIM2qs5VeeeleVXqnsVt25yCp4OZQhV6NX+ur2E5bFKA0TVOuO56YmmFBlOBM4LXYzjSllIzrAjkVJY9TBZL7slJxbp0+iRNknDZm7vycmNNZ6HIe2M6ZmqJdrM/O/Wicz0U0w4TLNDEq2+CjKBDEJmV1O+lwhM2JsgTLF7a6EDamizNh8ijYEb/nkVWjWqp7l+1q5fpHHUYBTOINL8OAa6nAHDfCBAYdneIU3RzovzrvzsWhdc/KZE/gj5/MHv7+N4Q==</latexit><latexit sha1_base64="yc1Q/d6h4IMWrSe3O3pd0EhIxZk=">AAAB7HicbZBNSwMxEIZn/az1q+rRS7BVPJXdIuix4MVjBbcttEvJprNtaDa7JFmxlP4GLx4U8eoP8ua/MW33oK0vBB7emSEzb5gKro3rfjtr6xubW9uFneLu3v7BYenouKmTTDH0WSIS1Q6pRsEl+oYbge1UIY1Dga1wdDurtx5RaZ7IBzNOMYjpQPKIM2qs5VeeeleVXqnsVt25yCp4OZQhV6NX+ur2E5bFKA0TVOuO56YmmFBlOBM4LXYzjSllIzrAjkVJY9TBZL7slJxbp0+iRNknDZm7vycmNNZ6HIe2M6ZmqJdrM/O/Wicz0U0w4TLNDEq2+CjKBDEJmV1O+lwhM2JsgTLF7a6EDamizNh8ijYEb/nkVWjWqp7l+1q5fpHHUYBTOINL8OAa6nAHDfCBAYdneIU3RzovzrvzsWhdc/KZE/gj5/MHv7+N4Q==</latexit>

T =

0 1 1 00 0 0 01 0 1 1

δx =

[x1 − x0 x2 − x1 x3 − x2 x4 − x3

]δy =

[y1 − y0 y2 − y1 y3 − y2

]

I Scan line-based representation, naturally supports easy computation of pattern complexity.

I Lossless feature representation.

I Easily feed into convolutional neural networks.

Problem Simplification

I Legal topology generation.

I Solving geometry constraints for DRC-clean patterns.

Methods

The Overall Flow

Existing Design

Pattern Library

3.1 Squish Pattern Extraction

3.2 Topology Generation

3.3 Legal Pattern Assessment

Detecting Critical

Dimensions

Applying Constraints

Solving Linear Sys.

Training TCAE

Perturbing Latent Space

Filtering Illegal

Topology

Transforming Convolutional Auto-Encoder

conv, 5x5x128

conv, 5x5x256

fc, 1024

fc, 32

x1, x2, ..., x32

fc, 1024

fc, 4x4x256

dconv, 5x5x128

dconv, 5x5x1

recognition:fgeneration:g

latent

conv, 5x5x128

conv, 5x5x256

fc, 1024

fc, 32

fc, 1024

fc, 4x4x256

dconv, 5x5x128

dconv, 5x5x1

x1, x2, ..., x32

recognition:fgeneration:g

perturbedlatent

x0

Input pattern to latent space,

l = f (T ; W f )

Topology reconstruction,

T ′ = g(l + ∆l ; W g)

Training objective,

minW f ,W g

||T −T ′||, s.t. ∆l = 0

I Inspired from TAE. [Hinton+,ICANN’11]

I Feature instantiation attains data set domainproperties.

I All capsules contribute together to producevariations of any input objects.

I The transformation in our framework appliesdirectly on the latent vector space that promises amuch larger diversity of the generated patternscompared to the limited transformation on thecoordinate system only in TAEs.

I Identity mapping in the training phase helps theTCAE capture the design rule properties of existingpatterns.

Perturbing the Latent Space

TCAE-Combine

I Generalization from existing topologies

T g = g(∑i

αi f (T i)),

where 0 < αi < 1,∀i are combination coefficientsand satisfy

∑i αi = 1.

I Sample results

T1<latexit sha1_base64="Rc3Gi5Q36BarX7gTPL4E1b63urs=">AAACEnicbZDNSsNAFIVv/K31r+rSzWARXEhJRNBlwY3LCv2DNpbJdNIOnUzCzI1QQt/ChVt9DHfi1hfwKXwFp20WtvXCwMc598KZEyRSGHTdb2dtfWNza7uwU9zd2z84LB0dN02casYbLJaxbgfUcCkUb6BAyduJ5jQKJG8Fo7up33ri2ohY1XGccD+iAyVCwSha6bEbURwGocCsPul5vVLZrbizIavg5VCGfGq90k+3H7M04gqZpMZ0PDdBP6MaBZN8UuymhieUjeiAdywqGnHjZ7PUE3JulT4JY22fQjJT/15kNDJmHAV2c5rSLHtT8V9PmHjBW4yB4a2fCZWkyBWbpwhTSTAm035IX2jOUI4tUKaF/QhhQ6opQ9ti0TbkLfexCs2rimf54bpcvcy7KsApnMEFeHADVbiHGjSAgYYXeIU359l5dz6cz/nqmpPfnMDCOF+/XLOd/Q==</latexit><latexit sha1_base64="Rc3Gi5Q36BarX7gTPL4E1b63urs=">AAACEnicbZDNSsNAFIVv/K31r+rSzWARXEhJRNBlwY3LCv2DNpbJdNIOnUzCzI1QQt/ChVt9DHfi1hfwKXwFp20WtvXCwMc598KZEyRSGHTdb2dtfWNza7uwU9zd2z84LB0dN02casYbLJaxbgfUcCkUb6BAyduJ5jQKJG8Fo7up33ri2ohY1XGccD+iAyVCwSha6bEbURwGocCsPul5vVLZrbizIavg5VCGfGq90k+3H7M04gqZpMZ0PDdBP6MaBZN8UuymhieUjeiAdywqGnHjZ7PUE3JulT4JY22fQjJT/15kNDJmHAV2c5rSLHtT8V9PmHjBW4yB4a2fCZWkyBWbpwhTSTAm035IX2jOUI4tUKaF/QhhQ6opQ9ti0TbkLfexCs2rimf54bpcvcy7KsApnMEFeHADVbiHGjSAgYYXeIU359l5dz6cz/nqmpPfnMDCOF+/XLOd/Q==</latexit><latexit sha1_base64="Rc3Gi5Q36BarX7gTPL4E1b63urs=">AAACEnicbZDNSsNAFIVv/K31r+rSzWARXEhJRNBlwY3LCv2DNpbJdNIOnUzCzI1QQt/ChVt9DHfi1hfwKXwFp20WtvXCwMc598KZEyRSGHTdb2dtfWNza7uwU9zd2z84LB0dN02casYbLJaxbgfUcCkUb6BAyduJ5jQKJG8Fo7up33ri2ohY1XGccD+iAyVCwSha6bEbURwGocCsPul5vVLZrbizIavg5VCGfGq90k+3H7M04gqZpMZ0PDdBP6MaBZN8UuymhieUjeiAdywqGnHjZ7PUE3JulT4JY22fQjJT/15kNDJmHAV2c5rSLHtT8V9PmHjBW4yB4a2fCZWkyBWbpwhTSTAm035IX2jOUI4tUKaF/QhhQ6opQ9ti0TbkLfexCs2rimf54bpcvcy7KsApnMEFeHADVbiHGjSAgYYXeIU359l5dz6cz/nqmpPfnMDCOF+/XLOd/Q==</latexit><latexit sha1_base64="Rc3Gi5Q36BarX7gTPL4E1b63urs=">AAACEnicbZDNSsNAFIVv/K31r+rSzWARXEhJRNBlwY3LCv2DNpbJdNIOnUzCzI1QQt/ChVt9DHfi1hfwKXwFp20WtvXCwMc598KZEyRSGHTdb2dtfWNza7uwU9zd2z84LB0dN02casYbLJaxbgfUcCkUb6BAyduJ5jQKJG8Fo7up33ri2ohY1XGccD+iAyVCwSha6bEbURwGocCsPul5vVLZrbizIavg5VCGfGq90k+3H7M04gqZpMZ0PDdBP6MaBZN8UuymhieUjeiAdywqGnHjZ7PUE3JulT4JY22fQjJT/15kNDJmHAV2c5rSLHtT8V9PmHjBW4yB4a2fCZWkyBWbpwhTSTAm035IX2jOUI4tUKaF/QhhQ6opQ9ti0TbkLfexCs2rimf54bpcvcy7KsApnMEFeHADVbiHGjSAgYYXeIU359l5dz6cz/nqmpPfnMDCOF+/XLOd/Q==</latexit>

T2<latexit sha1_base64="DzZuMFMIutEhnNMA8oEMc2NeNso=">AAACEnicbZDNSgMxFIUz9a/Wv6pLN8EiuJAyUwRdFty4rNA/aMeSSTNtaCYZkjtCGfoWLtzqY7gTt76AT+ErmGlnYVsvBD7OuRdOThALbsB1v53CxubW9k5xt7S3f3B4VD4+aRuVaMpaVAmluwExTHDJWsBBsG6sGYkCwTrB5C7zO09MG65kE6Yx8yMykjzklICVHvsRgXEQckibs0FtUK64VXc+eB28HCoon8ag/NMfKppETAIVxJie58bgp0QDp4LNSv3EsJjQCRmxnkVJImb8dJ56hi+sMsSh0vZJwHP170VKImOmUWA3s5Rm1cvEfz1u1JK3HAPCWz/lMk6ASbpIESYCg8JZP3jINaMgphYI1dx+BNMx0YSCbbFkG/JW+1iHdq3qWX64rtSv8q6K6Aydo0vkoRtUR/eogVqIIo1e0Ct6c56dd+fD+VysFpz85hQtjfP1C15Unf4=</latexit><latexit sha1_base64="DzZuMFMIutEhnNMA8oEMc2NeNso=">AAACEnicbZDNSgMxFIUz9a/Wv6pLN8EiuJAyUwRdFty4rNA/aMeSSTNtaCYZkjtCGfoWLtzqY7gTt76AT+ErmGlnYVsvBD7OuRdOThALbsB1v53CxubW9k5xt7S3f3B4VD4+aRuVaMpaVAmluwExTHDJWsBBsG6sGYkCwTrB5C7zO09MG65kE6Yx8yMykjzklICVHvsRgXEQckibs0FtUK64VXc+eB28HCoon8ag/NMfKppETAIVxJie58bgp0QDp4LNSv3EsJjQCRmxnkVJImb8dJ56hi+sMsSh0vZJwHP170VKImOmUWA3s5Rm1cvEfz1u1JK3HAPCWz/lMk6ASbpIESYCg8JZP3jINaMgphYI1dx+BNMx0YSCbbFkG/JW+1iHdq3qWX64rtSv8q6K6Aydo0vkoRtUR/eogVqIIo1e0Ct6c56dd+fD+VysFpz85hQtjfP1C15Unf4=</latexit><latexit sha1_base64="DzZuMFMIutEhnNMA8oEMc2NeNso=">AAACEnicbZDNSgMxFIUz9a/Wv6pLN8EiuJAyUwRdFty4rNA/aMeSSTNtaCYZkjtCGfoWLtzqY7gTt76AT+ErmGlnYVsvBD7OuRdOThALbsB1v53CxubW9k5xt7S3f3B4VD4+aRuVaMpaVAmluwExTHDJWsBBsG6sGYkCwTrB5C7zO09MG65kE6Yx8yMykjzklICVHvsRgXEQckibs0FtUK64VXc+eB28HCoon8ag/NMfKppETAIVxJie58bgp0QDp4LNSv3EsJjQCRmxnkVJImb8dJ56hi+sMsSh0vZJwHP170VKImOmUWA3s5Rm1cvEfz1u1JK3HAPCWz/lMk6ASbpIESYCg8JZP3jINaMgphYI1dx+BNMx0YSCbbFkG/JW+1iHdq3qWX64rtSv8q6K6Aydo0vkoRtUR/eogVqIIo1e0Ct6c56dd+fD+VysFpz85hQtjfP1C15Unf4=</latexit><latexit sha1_base64="DzZuMFMIutEhnNMA8oEMc2NeNso=">AAACEnicbZDNSgMxFIUz9a/Wv6pLN8EiuJAyUwRdFty4rNA/aMeSSTNtaCYZkjtCGfoWLtzqY7gTt76AT+ErmGlnYVsvBD7OuRdOThALbsB1v53CxubW9k5xt7S3f3B4VD4+aRuVaMpaVAmluwExTHDJWsBBsG6sGYkCwTrB5C7zO09MG65kE6Yx8yMykjzklICVHvsRgXEQckibs0FtUK64VXc+eB28HCoon8ag/NMfKppETAIVxJie58bgp0QDp4LNSv3EsJjQCRmxnkVJImb8dJ56hi+sMsSh0vZJwHP170VKImOmUWA3s5Rm1cvEfz1u1JK3HAPCWz/lMk6ASbpIESYCg8JZP3jINaMgphYI1dx+BNMx0YSCbbFkG/JW+1iHdq3qWX64rtSv8q6K6Aydo0vkoRtUR/eogVqIIo1e0Ct6c56dd+fD+VysFpz85hQtjfP1C15Unf4=</latexit>

↵ = 0.8<latexit sha1_base64="VYRAylFGV5NgPkPu8A9mEECKXIA=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUGRHsRii4cVnB/mA7lDtp2oZmkiHJCGXoO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww5kwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaIIrRPJpWqFqClngtYNM5y2YkUxCjlthqPbqd98okozKR7MOKZBhAPB+oygsdJjB3k8xBuvXOkWS17Zm427Cn4GJcim1i3+dHqSJBEVhnDUuu17sQlSVIYRTieFTqJpjGSEA9q2KDCiOkhniSfumVV6bl8q+4RxZ+rfixQjrcdRaDcjNEO97E3Ffz2m5YK3GMP0K0HKRJwYKsg8RT/hrpHutBu3xxQlho8tIFHMfsQlQ1RIjG2wYBvyl/tYhcZl2bd8f1WqXmRd5eEETuEcfLiGKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv7C+bgw==</latexit><latexit sha1_base64="VYRAylFGV5NgPkPu8A9mEECKXIA=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUGRHsRii4cVnB/mA7lDtp2oZmkiHJCGXoO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww5kwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaIIrRPJpWqFqClngtYNM5y2YkUxCjlthqPbqd98okozKR7MOKZBhAPB+oygsdJjB3k8xBuvXOkWS17Zm427Cn4GJcim1i3+dHqSJBEVhnDUuu17sQlSVIYRTieFTqJpjGSEA9q2KDCiOkhniSfumVV6bl8q+4RxZ+rfixQjrcdRaDcjNEO97E3Ffz2m5YK3GMP0K0HKRJwYKsg8RT/hrpHutBu3xxQlho8tIFHMfsQlQ1RIjG2wYBvyl/tYhcZl2bd8f1WqXmRd5eEETuEcfLiGKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv7C+bgw==</latexit><latexit sha1_base64="VYRAylFGV5NgPkPu8A9mEECKXIA=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUGRHsRii4cVnB/mA7lDtp2oZmkiHJCGXoO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww5kwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaIIrRPJpWqFqClngtYNM5y2YkUxCjlthqPbqd98okozKR7MOKZBhAPB+oygsdJjB3k8xBuvXOkWS17Zm427Cn4GJcim1i3+dHqSJBEVhnDUuu17sQlSVIYRTieFTqJpjGSEA9q2KDCiOkhniSfumVV6bl8q+4RxZ+rfixQjrcdRaDcjNEO97E3Ffz2m5YK3GMP0K0HKRJwYKsg8RT/hrpHutBu3xxQlho8tIFHMfsQlQ1RIjG2wYBvyl/tYhcZl2bd8f1WqXmRd5eEETuEcfLiGKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv7C+bgw==</latexit><latexit sha1_base64="VYRAylFGV5NgPkPu8A9mEECKXIA=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUGRHsRii4cVnB/mA7lDtp2oZmkiHJCGXoO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww5kwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaIIrRPJpWqFqClngtYNM5y2YkUxCjlthqPbqd98okozKR7MOKZBhAPB+oygsdJjB3k8xBuvXOkWS17Zm427Cn4GJcim1i3+dHqSJBEVhnDUuu17sQlSVIYRTieFTqJpjGSEA9q2KDCiOkhniSfumVV6bl8q+4RxZ+rfixQjrcdRaDcjNEO97E3Ffz2m5YK3GMP0K0HKRJwYKsg8RT/hrpHutBu3xxQlho8tIFHMfsQlQ1RIjG2wYBvyl/tYhcZl2bd8f1WqXmRd5eEETuEcfLiGKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv7C+bgw==</latexit>

↵ = 0.6<latexit sha1_base64="uovXqrsSC9IKsd7dop/bvOng/aM=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QSL4ELKjIi6EQpuXFawP9gO5U6aaUMzmSHJCKX0HVy41cdwJ259BZ/CVzBtZ2FbLwQ+zrkXTk6QCK6N6347uZXVtfWN/GZha3tnd6+4f1DXcaooq9FYxKoZoGaCS1Yz3AjWTBTDKBCsEQxuJ37jiSnNY/lghgnzI+xJHnKKxkqPbRRJH2/c8mWnWHLL7nTIMngZlCCbaqf40+7GNI2YNFSg1i3PTYw/QmU4FWxcaKeaJUgH2GMtixIjpv3RNPGYnFilS8JY2ScNmap/L0YYaT2MArsZoenrRW8i/utxHc958zFMeO2PuExSwySdpQhTQUxMJt2QLleMGjG0gFRx+xFC+6iQGttgwTbkLfaxDPXzsmf5/qJUOcu6ysMRHMMpeHAFFbiDKtSAgoQXeIU359l5dz6cz9lqzsluDmFunK9f6O2bgQ==</latexit><latexit sha1_base64="uovXqrsSC9IKsd7dop/bvOng/aM=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QSL4ELKjIi6EQpuXFawP9gO5U6aaUMzmSHJCKX0HVy41cdwJ259BZ/CVzBtZ2FbLwQ+zrkXTk6QCK6N6347uZXVtfWN/GZha3tnd6+4f1DXcaooq9FYxKoZoGaCS1Yz3AjWTBTDKBCsEQxuJ37jiSnNY/lghgnzI+xJHnKKxkqPbRRJH2/c8mWnWHLL7nTIMngZlCCbaqf40+7GNI2YNFSg1i3PTYw/QmU4FWxcaKeaJUgH2GMtixIjpv3RNPGYnFilS8JY2ScNmap/L0YYaT2MArsZoenrRW8i/utxHc958zFMeO2PuExSwySdpQhTQUxMJt2QLleMGjG0gFRx+xFC+6iQGttgwTbkLfaxDPXzsmf5/qJUOcu6ysMRHMMpeHAFFbiDKtSAgoQXeIU359l5dz6cz9lqzsluDmFunK9f6O2bgQ==</latexit><latexit sha1_base64="uovXqrsSC9IKsd7dop/bvOng/aM=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QSL4ELKjIi6EQpuXFawP9gO5U6aaUMzmSHJCKX0HVy41cdwJ259BZ/CVzBtZ2FbLwQ+zrkXTk6QCK6N6347uZXVtfWN/GZha3tnd6+4f1DXcaooq9FYxKoZoGaCS1Yz3AjWTBTDKBCsEQxuJ37jiSnNY/lghgnzI+xJHnKKxkqPbRRJH2/c8mWnWHLL7nTIMngZlCCbaqf40+7GNI2YNFSg1i3PTYw/QmU4FWxcaKeaJUgH2GMtixIjpv3RNPGYnFilS8JY2ScNmap/L0YYaT2MArsZoenrRW8i/utxHc958zFMeO2PuExSwySdpQhTQUxMJt2QLleMGjG0gFRx+xFC+6iQGttgwTbkLfaxDPXzsmf5/qJUOcu6ysMRHMMpeHAFFbiDKtSAgoQXeIU359l5dz6cz9lqzsluDmFunK9f6O2bgQ==</latexit><latexit sha1_base64="uovXqrsSC9IKsd7dop/bvOng/aM=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QSL4ELKjIi6EQpuXFawP9gO5U6aaUMzmSHJCKX0HVy41cdwJ259BZ/CVzBtZ2FbLwQ+zrkXTk6QCK6N6347uZXVtfWN/GZha3tnd6+4f1DXcaooq9FYxKoZoGaCS1Yz3AjWTBTDKBCsEQxuJ37jiSnNY/lghgnzI+xJHnKKxkqPbRRJH2/c8mWnWHLL7nTIMngZlCCbaqf40+7GNI2YNFSg1i3PTYw/QmU4FWxcaKeaJUgH2GMtixIjpv3RNPGYnFilS8JY2ScNmap/L0YYaT2MArsZoenrRW8i/utxHc958zFMeO2PuExSwySdpQhTQUxMJt2QLleMGjG0gFRx+xFC+6iQGttgwTbkLfaxDPXzsmf5/qJUOcu6ysMRHMMpeHAFFbiDKtSAgoQXeIU359l5dz6cz9lqzsluDmFunK9f6O2bgQ==</latexit>

↵ = 0.4<latexit sha1_base64="9eUcA35XlMWHNHJ1chf5eYjD0pI=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUGSnoRii4cVnB/mA7lDtp2oZmkiHJCGXoO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww5kwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaIIrRPJpWqFqClngtYNM5y2YkUxCjlthqPbqd98okozKR7MOKZBhAPB+oygsdJjB3k8xBuvXOkWS17Zm427Cn4GJcim1i3+dHqSJBEVhnDUuu17sQlSVIYRTieFTqJpjGSEA9q2KDCiOkhniSfumVV6bl8q+4RxZ+rfixQjrcdRaDcjNEO97E3Ffz2m5YK3GMP0r4OUiTgxVJB5in7CXSPdaTdujylKDB9bQKKY/YhLhqiQGNtgwTbkL/exCo3Lsm/5vlKqXmRd5eEETuEcfLiCKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv5aubfw==</latexit><latexit sha1_base64="9eUcA35XlMWHNHJ1chf5eYjD0pI=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUGSnoRii4cVnB/mA7lDtp2oZmkiHJCGXoO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww5kwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaIIrRPJpWqFqClngtYNM5y2YkUxCjlthqPbqd98okozKR7MOKZBhAPB+oygsdJjB3k8xBuvXOkWS17Zm427Cn4GJcim1i3+dHqSJBEVhnDUuu17sQlSVIYRTieFTqJpjGSEA9q2KDCiOkhniSfumVV6bl8q+4RxZ+rfixQjrcdRaDcjNEO97E3Ffz2m5YK3GMP0r4OUiTgxVJB5in7CXSPdaTdujylKDB9bQKKY/YhLhqiQGNtgwTbkL/exCo3Lsm/5vlKqXmRd5eEETuEcfLiCKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv5aubfw==</latexit><latexit sha1_base64="9eUcA35XlMWHNHJ1chf5eYjD0pI=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUGSnoRii4cVnB/mA7lDtp2oZmkiHJCGXoO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww5kwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaIIrRPJpWqFqClngtYNM5y2YkUxCjlthqPbqd98okozKR7MOKZBhAPB+oygsdJjB3k8xBuvXOkWS17Zm427Cn4GJcim1i3+dHqSJBEVhnDUuu17sQlSVIYRTieFTqJpjGSEA9q2KDCiOkhniSfumVV6bl8q+4RxZ+rfixQjrcdRaDcjNEO97E3Ffz2m5YK3GMP0r4OUiTgxVJB5in7CXSPdaTdujylKDB9bQKKY/YhLhqiQGNtgwTbkL/exCo3Lsm/5vlKqXmRd5eEETuEcfLiCKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv5aubfw==</latexit><latexit sha1_base64="9eUcA35XlMWHNHJ1chf5eYjD0pI=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUGSnoRii4cVnB/mA7lDtp2oZmkiHJCGXoO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww5kwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaIIrRPJpWqFqClngtYNM5y2YkUxCjlthqPbqd98okozKR7MOKZBhAPB+oygsdJjB3k8xBuvXOkWS17Zm427Cn4GJcim1i3+dHqSJBEVhnDUuu17sQlSVIYRTieFTqJpjGSEA9q2KDCiOkhniSfumVV6bl8q+4RxZ+rfixQjrcdRaDcjNEO97E3Ffz2m5YK3GMP0r4OUiTgxVJB5in7CXSPdaTdujylKDB9bQKKY/YhLhqiQGNtgwTbkL/exCo3Lsm/5vlKqXmRd5eEETuEcfLiCKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv5aubfw==</latexit>

↵ = 0.2<latexit sha1_base64="GgDez9hUrLanGjbStvKSML32RnE=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUmSLoRii4cVnB/mA7lDtppg3NJEOSEUrpO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww4Uwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaoIrRPJpWqFqClngtYNM5y2EkUxDjlthsPbqd98okozKR7MKKFBjH3BIkbQWOmxgzwZ4I1XrnSLJa/szcZdBT+DEmRT6xZ/Oj1J0pgKQzhq3fa9xARjVIYRTieFTqppgmSIfdq2KDCmOhjPEk/cM6v03Egq+4RxZ+rfizHGWo/i0G7GaAZ62ZuK/3pMywVvMYaJroMxE0lqqCDzFFHKXSPdaTdujylKDB9ZQKKY/YhLBqiQGNtgwTbkL/exCo1K2bd8f1mqXmRd5eEETuEcfLiCKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv4mmbfQ==</latexit><latexit sha1_base64="GgDez9hUrLanGjbStvKSML32RnE=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUmSLoRii4cVnB/mA7lDtppg3NJEOSEUrpO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww4Uwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaoIrRPJpWqFqClngtYNM5y2EkUxDjlthsPbqd98okozKR7MKKFBjH3BIkbQWOmxgzwZ4I1XrnSLJa/szcZdBT+DEmRT6xZ/Oj1J0pgKQzhq3fa9xARjVIYRTieFTqppgmSIfdq2KDCmOhjPEk/cM6v03Egq+4RxZ+rfizHGWo/i0G7GaAZ62ZuK/3pMywVvMYaJroMxE0lqqCDzFFHKXSPdaTdujylKDB9ZQKKY/YhLBqiQGNtgwTbkL/exCo1K2bd8f1mqXmRd5eEETuEcfLiCKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv4mmbfQ==</latexit><latexit sha1_base64="GgDez9hUrLanGjbStvKSML32RnE=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUmSLoRii4cVnB/mA7lDtppg3NJEOSEUrpO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww4Uwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaoIrRPJpWqFqClngtYNM5y2EkUxDjlthsPbqd98okozKR7MKKFBjH3BIkbQWOmxgzwZ4I1XrnSLJa/szcZdBT+DEmRT6xZ/Oj1J0pgKQzhq3fa9xARjVIYRTieFTqppgmSIfdq2KDCmOhjPEk/cM6v03Egq+4RxZ+rfizHGWo/i0G7GaAZ62ZuK/3pMywVvMYaJroMxE0lqqCDzFFHKXSPdaTdujylKDB9ZQKKY/YhLBqiQGNtgwTbkL/exCo1K2bd8f1mqXmRd5eEETuEcfLiCKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv4mmbfQ==</latexit><latexit sha1_base64="GgDez9hUrLanGjbStvKSML32RnE=">AAACDnicbZDNSgMxFIXv1L9a/6ou3QwWwYWUmSLoRii4cVnB/mA7lDtppg3NJEOSEUrpO7hwq4/hTtz6Cj6Fr2DazsK2Xgh8nHMvnJww4Uwbz/t2cmvrG5tb+e3Czu7e/kHx8KihZaoIrRPJpWqFqClngtYNM5y2EkUxDjlthsPbqd98okozKR7MKKFBjH3BIkbQWOmxgzwZ4I1XrnSLJa/szcZdBT+DEmRT6xZ/Oj1J0pgKQzhq3fa9xARjVIYRTieFTqppgmSIfdq2KDCmOhjPEk/cM6v03Egq+4RxZ+rfizHGWo/i0G7GaAZ62ZuK/3pMywVvMYaJroMxE0lqqCDzFFHKXSPdaTdujylKDB9ZQKKY/YhLBqiQGNtgwTbkL/exCo1K2bd8f1mqXmRd5eEETuEcfLiCKtxBDepAQMALvMKb8+y8Ox/O53w152Q3x7Awztcv4mmbfQ==</latexit>

TCAE-Random

I Introducing perturbation from certain distributionrandomly

T g = g(f (T i) + ∆x),

where ∆x ∼ N .

I Feature Sensitivity. Let l =[l1 l2 ... ln

]>be

the output of the layer associated with the latentvector space. The sensitivity si of a latent vectornode li is defined as the probability ofreconstructed pattern being invalid when aperturbation ∆li ∈ [−t, t] is added up on li witheverything else unchanged.

Legal Pattern Assessment

Creating DRC constraints for legal δxs and δys,

yi+1 − yi =p

2, ∀i ,

xi − xj = tmin, ∀(i , j) ∈ CT2T ,

xi − xj = lmin, ∀(i , j) ∈ CW ,xi+1 − xi > 0, ∀i ,

xmax − x0 = dx ,ymax − y0 = dy .

Results and Conclusion

Understanding Features in TCAE

Transformations Reconstructed Topologies

Extend or pull back line-ends

Create or destroy shapes

Control shape directions

TCAE-Random

Origin TCAE-Random

…

:

Comparison with State-of-the-ArtPerturbation with Gaussian exhibits greatest patterngeneration power with around 30% generated patternsare unique and DRC clean.

Existing Desig

n

Industry Tool

TCAE-Combine

TCAE-Random0

1

2

33.1

1.64

2.67

3.34

Div

ersi

ty(H

)

Conclusion

I Address the pattern library requirements in DFM flows/researches under advanced technology nodes.

I Propose a TCAE framework that can capture layout design rule characteristics.

I We show auto-learned features contribute to layout space locally or globally.

Haoyu Yang – CSE Department – The Chinese University of Hong Kong E-Mail: hyyang@cse.cuhk.edu.hk http://appsrv.cse.cuhk.edu.hk/∼hyyang/index.html