Maruya, Akihito

Zaidi, Qasim, Dr.

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Mental	geometry	for	estimating	relative	3D	size	Akihito	Maruya	&	Qasim	Zaidi	Graduate	Center	for	Vision	Research,	SUNY		A	3D	object	seen	from	different	views	forms	quite	different	retinal	images.	Humans	are	very	good	at	inferring	3D	pose	by	using	knowledge	of	projective	geometry	(Koch	et	al	PNAS	2018).	Shape	“constancy”	would	suggest	that	we	also	infer	correct	3D	relative	size/shape	despite	perspective	distortions,	but	is	that	true?	We	presented	frontal	views	of	rectangular	parallelepipeds	(3	lengths)	lying	on	the	ground	(16	poses).	Observers	(N=6)	adjusted	the	height	of	an	orthogonally	attached	narrow	cylinder	to	equate	the	physical	lengths	of	the	two	limbs	(Fig	1).	For	a	parallelepiped	of	length,	(𝐿"#)	the	projected	length,	(𝐿$)	changes	with	pose	as	a	distorted	sinusoid	(Viewing	elevation=Φ$,	focal	length=	𝑓$,	distance	from	the	object	=	𝑑$):		𝐿$ =𝐿"# ∙ 𝑓$ ∙ 𝑐𝑜𝑠- W + 𝑠𝑖𝑛- W ∙ 𝑠𝑖𝑛- 	Φ$ 	𝑑$ − 𝐿"# ∙ 𝑠𝑖𝑛 W ∙ cos 	Φ$		Observer’s	estimates	of	parallelepiped	length	were	close	to	veridical	for	front-parallel	poses	but	were	seriously	underestimated	for	poses	pointing	at	or	away	from	the	viewer	(Fig	2).	The	inverse	of	the	function	relating	the	projected	length	to	the	pose	gives	the	optimal	correction	factor	for	inferring	correct	physical	lengths	from	retinal	images.	Observers’	correction	factors	were	close	to	optimal	for	poses	close	to	fronto-parallel,	but	seriously	low	for	poses	close	to	line	of	sight.	Interestingly,	the	underestimation	increased	with	physical	length	of	the	parallelepiped.	Slant	matching	measurements	revealed	that	longer	objects	were	seen	as	slanted	down,	equivalent	to	an	increase	in	viewing	elevation.	Increased	viewing	elevation	requires	a	smaller	correction	factor,	so	we	tested	a	model	for	estimating	𝐿3D,		that	adds	a	free	parameter	𝑘	to	the	optimal	geometrical	back-transform,	where	𝑘>1	indicates	overestimates	of	viewing	elevation	(focal	length	of	the	retina	=	𝑓7,	distance	of	pupil	from	the	screen	=	𝑑7,	projected	length	on	the	retina	=	𝐿7):		𝐿3D=89∙:9∙:;89∙:9∙<=> ? ∙@A< B∙C; DE;∙E9∙ $FGH W DGIJH(W)∙GIJH(B∙	C;)		This	model	explains	the	underestimation	of	object	length	(Fig	3.)			These	results	show	that	observers	use	the	optimal	geometric	back-transform	for	estimating	object	length.		Since	illusory	change	in	relative	lengths	of	limbs	describes	one	class	of	shape	distortion,	shape	inconstancy	results	despite	correct	mental	geometry,	when	retinal	images	of	objects	evoke	misestimates	of	viewing	elevation.				Fig.1	Fig.2	Fig.3	

Mental geometry for estimating relative 3D size

https://docs.lib.purdue.edu/cgi/viewcontent.cgi?article=1143&amp;context=modvis

Mental geometry for estimating relative 3D size

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